WO2012056255A1

WO2012056255A1 - Method of mapping and control of surfaces of tunnels during the construction project

Info

Publication number: WO2012056255A1
Application number: PCT/GR2011/000040
Authority: WO
Inventors: Seraphim Amvrazis
Original assignee: Seraphim Amvrazis
Priority date: 2010-10-25
Filing date: 2011-09-21
Publication date: 2012-05-03
Also published as: GR1007395B

Abstract

Method of mapping and control of surfaces of tunnels in projective development w here it can be realized with the help of a computer. The method is characterized by the reception of the primary surveys by the total stations at the duration of constructional phases of a tunnel. It contributes to the forecasting of future excavations based according to the existing surface of excavations that were realized in the past. The current technical monitoring of tunnels under construction is not allowed (beyond the classical cross sections), a detailed and comprehensive monitoring of the project. As a result this contributes to the loss making of the project's quality, as well as to the to the bad cost estimation for a constructional company. The method of mapping and control of surfaces of tunnels in projective development, calculates cylindrically the points of the primary surveys with geometric adaptation of the proportional radius of each theoretical cross-section that is wished to be compared with the existing surface of a tunnel. The development of each arc of different radius with geometric passages of the proportional space where each point belongs, contributes in complete and precise development in the all length of arcs of radius until each point of measurement. As result of the transformation, it will be the completed information in the all length and width of a tunnel. The calculation of the algorithm of method is carried out with the aim to appear cylindrically but also absolutely at any point of the tunnel, information with regard to the future behavior of explosions. The future behavior of explosions examines the regions of explosions that were realized in the past. It presents how it could be a next explosion according to the geological criteria.

Description

METHOD OF MAPPING AND CONTROL OF SURFACES OF TUNNELS DURING THE CONSTRUCTION PROJECT

The invention is reported in the industry of manufacture of opening up of tunnels.

It can be used at the duration of manufacture, receiving the topographic primary surveys in the surfaces of tunnels.

It can be used as a controlling tool in cases where the manufacture has already been completed, receiving the topographic primary surveys in the surfaces of tunnels that were realized at the duration of manufacture.

Until now world methodology of topographic surveys in the frames of constructional levels of tunnels is the followings:

Export of primary measurements by the total station and their import in software that creates corresponding cross-sections.

One of the disadvantages of the existing methods is that there is not generalized revelation of the present situation that presents a tunnel under construction as a result there is a not generalized picture of the existing situation of surfaces.

Thus the engineers of work are called to examine each cross-section for long distances.

As an example if there is one kilometer information in a surface of a tunnel and has been printed out for each meter cross-section, then the engineers of the project in order to locate the problematic areas must examine 1000 pages.

This means chaotic and time-consuming processes, and many omissions.

In the final assessment of manufacture the accumulation of omissions will have as result in the increased considerably cases of constructional errors and erroneous forecasts.

As a final result of the existing until now methods, is that is not given the possibility by the responsible engineers of manufacture, of forecasting absolutely the constructional needs of the project.

The aim of method is the achievement in bigger degree of the control and quality of manufacture, the acceleration of work, and the reduction of cost of manufacture.

Method of mapping and control of surfaces of tunnels in projective development:

The invention of method of mapping and control of surfaces of tunnels in projective development depends from the reception of topographic primary surveys that can be taken by the total stations in the surfaces of tunnels at the duration of constructional phases.

The calculation of method can be realized with the support of computer.

Aim is the direct and generalized revelation of problematic regions in a single drawing. With this way can be achieved directed fast re-establishment in the surfaces that present big overbreaks, as well as in the underbreak surfaces. In this way does not exist the need of diagnosis of thousands of cross-sections that is, in effect, until now.

The general picture of mapping regions contributes in the forecast and evasion of future situations that are critical for the quality, the safety, and the cost of manufacture.

The cases where the invention of method of mapping and control of surfaces of tunnels in projective development offers technical solutions in the frames of manufacture are the followings:

1) In first phase should be given the points of topographic surveys of excavations that were collected by the total stations, in order to be incorporated in the algorithm of the invention were with the support of computer will be calculated the projective development of surfaces of a tunnel.

Then if there were cases of estimation errors of the quantity of explosives that were used in the past, or error of orientation of perforations, then will be presented in a form of united mapping any chance of abnormalities that were realized in the past in the surface of a tunnel. The presentation of problematic regions will be calculated in projective development in order to be determined in a united drawing of mapping the ground plan of existing surface of a tunnel.

As a result it will be achieved direct finding of faults of the past, and dissuasion from any chance of similar future estimate errors.

It, therefore, can be achieved an important reduction of cost and acceleration of the project working procedures. Reduction of cost because is avoided thus the chance of cover of any future overbreaks with concrete by the estimates errors. Acceleration of working procedures because the less quantities of overbreaks that should be overlapped with concrete exist, so much the work accelerates. Acceleration of working procedures mean also saving of cost. It, moreover, can be achieved the forecasted quality of manufacture, generally working procedures in less time, as well as the continuous qualitative control.

In this way is minimizing the probability of future erroneous estimates.

2) Important reduction of cost of work and increase of quality of manufacture at the process of re-establishment of surface of a tunnel before the final investment. Because when the responsible engineer of work or foreman, has printed the drawing of mapping, determines immediately the points that are problematic and without time-consuming topographic support becomes the work of re-establishment direct and in very less time compared to the traditional methods. The revelation of problematic regions in a single drawing of projective mapping as it is reported in paragraph 1, contributes in the better and faster planning of works in the tunnel. Furthermore, reduction of time that the operators and the employers should wait for, as in the traditional methods, because in the first phase of works at the beginning of re- establishment of the surface of a tunnel, the topographic support is not necessary.

3) Forecast of future excavations from blasting based according to the existing surface of excavations with explosions that were realized in the past.

The calculation forecasts virtual primary measurements in the space of the tunnel, and calculates the projective development of the virtual primary measurements as it is reported in paragraph 1, so that is attributed the mapping of projection of future excavations of forecast. The calculation of virtual primary measurements of excavation depends from the number of the topographic measurement points of the excavations with explosions that were realized in the past, with specific geologic grouping criteria. The engineers of manufacture should define the geologic grouping criteria.

The conditions of the geologic grouping criteria are:

A) Each cross-section of survey measurements of excavations that was realized in the past must have the same type of geology.

B) Each cross-section of survey measurements of excavations that was realized in the past must have the same quantity of explosives.

C) For each cross-section must have been used the same length of perforation.

In this way, the calculation and the merging of cross-sections will result to the creation of new cross-sections that they will begin from the point of the forehead of excavation of a tunnel. And their length will be the proportional length that will be selected by the responsible engineer of the project according to the length of perforation.

Aim of forecast of excavations is to give a solution for the technical issue for all experts of the subject in world level. The technical issue that concerns the engineers in the frames of manufactures of tunnels is because until now does not exist a method of forecast of explosions that would calculate and determine future cross-sections of excavations in the real space of a tunnel, mapped to the projective development. As a consequence of the problem is that there are not visual estimates of blasting before they are made. They thus are increased considerably cases of the estimate errors of quantity of explosives that should be used, as well as the length of perforation.

As a result from the creation of important overbreaks, the constructional company is called to overlap those areas with concrete. The result from the increasing of the cost of manufacture is the increasing of time of manufacture, the decrease of the quality of manufacture, as well as the decrease of the safety of workers. The safety of workers is decreased considerably because the bigger surfaces in-depth the concrete overlaps, so cases of collapses of concrete in unsuspected time are increasing.

4) Statistical analysis of forecast of future excavations with explosions.

Aim of statistical analysis of excavations is to determine the suitable quantity of explosives that will be used for a blasting.

That is to say, with the support of computer will be determining the quantities of explosives, and respectively will be calculating again the estimate of cross-sections of forecast of excavations that is reported in paragraph 3. It then will be calculating again the mapping of projective development of a tunnel as it is reported in paragraph 1.

Method of mapping and control of surfaces of tunnels in projective development, contributes to a great degree in the general depiction with form of projective mapping of surface of a tunnel, and the direct comprehension of the degree of the problems for correction and the dissuasion of future errors were in any case, would be loss-makings for a constructional company.

The projective development that results from the three-dimensional transformation of collegiate primary measurements is solved so that the result allows the easy and fast reading in all the real length of arcs from any theoretical cross-section.

The advantage of transformation is that the observer having the ground plan of mapping can read the analysis that includes all the real width of the developed cylindrical cross-sections. The resolution and dissuasion of time-consuming works, the reduction of cost of work, the increasing of quality and safety that are reported above, changes the classical methods of surveys and control of tunnels.

The method of mapping of control of surfaces of tunnels in projective development can be also used like a controlling tool from study services, from government manufacture services, and central offices of big constructional companies where worldwide will be able check the cost, the progress and the quality at the same time in each manufacture of a tunnel.

The invention analyzes the way of transformation of points of the primary measurements as the total station takes them.

The transformation that results from the cylindrical system in cylindrical projective development leads to the possibility of reading of surfaces of tunnels at all the length and real width, so that is revealed all information that is essential for the achievement of the aims above. So it is given technical solution for a fast re-establishment of the tunnel, forecast, and repair via the information of projective mapping that until now was impossible from the existing methods.

Explanation of method

The method of mapping and control of surfaces of tunnels is based on the reception of topographic points from the total station.

Aim is the analysis of the x, y, z of the points and transformation of them in projective level of development of a tunnel so that it creates the attribution of mapping comparing thus any theoretical cross-section concerning the existing surface.

Examples with drawings:

The explanation of method includes examples with drawings:

In drawing 1 are presented the primary measurements of points of a cross-section in the surface of a tunnel. In the same drawing is presented also the transformation of the cross- section that results afterwards from the calculation of the projective development of the primary points of the cross-section. In drawing 2 is presented a theoretical cross-section which includes the following geometric elements:

Theoretical values Rl, R2, and R3 of the radius of the circles where the cross-section is constituted.

Theoretical values AZ1, AZ2, and AZ3 of the azimuths of the arcs of the circles where the cross-section is constituted.

Value of azimuth AZ of the upper level of the arc of the circle where the point of example is included.

The value of azimuth Azp of the measured point of the example.

Value of azimuth Azpl of the upper level of the arc of the circle where the point of example is included until the point.

Value Rp of the distance of the point from the center of the circle where it belongs, until the point.

The coordinates XC10, YCIO of the center of the circle where the point belongs.

The coordinates in the two dimensional space of cross-section XA, YA of the point.

The Z value of the point.

The lengths of parallel arcs LI, L2 and L3 that precede from the point of measurement. The distance Lm of the total length that is calculated by the sum length of the parallel arcs LI, L2 and L3.

In drawing 3 is presented a case of a final attribution of method of mapping and control of surfaces of tunnels in projective development.

In drawing 4 is presented the same case of mapping as in drawing 3 in projective

development, with calculation of points that exceeds the limit of excavations according to the study.

In drawing 5 is presented a case of mapping of excavations in projective development and forecast of excavations that is proportional to grouping of geological criteria of excavations with explosions that were realized in the past.

In drawing 6 is presented the three-dimensional case of forecast of excavations as it results from the calculation of the virtual primary surveys of an excavation in the space of the tunnel.

In drawing 7 is presented a cross-section of an excavation with an explosion that was realized in the past.

In drawing 8 is presented a cross-section of excavation with an explosion that was realized in the past, and it has the same geology with the cross-section of drawing 7.

In drawing 9 is presented a cross-section that results from the merging of cross-sections of drawings 7 and 8.

In drawing 10 is presented the transformation in projective development of cross-section of drawing 9, as it is attributed in projective mapping as in drawing 5.

In drawing 11 is presented a three-dimensional option of a tunnel mapping in projective development.

In drawing 12 is presented the cross-section that results from the statistical analysis of forecast of future excavations with explosions. The statistical analysis forecasts the case of reduction of explosive quantities aiming at the better possible approach of future excavations, avoiding thus big overbreaks in the surface of the tunnel. The particular drawing presents again the optical output of cross-section of drawing 9 that results from the reduction of quantity of explosives concerning the quantities that have been realized. It is obvious the reduction of overbreaks that can be created. It then is calculated again the projective transformation as in drawing 10.

Requested elements for the calculation:

The transformation in development of projection for each point of measurement is dependent from:

A) The geometry of central axis of road construction.

B) The longitudinal profile of road construction.

C) The geometry of the theoretical cross-section so that becomes the comparison of the theoretical and the existing surface.

The aim is the collection of the specific elements so that can be calculated the development of projection.

The requested elements are the following:

A) For each point of measurement (Drawing 1 figure 1), the vertical projection on the axis of road construction. (Drawing 1 figure 2). That is to say, for each point is requested the coordinates (X, Y) on the alignment perpendicular where the point belongs.

B) For each point of measurement its vertical projection on the longitudinal profile of road construction. That is to say, for each point is requested the altitude of road construction perpendicular on the longitudinal profile position in which it belongs. (Drawing 1 figure 3). Consequently, the projection of point and the measured point will be supposed to find each other in the same kilometric place.

C) For each point of measurement the distance from the axis in which it belongs. (Drawing 1 figure 4). Thus for each point is requested its distance from the axis. Another condition is also the correct definition of its sign from the axis. That is to say, if at the serial flow of the axis the eccentricity of a point is on the left, then it will have a negative sign, while from the right positive. Therefore, if a point abstain from 6 meters from the axis and is from the left according to the serial flow of the axis, then it will be - 6 meters. If it is from the right it will be + 6 meters.

The calculation of the requested elements above can be calculated from any topographic software of road construction.

Calculation of method of mapping and control of surfaces of tunnels in projective development with examples.

The algorithm of calculation can be realized with the help of a computer.

STEP 1. Calculation in the two-dimensional space (X, Y cross-section coordinates).

That is to say, it will be calculated the height of a point from the horizon of road construction (Drawing 1 figure 5), and the distance of the point from the axis. (Drawing 1 figure 4).

The calculation of Y can become provided when the altitude of axis of road construction is known. That is to say, from the longitudinal profile where has become the projection of points.

Thus if the absolute Z(altitude) of a point is 49.45 m, and the corresponding absolute Z of road construction where the point belongs is 49.001 , then the height of the point from the horizon of road construction is [(Z of point) - (Z of road construction)] = -0.449m. Then the point is found concerning the cross-section hypsometrically - 0.449m from the horizon of road construction.

The theoretical cross-section belongs in the two-dimensional space (x, y), and the primary surveys emanate from the absolute three-dimensional space (x, y, z). It afterwards will be supposed it is fixed as X on the cross-section the distance of point from the axis (receiving the aspect that the sign is left or right according to the serial flow of the alignment), and as Y the difference of (Z of point - Z of road construction), that is to say - 0.449m. Suppose that the distance of a point is 7.569m right from the axis. It afterwards will result as Xa of cross- section= 7.569m and Ya of cross-section= - 0.449m. STEP 2. Grouping of measurement points in the cross-section.

It is taken into account from how many circles a theoretical cross-section is constituted, and for each point concerning the axis of road construction, must be found the hypsometric limit of the upper and lower level of each arc where each point belongs. Suppose that a point of measurement where the two-dimensional coordinates X, Y, have been calculated on the cross- section, and will be supposed it is compared to a theoretical cross-section. There is a specific case of a point according to drawing 2 of a cross-section with coordinates Xa= 7.569 and Ya= -0.449. The cross-section is included by five circles and aim is to be found in which of the five circles the point belongs.

More concretely the rules of grouping and determination of the circle that belongs to each point should be fixed as follows:

Suppose that it is known the coordinates of the center of the third circle right XC 10, Y C 10. (Drawing 2).

The upper level of the arc of the third circle right is known, hi =1.822. (Drawing 2)

The lower level of the arc of the third circle right is known. h2= -0.768. (Drawing 2) The values of hi, h2 are the distance from the horizon of road construction. Then:

If [Xa>0, Ya<hl, Ya>h2] then the point is grouped on the third circle right with circle center coordinates: XC 10^1.606, YC 10=1.247. (Drawing 2).

The combinations that will result from the relation above, will group each point of measurement for the specific cross-section.

STEP 3. Distance from the center of the circle until the point of measurement.

According to the relation above from the grouping of points (STEP 2), it has been found that the point with the cross-section coordinates Xa=7.569, Ya=-0.449, belongs to the third circle right with the center coordinates XC10=-1.606, YC10=1.247. The next step is to calculate the distance between the center of the circle and the point of measurement in the cross-section.

That is to say, the radius of the point. The calculation that will be used is the

following: SQRT [(Xa-XC 10) ^Λ2 + (Ya-YC 10) ^Λ2] THEREFORE:

SQRT [((7.569- (- 1.606)) ^Λ2 + (- 0.449-1.247) ^A2] =9.331. So the result is Rp=9.331m. (Drawing 2). In this way has been calculated the distance between the center of the third circle right and the point of measurement on the cross-section. This distance is also the existing radius of the point.

STEP 4. Calculation of the Z of the point of measurement.

The next step is the calculation of Z of point. The calculation of Z can be done when the theoretical radius of the circle where the point belongs is known. Thus if the third circle where the point belongs has a theoretical radius R3=9. 02m (Drawing 2), then the calculation will be as follows: Z= Rp-R3. The Rp was calculated in STEP 3. That is to say, Z= 9.331 -9.02 = 0.31 lm. With this calculation results from the difference of the point concerning the theoretical surface of cross-section. Hence, when becomes the measurement in the real space in the surface of a tunnel this calculation determines the distance of the point above the theoretical surface. In the opposite case if the value of Z is negative, then it is determined the distance beneath the theoretical surface. In the case of example the Z has positive value, and it was calculated that the point is more outside at 0.311 meters. The result of Z is the vertical distance of the point concerning the arc of the circle in which the point belongs.

STEP 5. Calculation of azimuth of cross-section.

The next step is to calculate the azimuth between the center of the third circle right where the point of measurement belongs, and the point of measurement. In this case, it will be supposed that the coordinates of the center of the third right circle are known. (XC10=-1.606,

YC 10=1.247 according to drawing 2), as well as the coordinates of the measured point

(Xa=7.569, Ya=-0.449.according to drawing 2).

The calculation in order to find the azimuth is the following:

Xa - XClO therefore we have: 7.569- (- 1.606) = -5.409. From this value will be

Ya - YCIO -0.449 - 1.247 calculated the arc of tangent. Therefore: ATAN (- 5.409) = - 88.361 grads. In order to find the correct quadrant in which the value - 88.361 grads belongs, will be examined the values of the result (Xa - XC10) =9.175 and (Ya - YCIO) = -1.696. And thus the results will be the following:

If (Xa - XC 10) >0 and (Ya - YC 10) <0. Then the point of measurement and consequently, the value of azimuth - 88. 361 grads that was calculated before belong on the second quadrant of the circle. And thus provided that the value of azimuth belongs on the second quadrant of the circle should:

AZp=-88.361grads + 200grads = 111.639grads. (Drawing 2)

It has been calculated the azimuth of point that belongs in the third circle right which is

AZp=l l 1.639grads.. Therefore, from the coordinates XC10, YCIO of the third circle right, has been calculated the horizontal dextral azimuth to the point of observation. (That is to say, is to say the point of measurement with coordinates Xa, Ya).

The next step is to calculate the correct azimuth that will be used for the projective development of the cross-section. That is to say : If the corner AZp is greater than 1 OOgrads, that is to say, greater than the first quadrant of the circle, then it will be as result AZp-400.

(First parameter of azimuth).

And also if (Azp-400) <-200 then Azp+ 400. (Second parameter of azimuth).

With the parameters of the azimuth is detered any chance of azimuth faults for the cross- section, and are used for the correct development of the algorithmic model without azimuthal errors and faults in the length of projective development.

STEP 6. Development of arcs in the cross-section.

The next step is to calculate the real length from the beginning of the upper hypsometric level hi of the arc that belongs on the third right circle, up to the point of measurement with code 2579. (Drawing 2)

As known it will be the coordinates Xa, Ya of the point of measurement on the cross-section. Thus we have Xa= 7.569, Ya= -0.449. In order to calculate the real length from the arc that belongs on the right third circle, up to the point of measurement, first must be calculated the azimuth of the upper hypsometric level hi of the arc of the third circle right, up to the point of measurement with code 2579. (Drawing 2). The azimuth of the upper hypsometric level of the arc of the third right circle should be given as known from the geometric characteristics of cross-section. Suppose that it has the azimuth AZ=95.938grads. (Drawing 2).

It then will be calculated the subtraction of the azimuth that was calculated in the STEP 5, which is the azimuth from the center of the third circle right, up to the point of measurement Azp=l 11.629 (Drawing 2), and the azimuth of the upper hypsometric level of the arc that belongs to the third right circle AZ=95.938grads. (Drawing 2).

Suppose that the name of the azimuth that is requested is Azpl . Then: Azpl = 1 1 1.639- 95.938= 15.701grads. (Drawing 2).

Therefore, it has been calculated that the azimuth from the beginning of the upper hypsometric level of the arc of the third circle right hi until the point of measurement is Azpl=15.701grads. (Drawing 2).

Now having the calculated azimuth above, it can be calculated the length of arc. That is to say, the length of arc from the upper hypsometric level of arc hi that it belongs on the third right circle, until the point of measurement that belongs also on this circle.

In order to calculate it, there must be as known:

The distance from the center of the circle until the point of measurement, where in STEP 3 were calculated that it is Rp= 9.331 m.

The azimuth from the beginning of the upper hypsometric level hi of the arc that belongs on the third right circle, until the point of measurement, which is: Azpl= 15. 701 grads the calculation in order to find the arc length will be:

L3= PI x Rp x Azpl

200

Thus L3=2.301m. (Drawing 2). STEP 7. Development of points of measurement in the cross-section.

Development of points of measurement in the cross-section will be used for each point of measurement that exists in the cross-section in order to attribute the final form of the sections total development. It, consequently, will be calculated the two-dimensional transformation of the cross-sections. The transformation will be two-dimensional because according to the explanation of Step 1 the coordinates are found in the frames of the cross-section coordinates (X, Y) and not in the absolute coordinates system (X, Y, Z).

The third right circle is the continuity of the top circle and the second right circle. Thus it will be calculated the total length of arc Lm (Drawing 2) from the axis of the top until the point of measurement, which belongs on the third right circle. This arc will be parallel with the theoretical arcs of each circle, and the distance from each other will be the Z value of the point that was calculated in Step 4. (Z=0.31 lm). First must be calculated the lengths of the two arcs that precede from the arc where the point belongs, and their length sum will be added in the result that was calculated in Step 6. L3=2.30.

As known values must be the followings:

The theoretical values of Radius of the arcs of the circles that precede from the arc where the point of measurement belongs.

Suppose that the theoretical radius of the top circle has value Rl =7.90m. (Drawing 2).

The theoretical radius of the second circle right has value R2=5.30m. (Drawing 2).

It will be supposed also that i s given as known the azimuth of the lower hypsometric level of each arc that precede from the arc where the point of measurement belongs.

Suppose that the azimuth of the lower hypsometric level of the right arc of the top circle is:

AZl=60.120grads. (Drawing 2).

To azimuth of the lower hypsometric level of the right arc of the second right circle is:

AZ=95.938grads. (Drawing 2).

Thus for the right arc that belongs on the top circle will result from the following calculation: Ll= PI x Rl x AZl

200

Thus PI x 7.90 x 60.120 = 7.4605. Thus Ll=7.4605. (Drawing 2).

200

For the right arc that belongs on the second circle will result the following calculation:

12= PI X R2 X (AZ-A Z1 ). The result of AZ-AZ1 is azimuth of AZ2 (Drawing 2).

200

Thus PI x 5.30 x (95.938-60.120') = 2.9819. Thus L2=2.9819. (Drawing 2).

200

In order to calculate the total length of the parallel arc up to the point of measurement in the cross-section, it will be the following calculation: Lm= Ll+L2+L3=12.7434[m]. (Drawing 2).

The sequence of the algorithm will be, in effect, with the same way for each point. For example, if a point it belongs on the fourth right circle then according to Step 6, (development of arcs in the cross-section), would calculate the corresponding L4. It would result the same calculation as for LI, L2. The value L3 will be calculated as follows:

L3= PI x R3 x AZ3 (Drawing 2).

200

The value of azimuth AZ3 will be found from the difference of the azimuth of the lower level of the arc of the third right circle, and the azimuth of the lower level of the arc of the second right circle.

Then in order to calculate the total length of the parallel arc up to the point of measurement in the cross-section, it will be calculated as follows: Lm=Ll+L2-l-L3+L4. STEP 8. Three-dimensional transformation of the coordinates for the attribution of mapping of cylindrical development of a tunnel.

It will be calculated the transformation of the coordinates X, Y, Z of the point of the primary topographic survey (Drawing 1 figure 6),

So that to become import in the final calculating attribution of mapping.

As known values must be the followings:

The XO, YO of the axis where according to the requested elements needed for the calculation, is the vertical projection of the point on the axis. (Drawing 1 figure 2). That is to say, the XO, YO is the coordinates from the position that the measured point is perpendicular on the axis. Suppose that X0=378990.256, Y0=4429174.972.

The calculated Z of point where according to Step 4 it was found that is Z=0.31 1.

The length of development of point where according to Step 7 it was calculated that is Lm=12.7434.

The absolute XI, Yl of point (Drawing 1 figure 1) as come from the total station reception of the existing surface of a tunnel. Suppose that Xl= 378997.025, Yl=4429178.358.

Absolute azimuth AZ1 of point of measurement from the axis. (Drawing 1 figure 7) By calculating the azimuth concerning the development, can be calculated rightly and absolute the projective mapping according to the direction of the tunnel axis. In order to calculate the absolute azimuth it will be supposed that is known the XO, YO, and the XI, Yl. Thus the calculation will be as follows:

XI - X0 = 1.999. From this value

Yl - Y0

It will be calculated the arc of the tangent. Therefore: ATAN (1.999) = 70.473 grads.

In order to find the correct quadrant in which the value 70. 473grads belongs, will be examined the values of the result (XI - X0) = 6.769 and (Yl - Y0) = 3.386.

And thus the regulation that will result will be the following:

If (XI - X0) >0 and (Yl - Y0) >0, then the point of measurement and consequently, the value of azimuth 70. 473grads that was calculated before belongs in the first quadrant of the circle.

And therefore provided that the point of measurement belongs in the first quadrant of the circle should: AZl=70.473grads.

The calculation that will be used in order to become the transformation for the X will be:

X of (cylindrical development) =X0+Lm (always with positive sign) x SIN (AZ1)

Thus: X=378990.256+ 12.7434 x SIN (70.473)

Thus it results: X= 379001.6531,

The calculation that will be used in order to become the transformation for the Y will be:

Y of (cylindrical development) =Y0+Lm (always with positive sign) x COS (AZ1)

Thus: Y=4429174.972+ 12.7434 x COS (70.473)

Thus, it results: Y = 4429180.6728

The final results of the three-dimensional transformation of the point (Drawing 1 figure 6) that will be included in the mapping will be following:

X= 379001.6531

Y = 4429180.6728

Z = 0.311

The crowd of points that will be converted in cylindrical development will create the total mapping of the surface of a tunnel.

Drawing 3.

In drawing 3 is presented a case of mapping that result from the projective transformation of the crowd of primary topographic points, and is visualized from ground terrain software. In figure 1 is presented the line of cross-section of example in projective development. In figure 3, 4 are presented regions without bracing of type invert. The characteristic of those regions is that their width is smaller concerning the remaining mapping because the theoretical cross- section includes fewer arcs. In figure 2 is presented the axis of the tunnel in which the attribution of projective mapping is directed. It is presented also different chromatic areas as well as contour lines from the proportional altitudes of the points that results from the calculation of Z of points as it is reported in Step 4.

The values on the projective mapping are values in centimeters that result from the theoretical cross-section concerning the existing surface of a tunnel, in random areas of the map.

Drawing 1 1.

In figure 1 is presented the projective development of a surface of a tunnel that results from the transformation of the crowd of the primary surveys from the total stations on the existing surface. The primary surveys are presented in figure 6. In figure 3 is presented the line of the cross-section from the primary surveys. The line of cross-section includes the point that was calculated in the steps 1-8. In figure 2 is presented the three-dimensional projective development of the line of the cross-section as it was calculated in Step 8. In figure 4 is presented the axis of the tunnel on which the attribution of mapping is directed. In figure 5 is presented the theoretical surface of the tunnel.

Calculation of limit of excavations on the mapping based from drawing4.

Is a usual phenomenon to require reinstatement of surface of a tunnel before the final investment because of the excessive overbreaks of the excavations (Usually with explosives). Thus in these cases the study gives some limits about how much it will be supposed that the most overbreak can be on the surface. Aim of the calculation of the limit of excavations on the mapping is to appear the areas that need covering with concrete (gunite) in a single drawing so that is realized the work of re-establishment of tunnel surfaces in less time than the classical methods until now.

The acceleration of this working procedure contributes also to the reduction of cost of the manufacture. Furthermore, it contributes in the quality of manufacture. In the quality of manufacture contributes because the mapping reveals all the areas that need to be repaired and does not leave margins of omissions. Omissions from the existing methods are logical to exist, because of the thousands of cross-sections that are printed.

As a result of the traditional methods it is that does not exist a generalized picture of the problematic regions. So the cases of many omissions are increased. The reason is the chaotic situation that prevails from the crowd of cross-sections.

Suppose that according to the study on the surface of a tunnel is not allowed to exist overbreaks from the theoretical cross-section above 40 centimeters. Then from the crowd of points that belongs on the projective mapping and has been calculated according to the Step 1-8, will be fixed the following regulation:

Suppose that exists a point with Z>0.40m then must be Z- 0.40m. Also the points that have Z<0.40m the value of Z that they have to will become 0.000m. In this way are revealed automatically the regions that exceed the limit that was given by the study because the surface in this limit is visualized from the subtraction of Z of the existing points of mapping with the value of Z of the limit. The points that are inside the limit, the Z value is annihilated. In final analysis results again a new surface of projective mapping with the level regions updated. Where the points are inside the limit the altitude in these regions will be 0.000m. Where regions of overbreaks exist, will be proportional with the minimal values according to the study who is supposed to overlap with concrete. They can also be revealed the regions that are beneath the theoretical cross-section. That is to say, the regions where the Z of the points is less than zero. Thus it will be fixed as regulation the following: Where exists a point with Z>0 then the value of Z to become 0.000 m and where the Z<0 is maintained the existing value of Z. So it results again a new surface of projective mapping that reveals immediately which regions have Z<0.

In drawing 4 is presented a case of mapping that is visualized by software of terrain model creation. The drawing presents the surfaces that exceed the limit of overbreaks (figure 2). In figure 1 is presented the line of cross-section of the example. Advantages of method of mapping and control of surfaces of tunnels in projective development.

A) Qualitative control of manufacture.

B) Geometric control.

C) Control of cost of manufacture.

D) Topographic control.

E) Instant and completed control of manufacture according to the models of study.

F) Instant prevention and re-establishment of problematic regions at the duration of manufacture.

G) Reduction of the cost and the time schedule of works.

H) Generalized picture for the situation of the manufacture and what problems it faces.

I) Updating of the mapped regions at any moment and follow-up of manufacture in any stage with continuous briefings.

J) Complete comprehension of the situation of a tunnel directly in a single drawing of mapping, avoiding thus the reading of classic cross-sections. For example, if there are primary surveys for one kilometer length and the frequency of reception of that is each meter then will be supposed are printed out 1000 pages of cross-sections. It therefore, exists a difficulty of comprehension.

K) Easy reading of the mapped regions without the need of specialized knowledge.

Forecast of future excavations with explosions based according to the existing surface of excavations with explosions that were realized in the past.

The forecast of future excavations with explosions is based on the reception of topographic primary surveys that were realized in the tunnel in the frames of excavations with explosions. That is to say, by the topographic primary surveys that are realized, is surveyed the existing morphology of the forehead of a tunnel immediately after each explosion. That is to say, before the covering with concrete.

Problems that are called to be solved with the particular analysis:

A) Many times because of the bad estimate of quantity of explosives, are created big overbreaks in the surface of the forehead of a tunnel. So in order to overlap those regions with concrete the company is called to pay considerably the cost of this bad estimation. The cost of is proportional to quantity in cubic meters.

In drawing 5 is presented the mapping of excavations that was realized in a tunnel.

The way of calculation of excavations in projective development of mapping, is reported in Step 1-8. In drawing 5 and in figures 1, 2 are presented two cross-sections that present big overbreaks after excavation with explosives.

B) The bad estimation that is reported above, will overload also the work time schedule of the company. That is to say, further time of work of personnel, and more generally the elongation of the timetable of completion of work.

C) The bad estimate of geology will involve the problems that are described in paragraphs A, B.

D) The bad estimate of length of an explosion in combination with the bad estimate of geology will involve the problems that are described in paragraphs a, b.

Aim of forecast of future excavations with explosions is the research of explosions that has been already realized in the past. They thus are investigating similar situations of the past, from the start of the manufacture presenting the problems.

In this way is achieved the correct future prevention, avoiding thus similar situations. The way in order to achieve the correct prevention is the identification of previous cross- sections that has been realized by the primary surveys after each explosion.

The conditions in order to realize the correct identification are the following:

a) The cross-sections that will be identified must have the same geology.

b) For each cross-section has been used the same quantity of explosives.

c) For each cross-section has been realized the same length of perforation.

Provided that the conditions above are fixed by the responsible engineers of manufacture, then will be calculated future cross-sections. That is to say, it will be calculated virtual points of primary surveys. It thus will be created virtual cross-sections in the space of a tunnel. An example is presented in drawing 5. In drawing five figure 3, is presented with form of projective mapping how it could be the morphology of ground in the next explosions, always based on the explosions of the past. So In the figure 3 of drawing 5 is presented the merging of two cross-sections of figure 1, 2. The calculation of projective mapping of forecast of excavations with explosions can be realized as it is reported in Steps 1-8.

Requested elements for the calculation of forecast of future excavations with explosions: The requested elements for the calculation that will be needed are based on the steps of 1 -8. Suppose that in two cross-sections of excavation with explosives and for each point on the cross section that has been surveyed, according to the Steps 1-5, the calculation of the following elements has been done:

Step 1. Calculation in the two-dimensional space (X, Y of cross-section).

Step 2. Grouping of measurement points on the cross-section.

Step 3. Distance from the center of the circle until the point of measurement.

Step 4. Calculation of Z of the points of measurement.

Step 5. Calculation of azimuth for each point on the cross-section.

Algorithm of calculation of forecast of future excavations with explosions.

The algorithm of calculation of forecast of future excavations with explosions can be realized with the help of a computer.

When the calculations above according to Step 1-5 have been realized for each cross-section of excavation and for each point of measurement on the cross-section that was realized in the past, the next phase is the grouping of cross-sections according to the following criteria.

1) The excavations must have same geology.

2) For each excavation has been used the same quantity of explosives.

3) For each excavation was realized the same type of perforation.

Suppose that is requested by the responsible engineers of manufacture to realize the forecast of excavation with explosions in the forehead of a tunnel. It has been found that the geology of the forehead is constituted mainly by marble and will be realized perforation of length of three meters and will be used 200 kilos of explosives.

Then it will be done a research from the responsible engineers of manufacture in previous cross-sections, in order to investigate if there have been realized topographic primary surveys of cross-sections with similar cases according to the criteria that are reported above.

When the research is realized and will be found that has been already realized similar explosions in cross-sections as it presents in Drawing 5 figures 1, 2, then the steps that will follow for the calculation of each point of measurement on each cross-section, will be the followings:

Distance from the center of the circle where each point belongs until the point of measurement according to Step 3. Suppose that for the cross-sections of figure 1 , 2 of drawing 5, the points that belong on the top circle have been grouped and the theoretical radius of the circle is: R=7.85.

According to Step 3 the radius of each point is calculated.

The difference that results from the radius of each point and the theoretical radius is the Z of the points according to the calculation of Step 4.

The results are portrayed in tables 1, 2, and they concern the analysis of the cross-sections that are shown in the drawings 7 and 8. The figure 1 in the drawings 7 and 8 presents the radius of the points as they have been analyzed in tables 1 and 2. The figure 2 in the drawings 7 and 8 presents the arc of the top circle where the points of the tables 1 and 2 belongs. The figure 3 in the drawings 7 and 8 presents the code of each point as they are presented in tables 1 and 2.

Calculation of azimuth of cross-section according to Step 5.

Suppose that the azimuth according to Step 5, for each point of cross-sections of drawings 7 and 8 that belongs on the top circle, has already been calculated. The azimuths of the points of the cross-sections are presented in tables 1 and 2. Negative sign on the values of the azimuths is presented when the measured points are found left from the axis, according to the serial flow of the alignment.

TABLE I . CROSS-SECTION OF DRAWING 7. THE CROSS-SECTION IS REPORTED TO THE CROSS-SECTION OF FIGURE 1 OF DRAWING 5. KILOMETRIC POSITION: 2785.166

AZIMUTH OF POINT OF DETERMINATION OF RADIUS OF THEORETICAL Z OF

POINT CROSS-SECTION CIRCLE POINT RADIUS POINTS

7193 59.563 TOP CIRCLE 8.344 7.85 0.494

7194 54.625 TOP CIRCLE 8.554 7.85 0.704

7195 50.645 TOP CIRCLE 8.599 7.85 0.749

7196 44.343 TOP CIRCLE 8.375 7.85 0.525

7197 38.951 TOP CIRCLE 8.503 7.85 0.653

7198 33.047 TOP CIRCLE 8.629 7.85 0.779

7199 28.746 TOP CIRCLE 8.537 7.85 0.687

7200 24.810 TOP CIRCLE 8.325 7.85 0.475

7201 20.287 TOP CIRCLE 8.621 7.85 0.771

7202 15.400 TOP CIRCLE 8.616 7.85 0.766

7203 9.117 TOP CIRCLE 8.632 7.85 0.782

7204 2.496 TOP CIRCLE 8.292 7.85 0.442

7205 -2.567 TOP CIRCLE 8.485 7.85 0.635

7206 -8.825 TOP CIRCLE 8.554 7.85 0.704

7207 -14.365 TOP CIRCLE 8.640 7.85 0.790

7208 -19.171 TOP CIRCLE 9.460 7.85 1.610

7209 -22.716 TOP CIRCLE 9.359 7.85 1.509

7210 -26.318 TOP CIRCLE 9.193 7.85 1.343

7211 -30.570 TO CIRCLE 9.103 7.85 1.253

7212 -34.854 TOP CIRCLE 8.994 7.85 1.144

7213 -40.870 TOP CIRCLE 9.055 7.85 1.205

7214 -45.907 TOP CIRCLE 8.838 7.85 0.988

7215 -51.752 TOP CIRCLE 8.908 7.85 1.058

7216 -57.887 TOP CIRCLE 9.085 7.85 1.235 TABLE 2. CROSS-SECTION OF DRAWING 8. THE CROSS-SECTION IS REPORTED TO THE CROSS-SECTION OF FIGURE 2 OF DRAWING 5. KILOMETRIC POSITION: 2739.173

AZIMUTH OF POINT OF DETERMINATION OF RADIUS OF THEORETICAL

OF POII

POINT CROSS-SECTION CIRCLE POINT RADIUS

8515 60.5773 TOP CIRCLE 8.277 7.85 0.427

8516 56.1641 TOP CIRCLE 8.503 7.85 0.653

8517 51.1861 TOP CIRCLE 8.402 7.85 0.552

8518 47.7680 TOP CIRCLE 8.824 7.85 0.974

8519 44.5169 TOP CIRCLE 9.107 7.85 1.257

8520 39.0804 TOP CIRCLE 8.689 7.85 0.839

8521 36.7360 TOP CIRCLE 9.061 7.85 1.211

8522 33.5748 TOP CIRCLE 8.945 7.85 1.095

8523 28.6689 TOP CIRCLE 8.494 7.85 0.644

8524 25.3291 TOP CIRCLE 8.620 7.85 0.770

8525 20.9772 TOP CIRCLE 8.684 7.85 0.834

8526 17.6736 TOP CIRCLE 8.691 7.85 0.841

8527 12.9136 TOP CIRCLE 8.513 7.85 0.663

8528 8.8814 TOP CIRCLE 8.457 7.85 0.607

8529 4.1165 TOP CIRCLE 8.388 7.85 0.538

8530 -0.5408 TOP CIRCLE 8.358 7.85 0.508

8531 -4.2020 TOP CIRCLE 8.399 7.85 0.549

8532 -7.3406 TOP CIRCLE 8.440 7.85 0.590

8533 -10.8436 TOP CIRCLE 8.790 7.85 0.940

8534 -14.3243 TOP CIRCLE 9.215 7.85 1.365

8535 -18.7422 TOP CIRCLE 9.291 7.85 1.441

8536 -22.6278 TOP CIRCLE 8.977 7.85 1.127

8537 -25.4523 TOP CIRCLE 8.576 7.85 0.726

8538 -28.7882 TOP CIRCLE 8.637 7.85 0.787

8539 -31.9552 TOP CIRCLE 8.719 7.85 0.869

8540 -35.2237 TOP CIRCLE 8.710 7.85 0.860

8541 -38.3913 TOP CIRCLE 8.689 7.85 0.839

8542 -41.4198 TOP CIRCLE 8.641 7.85 0.791

8543 -45.0747 TOP CIRCLE 8.383 7.85 0.533

8544 -48.6745 TOP CIRCLE 8.461 7.85 0.611

8545 -52.0896 TOP CI CLE 8.536 7.85 0.686

8546 -55.2175 TOP CIRCLE 8.496 7.85 0.646

8547 -59.5371 TOP CIRCLE 8.320 7.85 0.470

Azimuthal determination:

The crowds of points from the two different cross-sections that are presented on the tables above belong to the top arc of the circle. Suppose that the lower hypsometric level of the left arc of the top circle has the azimuth - 60.120grads, and from right respectively 60.120grads. (Drawing 7, 8). Then it is determined per how many grads are requested to calculate the average of the radius of the points that are reported on the two tables above. The radius of each point is the distance from the center of the circle on which they belong until the measured points. For example, suppose that it will be calculated the average of the radius of the points per lOgrads. From the two tables than from the beginning of the left arc of the top circle, and counterclockwise it will be as follows: O.OOOgrads, -lO.OOOgrads, -20.000grads, - 30.000grads, - 40.000grads, - 50.000grads, - 60.000grads, - 60.120grads.

Respectively, from the beginning of the right arc of the top circle and clockwise will be, in effect, the azimuthal range:

O.OOOgrads, lO.OOOgrads, 20.000grads, 30.000grads, 40.000grads, 50.000grads, 60.000grads.

For each range per 1 Ograds is calculated the average of the radius of the points that comes from both tables. All points of the example belong on the top circle of the theoretical cross section.

The azimuthal determination should be determined from the engineers of manufacture.

The azimuthal determmation will be realized for each point that belongs to any circle.

Merging of cross-sections of forecast of excavations with explosions.

The final results of the azimuthal determmation incorporate the cross-sections from the tables 1, 2 and create a single table with the average of the radius of the points from the two cross- sections per 1 Ograds.

The merging of cross-sections can be realized from unlimited cross-sections of topographic primary surveys that were realized on the past.

An example from the calculation of merging of cross-sections is presented on table 3.

Table 3 emanates from the merging of the two cross-sections of drawings 7,8, which the elements of the topographic primary surveys that are constituted, were analyzed on tables 1,2. The elements that present on the table 3 are the followings:

Column 1 is reported to the code of each point as it was realized from the topographic

primary surveys.

Column 2 is reported to the azimuth of each point of the cross-section. The azimuth for each point of cross-section is calculated according to Step 5.

Column 3 is reported to the radius of each point of the cross-section. The radius of each point of cross-section is calculated according to Step 3.

Column 4 is reported to the theoretical radius of the circle where each point belongs. The grouping of topographic points from the primary survey is reported in Step 2.

Column 5 is reported to the Z of the points that come from the primary survey. The

calculation the Z of the points is reported to Step 4.

Column 6 is reported to the result that comes from the average of the radius of the points, with the azimuthal range per 1 Ograds as it has been reported above.

Column 7 is reported to the new Z of the points that comes from the difference of average of the existing points radius with the azimuthal range 1 Ograds, and their theoretical radius.

Column 8 is reported to the determination of the circle where each point of the primary survey belongs. The determination of the circle where each point of the primary survey belongs is reported in Step 2.

Column 9 is reported to the kilometric position of the points.

TABLE 3. MERGING OF CROSS-SECTIONS 2785.17 AND 2739.17.

2 9

3

1 AZIMUTH 5 7 8 CROSS-

RADIUS AVERAGE OF SECTIONS

POINT OF POINT Z OF AVERAGE OF RADIUS DETERMINATION _K,_LOMETRIC

OF THEORETIC POINTS PER 10

OF CROSS- POINTS PER 10GRADS OF CIRCLE POSITION

POINT AL RADIUS GRADS SECTION

7217 -62.9138 8.749 7.85 0.899 8.644 0.794 TOP CIRCLE 2785.17

8548 -62.6819 8.411 7.85 0.561 8.644 0.794 TOP CIRCLE 2739.17

8547 -58.8337 8.320 7.85 0.470 8.669 0.819 TOP CIRCLE 2739.17

7216 -57.8872 9.085 7.85 1.235 8.669 0.819 TOP CIRCLE 2785.17

8546 -55.2175 8.496 7.85 0.646 8.669 0.819 TOP CIRCLE 2739.17

8545 -52.0896 8.536 7.85 0.686 8.669 0.819 TOP CIRCLE 2739.17 7215 -51.7516 8.908 7.85 1.058 8.669 0.819 TOP CIRCLE 2785.17

8544 -48.6745 8.461 7.85 0.611 8.581 0.731 TOP CIRCLE 2739.17

7214 -45.9070 8.838 7.85 0.988 8.581 0.731 TOP CIRCLE 2785.17

8543 -45.0747 8.383 7.85 0.533 8.581 0.731 TOP CIRCLE 2739.17

8542 -41.4198 8.641 7.85 0.791 8.581 0.731 TOP CIRCLE 2739.17

7213 -40.8696 9.055 7.85 1.205 8.833 0.983 TOP CIRCLE 2785.17

8541 -38.3913 8.689 7.85 0.839 8.833 0.983 TOP CIRCLE 2739.17

8540 -35.2237 8.710 7.85 0.860 8.833 0.983 TOP CIRCLE 2739.17

7212 -34.8543 8.994 7.85 1.144 8.833 0.983 TOP CIRCLE 2785.17

8539 -31.9552 8.719 7.85 0.869 8.833 0.983 TOP CIRCLE 2739.17

7211 -30.5704 9.103 7.85 1.253 8.974 1.124 TOP CIRCLE 2785.17

8538 -28.7882 8.637 7.85 0.787 8.974 1.124 TOP CIRCLE 2739.17

7210 -26.3177 9.193 7.85 1.343 8.974 1.124 TOP CIRCLE 2785.17

8537 -25.4523 8.576 7.85 0.726 8.974 1.124 TOP CIRCLE 2739.17

7209 -22.7156 9.359 7.85 1.509 8.974 1.124 TOP CIRCLE 2785.17

8536 -22.6278 8.977 7.85 1.127 8.974 1.124 TOP CIRCLE 2739.17

7208 -19.1710 9.460 7.85 1.610 9.152 1.302 TOP CIRCLE 2785.17

8535 -18.7422 9.291 7.85 1.441 9.152 1.302 TOP CIRCLE 2739.17

7207 -14.3652 8.640 7.85 0.790 9.152 1.302 TOP CIRCLE 2785.17

8534 -14.3243 9.215 7.85 1.365 9.152 1.302 TOP CIRCLE 2739.17

8533 -10.8436 8.790 7.85 0.940 8.534 0.584 TOP CIRCLE 2739.17

7206 -8.8246 8.554 7.85 0.704 8.534 0.684 TOP CIRCLE 2785.17

8532 -7.3406 8.440 7.85 0.590 8.534 0.684 TOP CIRCLE 2739.17

8531 -4.2020 8.399 7.85 0.549 8.534 0.684 TOP CIRCLE 2739.17

7205 -2.5666 8.485 7.85 0.635 8.534 0.684 TOP CIRCLE 2785.17

8530 -0.5408 8.358 7.85 0.508 8.426 0.576 TOP CIRCLE 2739.17

7204 2.4957 8.292 7.850 0.442 8.426 0.576 TOP CIRCLE 2785.17

8529 4.1165 8.388 7.850 0.538 8.426 0.576 TOP CIRCLE 2739.17

8528 8.8814 8.457 7.850 0.607 8.426 0.576 TOP CIRCLE 2739.17

7203 9.1175 8.632 7.850 0.782 8.426 0.576 TOP CIRCLE 2785.17

8527 12.9136 8.513 7.850 0.663 8.607 0.757 TOP CIRCLE 2739.17

7202 15.3997 8.616 7.850 0.766 8.607 0.757 TOP CIRCLE 2785.17

8526 17.6736 8.691 7.850 0.841 8.607 0.757 TOP CIRCLE 2739.17

7201 20.2868 8.621 7.850 0.771 8.547 0.697 TOP CIRCLE 2785.17

8525 20.9772 8.684 7.850 0.834 8.547 0.697 TOP CIRCLE 2739.17

7200 24.8096 8.325 7.850 0.475 8.547 0.697 TOP CIRCLE 2785.17

8524 25.3291 8.620 7.850 0.770 8.547 0.697 TOP CIRCLE 2739.17

8523 28.6689 8.494 7.850 0.644 8.547 0.697 TOP CIRCLE 2739.17

7199 28.7462 8.537 7.850 0.687 8.547 0.697 TOP CIRCLE 2785.17

7198 33.0474 8.629 7.850 0.779 8.765 0.915 TOP CIRCLE 2785.17

8522 33.5748 8.945 7.850 1.095 8.765 0.915 TOP CIRCLE 2739.17

8521 36.7360 9.061 7.850 1.211 8.765 0.915 TOP CI CLE 2739.17

7197 38.9505 8.503 7.850 0.653 8.765 0.915 TOP CIRCLE 2785.17

8520 39.0804 8.689 7.850 0.839 8.765 0.915 TOP CIRCLE 2739.17

7196 44.3427 8.375 7.850 0.525 8.769 0.919 TOP CIRCLE 2785.17

8519 44.5169 9.107 7.850 1.257 8.769 0.919 TOP CIRCLE 2739.17 8518 47.7680 8.824 7.850 0.974 8.769 0.919 TOP CIRCLE 2739.17

7195 50.6448 8.599 7.850 0.749 8.481 0.631 TOP CI CLE 2785.17

8517 51.1861 8.402 7.850 0.552 8.481 0.631 TOP CIRCLE 2739.17

7194 54.6253 8.554 7.850 0.704 8.481 0.631 TOP CIRCLE 2785.17

8516 56 1641 8.503 7.850 0.653 8.481 0.631 TOP CIRCLE 2739.17

7193 59.5634 8.344 7.850 0.494 8.481 0.631 TOP CIRCLE 2785.17

8515 60.5773 8.277 7.850 0.427 8.277 0.427 TOP CIRCLE 2739.17

It will be calculated as an example the points of the two different cross-sections that belong in the azimuthal range - 50.000grads until - 60.000grads.

After the grouping of the points that has preceded from the example above, the points of the two tables were found that belong to the arc of the top circle. The azimuthal determination was fixed per lOgrads. At the next step is calculated the average of the radius of the points of the arc of the top circle per lOgrads. It will be calculated points of the two different cross- sections that belong in the azimuthal range - 50.000grads until - 60.000grads.

For table 1 the points that they belong in the particular range are the followings:

OF

POINT AZIMUTH OF POINT OF DETERMINATION OF ADIUS OF THEORETICAL z

OINTS CROSS-SECTION CIRCLE OINT RADIUS

7215 -51.752 TOP CIRCLE 8.908 7.85

7215 -57.887 TOP CIRCLE 9.085 7.85

For table 2 the points that they belong in the particular range are the followings

POINT AZIMUTH OF POINT OF DETERMINATION OF RADIUS OF THEORETICAL

OF POINTS CROSS-SECTION CIRCLE POINT RADIUS

8545 -52.0896 TOP CIRCLE 8.536 7.85 0.686

8546 -55.2175 TOP CIRCLE 8.496 7.85 0.646

8547 -59.5371 TOP CIRCLE 8.320 7.85 0.470

The points of the two tables belong in the azimuthal ranges between - 60.000grads and - 50.000grads. In the next step is calculated the average of the radius of the points from both tables. Thus it will result: R2= 8.908+ 9.085+8.536+8.496+8.320 = 8.669.

5

The value of R2=8.669 is included in table 3 for the particular azimuthal ranges, and is reported in the column of 6 table 3.

Z of points that results from the mean of the point's radius.

The Z of points will be the difference from the mean of radius that was calculated above, with the value of the theoretical radius of the top circle. That is to say, Z2=8.669-7.85=0.819.

The value of Z2=0.819 is included in table 3 for the particular azimuthal ranges, and is reported in the column of 7 table 3.

In drawing 9 is presented in figure 1 the radius R2 of the points that resulted from the mean of radius of the points from the two cross-sections. In the figure 2 of drawing 9 is presented the Z of points that results from the difference of the theoretical value of the radius that comes from the top circle. In the same drawing is presented as well the value of R2. In figure 3 of drawing 9 is presented the beginning of the lower level of the azimuth of the left arc that belongs to the top circle. This azimuth is - 60.120grads. In the figure 4 of drawing 9 is

presented the first alternation of the azimuthal ranges that was fixed per 1 Ograds, beginning from the azimuth - 50.000grads and - 60.000grads counterclockwise. In figure 5 of drawing 9 is presented the next azimuthal alternation that will result according to the calculations above. More generally drawing 9 presents the behavior of the surface of a cross-section after the merging of previous cross-sections of excavations from the past. At the next steps of the algorithm are calculated the followings:

A) The two-dimensional coordinates from the points that are attributed to the cross-section of the merging of the two cross-sections that were realized in the past.

B) The three-dimensional absolute coordinates of the future points of forecast in the cylindrical surface of a tunnel. That is to say, it will be calculated coordinates of virtual points.

C) The three-dimensional development of the merged cross-section to attribute the projection mapping of the forecast, according to Step 1-8.

It will be supposed that the azimuth of the direction of the axis of the tunnel has already been calculated and concerns the point that has been selected in order to calculate the forecasting mapping projection. Suppose that the azimuth of the direction of axis in the region that is selected to create the forecasting of excavations, is AZ=162.329grads. Also it is known that the points that are reported above belonging to the left arc of the top circle. Then the rule should be fixed as follows:

For each point of forecast of excavations that has a negative sign, that is to say, is found left from the axis according to the serial flow of the tunnel alignment, then the azimuth for each point of a cross-section from the axis will be 162.239grads+300grads.=462.329grads. In the opposite case, that is to say, if the point is found right from the axis then it will be, in effect, 162.239+100 grads=262.239grads.

Thus azimuth for the two points of the example will be 462.329grads.

Because the value of azimuth is greater than 400grads then it will be supposed is calculated the parameter 462.329-400=62462. Therefore the azimuth AZ1 of the axis until the cross- section will be AZ 1=62.329.

This azimuth is the absolute azimuth from the axis concerning the points that belong to the forecasting cross-section which its azimuth will be used later for the calculation of the absolute coordinates of XI, Yl of the forecasting points.

Thus as known are the followings:

Coordinates of the center of the top circle where the points belong:

XC 1=0.000, YC 1 =0.004. Drawing 9.

The mean of radius of points R2=8.669. Drawing 9 figure 1.

The azimuth of cross-section for each point as they are presented on the tables above.

Suppose that a point that results from the merging of cross-sections has the azimuth Azp=- 59.537grads. Drawing 10 figure 2. The azimuth for each point of cross-section is calculated according to Step 5.

Calculation of cross-section coordinates for the point.

In order to calculate the cross-section coordinates Xa, Ya for the point of Drawing 10 figure 1, it will be realized according to the following calculation: Xa= XC1+R2*SIN (Azp) It therefore, results in Xa=0.000+8.6795*STN (- 59.537) = -6.985.

Thus the X of cross-section for the point, is Xa=-6.985. Consequently, the point is found left from the axis 6.985 meters. (Drawing 10 figure 3).

For the calculation of Ya of the point will be in effect the following calculation:

Ya= YCl+R2*COS (Azp). It therefore, results in Ya=-0.04+8.6795*COS (- 59.537) = 5.1 126.

Thus the Y of cross-section for the point is Ya=5.1126. Consequently, the point is found right from the axis 5.1 126 meters. (Drawing 10 figure 4).

Calculation of the absolute coordinates X, Y of the points so that the calculation of the three- dimensional transformation of projective development according to Step 8 can be continued. In this way will be calculated the three-dimensional transformation of the coordinates that will be used for the attribution of mapping of cylindrical development of a tunnel.

That is to say, they will be calculated as points of primary measurements, which in the substance have never been really measured, but they will be used for the final mapping of forecasting where the position of the future blasting has been selected.

In order to calculate the absolute coordinates of the forecasting points it will be supposed first it is calculated the two-dimensional coordinates of the points. The calculation in the two- dimensional space (Xa, Ya of a cross-section), can be realized according to Step 1.

Consequently, for each point will be calculated the absolute coordinates for X, Y (Drawing 1 figure 1). That is to say, the absolute coordinates of the points in the particular region where it has been selected by the responsible engineers of the project, so that to continue the steps of calculations of the three-dimensional development of forecast of future explosions. In the substance the X, Y will be the coordinates of points where according to Step 8, they are given by the topographic primary surveys in the interior of a tunnel. In this case, however, that coordinate does not exist, and they need to be calculated, because they belong in a region. Where has not yet been realized excavation of a tunnel.

Thus it will be supposed that the following elements are already known:

Coordinates of the axis X0, Y0 where it has been selected by the responsible engineers of manufacture as the beginning of forecast of excavations. The coordinates X0. Y0 is the vertical projection of the virtual points of forecast to the axis. (Drawing 1 figure 2). Suppose that X0= 374258.762, Y0= 4412377.287.

Xa of point of cross-section is Xa=-6.985. (Drawing 10 figure 3)

The azimuth of the points to the axis was calculated above and was found AZ1 =62.329 (Drawing 1 figure 7).

Thus the calculation of the absolute coordinates of XI of the point will be as follows:

XI (absolute) = X0+Xa*SIN (AZ1 )

It therefore, results in XI (absolute) =374258.762+ (- 6.985) *S1 (62.329) = 374252.965.

The calculation of the absolute Yl of the point will be as follows:

Yl (absolute) = Y0+Xa*COS (AZ1 )

It therefore, results in Yl (absolute) =4412377.287+ (- 6.985) *COS (62.329) = 4412373.391.

In order to be calculated the absolute altitude of the forecasting point it will need to be found its vertical projection to the longitudinal profile of the road construction. That is to say, for the point is requested the altitude of road construction vertical in the axis in which it belongs.

(Drawing 1 figure 3). Suppose that the projection of the axis for this point has altitude Z=45.08. Then the sum of the altitude of road construction and then Ya (Drawing 10 figure 4) that was calculated above will be absolute Z of the forecasting point. That is to say, Z of forecasting point = 45.08+5.1126=50.1926.

In drawing 6 figure 8 is presented the three-dimensional output of a cross-section that includes the point where it has been calculated above.

In the figure 4 of drawing 6 is presented the beginning of forecast of excavations as the responsible engineers could request it. In the particular case of the example, the length of perforation was requested to be 3 meters.

In the figure 5 of drawing 6 is presented the finishing of the three meters of forecast that was requested. In the figure 3 of drawing 6 is presented the three-dimensional model of excavations of a tunnel that was realized in the past.

In the figures of 1,2 of drawing 6 are presented the two cross-sections of the examples above that was realized the merging to be calculated the cross-section of forecast.

In the figure 6 of drawing 6 is presented the three-dimensional realistic simulation of surface of excavations that was realized in the past.

In the figure 8 of drawing 6 are presented the interior of a tunnel, and the forecast of excavations where the point of the example is included as well, and was calculated above. In the figure 7 of drawing 6 is presented the finishing of the three meters of forecast of excavations in the interior of a tunnel. The presentation and the three-dimensional visualization of cross-section of forecast of excavations as it is presented in drawing 6 and in figures 4,5,7,8 are the corresponding cross- section of forecast of excavations that was calculated in the two-dimensional space, and it is presented in drawings 9, 10.

In order to attribute the surface of projective mapping in the real space of the project, it will be supposed that has been realized a request by the responsible engineers about the length that will need for the forecasting calculation. Thus following the steps until now that concerns the explanation of method of mapping and control of surfaces of tunnels in projective development, depending on the length of perforation it will be supposed that become respectively the forecasting calculations.

For example, if it has been requested that the particular cross-section of forecast of excavation will run for 3 meters, creating cross-sections for each meter, then will be changed the following elements:

The X0, Y0, Z0 of axis for each next cross-section.

The azimuth of direction of the axis for each next cross-section. (Consequently, the azimuth of the points to the axis).

It will remain the distance from the center of each circle until each point that results from the mean of merging of cross-sections per 10 grads. That is to say, R2= 8. 6795m will remain the same, as well as the cross-section azimuth for each point.

It will be more concretely realized new absolute coordinates of X, Y, Z of the points for each forecasting cross-section that will run to the requested position and length. That is to say, for each meter for the next 3 meters, and from the beginning of the first forecasting cross-section. Thus from the crowd of the new virtual points that will be calculated will be created a single surface in which it will present the forecasting of the next explosion that will be realized in the length of 3 meters.

The forecasting example as it is reported above is linear.

It is linear because the final surface of mapping that will be created by the cross-section that will run per one meter, will present the same Z for each point in the all length. The not linear method can be achieved from the following way:

Suppose that has been realized primary surveys of explosions where according to the criteria above, has been grouped according to the geology of the ground and the quantity of explosives that was used. Then it will be supposed that the merging of cross-sections has been calculated according to the range that has been realized by the primary surveys.

For example, suppose that have been realized two explosions with the same criteria and had length 3 meters each. In the space of each explosion have been surveyed cross-sections every meter.

It consequently, will be calculated the merging of cross-sections with the following way: The first cross-section of the explosion A with the first cross-section of the explosion B.

The second cross-section of the explosion A with the second cross-section of the explosion B.

The third cross-section of the explosion A with the third cross-section of the explosion B.

It thus is created not linear forecasting of mapping of excavation, which has important resemblances of erratic overbreaks of explosions of the real excavation and thus can be achieved more analytical forecasting.

The precision of forecast will be higher each time. Where will be realized new explosions.

The reason is because of the increasing for each time of the crowd of the primary surveys of the real excavation. In this way will be determined more the behavior of the tunnel forecasted surface model after each explosion. So that it will be dramatically minimized the estimation error of blasting. Offering thus to the constructional company a priceless tool of reduction of cost of excavations, as well as better quality of the manufacture and control of tunnels. According to Step 6 will be calculated the development of the cross-section arcs.

As known is:

Azp = - 59.537grads. (Drawing 10 figure 2)

The distance from the center of the top circle until the point. The distance is calculated by the mean of the radius of the points that emanates from the merging of cross-sections. That is to say, R2= 8.669m (Drawing 9 figure 1).

The calculation that will be realized is the following: Ll= PI x 8.669 x -59.537

200

Thus the Lm of the first point is: Lm= -8.1171m. (Drawing 10 figure 5).

There is no other circle that precedes from the position that the point belongs. So the specific value that has been calculated above belongs to the top circle.

According to Step 8 (three-dimensional coordinates transformation for the attribution of mapping of cylindrical development of a tunnel), will be calculated the cylindrical development of the forecasting point that was calculated above. As known will exist the following elements:

A) The coordinates of the central axis X0, Y0 where they are the vertical projection of the points to the axis. X0= 374258.762, Y0= 4412377.287. (Drawing 1 figure 2)

B) The absolute coordinates XI (absolute) = 374252.965, Yl (absolute) = 4412373.391. Those coordinate concerns the forecasting point that was calculated above.

C) The Lm of the point as it has been calculated above is: Lm=-8.1171m (Drawing 10 figure5).

D) The absolute azimuth from the cross-section that the forecasting points included, until the axis.

Above was needed to be as known the azimuth from the axis, to the forecasting points, so that to be calculated the absolute XI, Yl of the forecasting point. The azimuth that had been calculated above, was AZ1=62.329. Thus in order to calculate the azimuth to the opposite direction will be 62.329+200=262.329grads.

That is to say, the azimuth from the forecasting point to the axis X0, Y0 will be

AZ2=262.329grads.

Thus the steps of the calculation for the three-dimensional forecasting point will be as follows:

The algorithm that will be used in order to calculate the transformation for the XI will be: X of (cylindrical development) =X0+Lm (always with positive sign) x SIN (AZ2)

Thus: Xl=374258.762 + (- 8.1171) x SIN (262.329) = 374265.499

The type that will be used in order to calculate the transformation for the Yl will be:

Yl of (cylindrical development) =Y0+Lm (always with positive sign) x COS (AZ2) Thus: Yl=4412377.287+ (- 8.1171) x COS (262.329) = 4412381.815

The Z of the points was calculated above and emanates from the mean of the radius of the two points from the center of the top circle. That is to say, R2= 8.6795m.

The theoretical radius has value 7.85m. Therefore, 8.6795-7.85=0.8295m.

Thus the Z1=0. 8295m

The final results of the projective three-dimensional transformation of the first point where it will be included in the mapping of forecast of excavation will be the following:

Xl= 374265.499

Yl = 4412381.815

Zl=0.8295. The projective transformation of point in the two-dimensional space of cross-section presents in drawing 10 figure 6.

Example of the projective three-dimensional transformation of the point is presented in drawing 1 figure 6.

Calculation of statistical analysis of forecast of excavations.

Conditions in order to be realized the correct identification are the following:

A) The cross-sections that will be identified, they have same geology.

B) It has been used the same length of perforation.

C) It has been calculated the merging of the forecasting cross-sections according to drawings 9, 10.

The algorithm of the statistical analysis of forecast of excavations can be realized with the help of a computer.

According to the calculation of forecast of excavations that was realized, was calculated the mean of radius of the points from the two tables. It was found that they belong to the left arc of the top circle. The value was R2=8.669. (Drawing 9 figure 1). It was selected as an example the azimuthal ranges of the top circle between - 60.000grads and - 50.000grads (Drawing 9 figure 4).

Suppose that the quantity of explosives that was used for the cross-section of drawing 7 is 200 kilos.

Suppose that the quantity of explosives that was used for the cross-section of drawing 8 is 180 kilos.

Then as known it will be supposed that was found the following:

The mean of the radius of the forecasting points R2=8.669 (Drawing 9 figure 1).

The mean of the quantity of the explosives of the cross-sections that in the particular case is E= 200+180 = 190 kilos.

2

The value of the radius of the top circle as it is reported in table 3 is R=7.85. Consequently, the mean of Z of the points for the azimuthal ranges between - 60.000grads and - 50.

OOOgrads was calculated as follows: Z=R2-R=0.819 (Drawing 9 figure 2).

In this way has been realized a forecasting according to the cross-sections that were realized by the primary surveys, as well as according to the individual geological criteria. For the particular position of the cross-section will exist overbreak in the surface 0.819 meters out from the theorist. (Drawing 9 figure 2). That is to say, according to the mean E=190 of the quantities of the explosives that has been calculated will exist this result.

It will be calculated the possibility of reduction of the quantity of explosives, in order to avoid similar overbreak in this area and more generally to the entirety of the forecasting cross- section. Thus if finally has been selected by the responsible engineer the reduction of explosives to be 16.98 kilos, then the calculation that will be used will be the following: R3 = 173.02*R2 = 7.894.

E

In this way has been recalculated the radius of the forecasting points in the particular region of the cross section. According to figure 6 of drawing 12, it will be R3=7.894.

The new Z of the forecasting points that will result will be the difference between the R3 and the theoretical value of the radius of the top circle. That is to say, Z2 = R3-R = 0.044 as it is presented in the figure 5 drawing 12.

It will be calculated the difference between the initial Z and the Z2 of the forecasting points that resulted from the mean of the radius of the forecasting points. That is to say, it will be: DZ=Z-Z2., That is to say, DZ=0.819-0.044=0.775.

According to the grouping of points, it was found that the points belong to the top circle with theoretical value of radius R=7.85. The next step is to become subtraction 0.775 meters.

That is to say, it will become subtraction 0.775 meters of the radius for each forecasting point of the cross-section where was realized the merging, and belong to the top circle, to the azimuthal ranges - 60.000grads until - 50.000grads.

The same steps for the explanation until now of the statistical analysis of forecast of excavations will be done for the points of cross-section that belong to any circle of the cross- section. That is to say, for each circle of the cross section will be realized autonomous behavior of the calculation according to the statistical criteria above. That is to say, the DZ of points that was calculated above will be proportional with the mean of radius of the forecasting points .

Visualization of statistical analysis of forecast of excavations.

a) Calculation of the new forecasting point coordinates of the statistical analysis in the two- dimensional space of cross-section.

b) Calculation of the new forecasting point coordinates of the statistical analysis in the three- dimensional space of a tunnel.

c) Calculation of the three-dimensional projective development of the new forecasting points of the statistical analysis.

A) It will be realized the calculation of the coordinates for the new forecasting points of the statistical analysis in the two-dimensional space of cross-section. It will be followed the same steps like for the calculation of the excavation virtual forecasting points, with the difference that it will not be included in the calculation the value of R2 (mean of radius of points that resulted from the merging of two cross-sections).

It will be included the value of R3=7.894 that was calculated according to the figure 6 drawing 12.

That is to say, the calculation for the new Xa of the cross-section that will result is the following:

Xa= XCl+R3*SrN (Azp)

It therefore, results in Xa=0.000+7.894*SIN (- 59.537) = -6.352.

Thus the X of cross-section for the point is Xa=-6.352. Consequently, the point is found left from the axis 6.352 meters. (Figure 2 drawing 12).

For the calculation of the new Ya of point that will result is in effect the following calculation:

Ya= YCl+R3*COS (Azp). It therefore, results in Ya=-0.04+7.894*COS (- 59.537) = 4.646. Thus the Y of cross-section for the point is Ya=4.646. Consequently, the point is found 4.646 meters from the axis. (Figure of 3 drawing 12).

In drawing 12 is presented the cross-section of excavation as it results after the statistical analysis of forecast of excavations. The difference concerning the cross-section of drawing 10 is obvious, and this difference concerns the morphology of surface of excavation.

In drawing 12 figure 2 is presented the new Xa of the forecasting point that results according to the calculation above.

In drawing 12 figure 3 is presented the new Ya of the forecasting point that results according to the calculation above.

In drawing 12 figure 6 is presented the new radius R3 of the forecasting points that results according to the calculation of statistical analysis of excavations with possible reduction in the quantity of explosives.

In drawing 12 figure 4 is presented the azimuth Azp of the point that was used for the calculation. The azimuth Azp remains same as in drawing 10 figure 2.

B) It will be realized the calculation of the coordinates of the new forecasting points of the statistical analysis in the three-dimensional space of the tunnel. It will be followed the same steps, with the difference that will be replaced the Xa=- 6.985 of the forecasting point as it is presented in drawing 10 figure 3, with the new Xa= of -6.352 of the forecasting point that results from the statistical analysis of forecast of excavations. (Drawing 12 figure 2). Thus as known it will be supposed that exist the following elements:

The coordinates of axis XO, YO where it has been selected by the responsible engineers of manufacture as the beginning of forecast of excavations.

The coordinates XO. YO is the vertical projection of the virtual forecasting points to the tunnel axis. (Drawing 1 figure 2). The axis coordinates are already given. Thus X0= 374258.762, and Y0= 4412377.287.

The new Xa of the forecasting point of the cross-section Xa=-6.352. (Drawing 12 figure 2) The azimuth from the forecasting points to the axis. It was calculated above: AZ1=62.329 (Drawing 1 figure 7).

Thus the new calculation of the absolute XI of the forecasting point will be realized as follows:

XI (absolute) = X0+Xa*SFN (AZ1)

It therefore, results in XI (absolute) =374258.762+ (- 6.352) *SI (62.329) = 374253.4899. The calculation of absolute Yl of the forecasting point will be realized as follows:

Yl (absolute) = Y0+Xa*COS (AZ 1 )

Therefore, Yl (absolute) =4412377.287+ (- 6.352) *COS (62.329) = 4412373.7438.

In order to calculate the new absolute altitude of the forecasting point it will be requested the value of its vertical projection to the longitudinal profile of the tunnel axis. That is to say, for the point is asked the altitude of the road construction vertical in the axis in which it belongs. (Drawing 1 figure 3). Suppose that the axis of road construction for this point has altitude Z=45.08. Then the sum of the altitude of the axis and the new Ya that results from the new forecasting points of the statistical analysis in the three-dimensional space of a tunnel (Drawing 12 figure 3) that was calculated above, will also be the new absolute Z of the forecasting point. That is to say, Z=45.08+4.646=49.726.

C) In order to realize the calculation of the three-dimensional projective development for the new forecasting points of the statistical analysis, first will precede the following calculations:

According to Step 6 will be calculated the development of the cross-section arcs.

As known elements are the followings:

Azp = - 59.537grads. (Drawing 12 figure 4)

The new distance from the center of the top circle, until the point that results from the statistical analysis of the forecast of excavations. That is to say, R3= 7. 894m, as it was calculated and is presented in drawing 12 figure 6.

The calculation will be: Lm= PI x 7.894 x -59.537

200

Thus Lm of first point =-7.3825m. (Drawing 12 figure 7).

The forecasting points belong to the top circle. So there is no other circle to precede from the forecasting points. So the rule of Step 7 will not be included in this calculation.

According to Step 8 (three-dimensional coordinates transformation for the attribution of mapping of cylindrical development of a tunnel), will be calculated the cylindrical development of the forecasting point above. As known will exist the following elements: a) The of coordinates of the axis X0, Y0 where they are the vertical projection of the forecasting points to the axis. Suppose that X0= 374258.762, Y0= 4412377.287. (Drawing 1 figure 2).

b) The absolute coordinates XI (absolute) = 374253.4899, Yl (absolute) = 4412373.7438, from the new forecasting point of the statistical analysis of forecast of excavations where it was calculated above.

c) The new Lm of the forecasting point that results from the statistical analysis of forecast of excavations. That is Lm = -7.3825m (Drawing 12 figure 7). d) The absolute azimuth from the cross-section where the forecasting points are included, until the axis.

Before it needed to exist as known the azimuth from the axis to the forecasting cross-section. So that to calculate the absolute XI , Yl of the forecasting point. The azimuth had been calculated to be AZ1=62.329. Thus in order to calculate the azimuth for the opposite direction will be 62.329+200=262.329grads.

AZ2=262.329grads.

Thus, the steps of calculation for the three-dimensional calculation of projective development for the point will be as follows:

The algorithm that will be used in order to calculate the transformation for the XI will be: XI of (cylindrical development) =X0+Lm (always with positive sign) x SIN (AZ2)

Thus, we have: Xl=374258.762 + (- 7.3825) x SIN (262.329) = 374264.889

The algorithm that will be used in order to calculate the transformation for the Yl will be: Yl of (cylindrical development) =Y0+Lm (always with positive sign) x COS (AZ2) Thus, we have: Yl=4412377.287+ (- 7.3825) x COS (262.329) = 4412381.405

The new Z of the forecasting point was calculated above and emanates from the algorithm of the statistical analysis of forecast of excavations. It has value:

Z2= 0.044 (Drawing 12 figure 5).

The final results of the projective three-dimensional transformation of first forecasting point where it will be included in the mapping of forecast of excavation will be the followings: Xl= 374264.889

Yl = 4412381.405

Z2= 0.044.

The projective transformation of the forecasting point in the two-dimensional space of cross- section that emanates from the algorithm of statistical analysis of forecast of excavations is presented in drawing 12 figure 7.

Example of projective three-dimensional transformation of the forecasting point is presented in drawing 1 figure 6.

The statistical analysis of forecast of excavations contributes in the prevention of future explosions and the dissuasion from any chance of estimate errors that in any case would be loss making at the duration of manufacture. They would be loss-making because in the cases of important overbreaks that would result without the forecasting, would need of re- establishment of overbreaks with big quantities of concrete. This fact increases the cost of production, reduces the quality of manufacture, as well as the safety of the employers. It reduces the employers safety because the more overbreaks exists on those surfaces, the more the obligatorily of overlapping with big quantities of concrete will be. So it is increased the probability of collapses of those surfaces in unsuspecting time.

Claims

C L A I M S.

1 ) Method of mapping and control of surfaces of tunnels in projective development that can be realized with the help of a computer where at the duration of manufacture are received by the topographic primary surveys from the total stations in the surfaces of tunnels at any phase of manufacture. It results in the transformation of the crowd of the topographic points from the cylindrical absolute system, to the three-dimensional cylindrical projective development. The following steps characterize the preparation and algorithm that will be used:

Preparation of the necessary data required in order to calculate the number of the primary surveying points from the cylindrical system to the absolute three-dimensional cylindrical projection.

The calculations of preparation steps can be realized with the help of a computer, using the classical methods of resolving road axis geometry.

STEP 1 OF PREPARATION

For the calculation of the requested data, should be given by the initial study the geometrical features of the axis of the road, as well as the longitudinal profile for each tunnel where the invention will be applied.

The geometric data features that should be known by the initial study are the followings: a) Geometry of the central axis of the road. Therefore, should be known the following features:

1 ) Polygonal coordinates.

2) Radius of the assembly curves.

2) Arcs of the circles.

3) Parameters of the clothoid curves.

b) Geometry of the longitudinal profile of the road (Radius, elevations).

STEP 2 OF PREPARATION

Since the above geometric features will be given, it will be followed the importation of the coordinates of the primary topographic points as they have been recorded by the

measurements of the total stations on the surface of the tunnel, with coordinates XI , Yl , Zl, for each point.

The data that should then be calculated are the followings:

a) For each primary measured point with coordinates XI, Yl , is requested the calculation of its perpendicular projection on the road axis.

That is to say, for each measured point is calculated the corresponding coordinates X0, Y0 of the axis in which has been realized perpendicular projection of each measured point.

From the calculation of the coordinates XO, Y0, that are corresponding to the perpendicular projection of each measured point on the axis of the road, will result in the calculation of the distance between X0, Y0 and XI , Yl . This distance will be considered, as sections Xa coordinates in the following steps of the calculation of the invention's algorithm.

The negative or positive sign of Xa depends from the direction of the axis mileage flow. b) Consequently, for every measured point XI, Yl that had been calculated its projective axis coordinates X0, Y0, could be calculated the elevation Za of the axis longitudinal profile, which corresponds to the same mileage position of the X0, Y0.

The calculations as mentioned to 1 , 2 preparation steps, can be realized with the help of a computer, using the classical methods of resolving road axis geometry. The algorithm of method of mapping and control of surfaces of tunnels in projective development that will be used for the transformation of the number of topographic primary measured points from the absolute cylindrical system into the three dimensional projective development, is characterized by the following range of steps:

STEP 1. Calculation of coordinates in the two-dimensional cross-sectional space Xa, Ya section coordinates.

Namely, Xa section coordinates is the perpendicular distance of each measurement point from the axis as it is mentioned on the preparation step 2 paragraph (a). Ya is the hypsometric difference from each point regard to each level Za of the longitudinal profile, which has been realized the perpendicular projection for each point as mentioned on the preparation step 2 paragraph (a).

In case where distance Xa has not been calculated as it is referred in paragraph (a) preparation of step 2, then since they are known the coordinates for each measurement point XI, Yl, and the coordinates X0, Y0 of its perpendicular projection to the axis, then the distance Xa will result from the following formula:

Xa = SQ T [(XI -X0) ^Λ2 + (Yl -Y0) ^Λ2].

The calculation of Ya can be realized where the elevation Za is known as it is referred in paragraph (a) step 2 of preparation. Namely, the absolute elevation of the longitudinal profile for each point that has been projected perpendicular to the axis of the road.

Therefore, if the elevation of a primary measured point is Zl, then the calculation to find its Ya section coordinates is the following: Ya= Zl-Za.

STEP 2. Grouping of measured points on the cross-sections.

The aim is to be found for each point the coordinates XC, YC of the center of the circle in which each measuring point belongs, as well as the theoretical radius Rl of each circle as it had given by the initial study in accordance to the theoretical section.

The method of calculating the grouping of the measurement points on section is as follows: It first is taken into consideration by how many circles a theoretical section constituted.

Then for each point will be compared the Ya section coordinates of each, with the elevation limits of the upper and lower levels of each arc where a theoretical cross section is composed. So that each measured, point will be grouped in accordance to the circle to which it belongs. Consequently, each measured point will be grouped in accordance to the arc to which it belongs.

Therefore, for each measured point where have been calculated the two-dimensional sectional coordinates Xa, Ya as it is mentioned in step 1, will be found in which circle belongs, considering the upper and lower elevation limits of the arcs of the circles where a theoretical cross section is composed.

Elevation limit of the upper and lower level of each arc of a circle is called the hypsometric difference between the road axis, and the beginning-end of each arc, that a theoretical cross section composed.

To perform the grouping of measurement points based on the elevation limits of each arc, there should be as known the following elements given by each initial study:

A) Coordinates XC, YC of the centers of the circles, which a theoretical cross section is composed.

B) The theoretical radius Rl for each circle on the cross-section as it is given by the initial study.

C) The Xa, Ya cross section coordinates of each measured point as have been calculated in step 1.

D) The value of azimuth AZR with direction reference from the center of each circle with center coordinates XC, YC, up to the lower and upper elevation limit of each arc where each circle is composed on the theoretical cross section. The azimuth must be in grads. Where zero grads are the axis of the theoretical cross-section. The azimuth direction can be counterclockwise or clockwise depended whether the cross- section coordinates of Xa has a negative or positive sign.

To calculate the elevation limits of the arcs of circles, will be realized the following calculations:

For a circle with a known azimuth, AZRlwith direction reference from its center coordinates XC, YC up to the upper level of its arc, will be applied the following formula:

hi (upper arc level) = YC + Rl x COS (AZR1 x PI () / 200).

For the same circle with direction reference from its center coordinates XC, YC up to the lower level of arc with azimuth AZR2, will be applied the following formula:

h2 (lower arc level) = YC + Rl x COS (AZR2 x PI () / 200).

The values hi, h2 represents the hypsometric difference between the axis of the road and the beginning and end of each arc of the circles that a theoretical section is contained.

In the same way can be calculated the remaining elevation limits of the begin and end of the arcs of the remaining circles.

The values AZR1, AZR2, represent the azimuthal values from the upper and lower elevation limit of the arc of each circle with direction reference from each center of a circle.

In the same way can be calculated the remaining elevation's limits of the begin and end of the arcs of the remaining circles.

The azimuthal values AZR1, AZR2, are given by the initial study of the theoretical section.

As Rl is the theoretical radius of each circle on the theoretical cross section, based on the initial study.

When it will be realized the calculation of the elevation limits of all arcs of circles that consist of a theoretical section, then it will be grouped for each measurement point on which circle it belongs, whether is on the left or right of the cross-section axis. This is determined by the sign of the Xa of cross section as mentioned in step 2, paragraph (a) of preparation.

Thus, the assumptions that will follow will determine the circle on which each measuring point with cross-section coordinates Xa, Ya.

For the upper and lower elevation HI, H2 of the arc AC 1R where is on the right of the theoretical section, and has a theoretical radius Rl and circle center coordinates XC1, YCl, will be applied the following formula:

If [Xa>0 αι Ya<Hl και Ya>H2], then the measurement points are determined in the particular arc with center coordinates XC1, YCl, and theoretical radius Rl.

For the upper and lower elevation H2, H3 of the arc AC1L which is on the left of the theoretical section, and has a theoretical radius R2 and circle center coordinates XC2, YC2, will be applied the following formula:

If [Xa<0 και Ya<H2 και Ya>H2], then the measurement points are determined in the particular arc with center coordinates XC2, YC2, and theoretical radius R2.

In order to group the measured points that belong on the arc ACL2, which is below and in continuity of the arc AC 1 L, then it will be as follows:

The elevation limit H3 which is found on the junction between the arc AC1L and the arc

AC2L, is also the lower level of arc AC1L, and the upper level of arc AC2L.

If the value of the lower level of the arc ACL2 is H4, then it will be as follows:

If [Xa<0 and Ya<=H3 and Ya>H4], then the measured points will be determined on the particular arc with center circles coordinates XC3, YC3. Therefore, it can be determined also the theoretical radius R3 of the circle.

The combinations that will result from the formulas above, will determine each measured point to the circle in which it belongs whether is on the left or right on the cross section. Therefore, it can be determined in which circle each measured point belongs, since it has been calculated the cross section coordinates Xa, Ya as it is referred on step 1 of the claim.

STEP 3. Distance from the center of the circle until the point of measurement.

Aim is to calculate the distance between the center of the circle where each point belongs according to Step 2, until the points of measurement on the theoretical cross-section of a tunnel. That is to say, it will be calculated the radius R2 of each point. The calculation that will be used is the following:

R2=SQRT [(Xa-XC) ^Λ2 + (Ya-YC) ^Λ2]. Where Xa, Ya are the cross-section coordinates for each point. Where XC, YC is the coordinates of the center of each circle where the point belongs according the points grouping that were realized on step 2.

STEP 4. Calculation of Z of each measured point.

The calculation of Z can be realized since it is given by the initial study the theoretical value of radius Rl for each arc of the circles were the measured points have been grouped. The method of grouping for the measured points is referred in STEP 2. The theoretical value of radius Rl of each arc of the circles on the theoretical section should be given by the initial study as it is referred on STEP 2 paragraph (B). It is required also the value of the radius R2 of each measured point, as it is calculated on STEP 3.

Thus Z=R2-R1. With the calculation of Z it is notified how much abstains each point of measurement from the theoretical surface of a tunnel.

STEP 5. Calculation of azimuth of cross-section.

It is calculated the azimuth from the center of the circle where each point of measurement belongs on the cross-section, until the point of measurement. In this case, it will be supposed they exist as known the coordinates XC, YC of the center of each circle, as well as the coordinates of the point of measurement on the cross-section of a tunnel Xa, Ya.

The calculation in order to find the azimuth is the following:

F= Xa - XC

Ya - YC

From this value will be calculated the arc of the tangent (AT AN), as well as the finding of the correct quadrant of the circle.

STEP 6. Development of arcs in the cross-section.

It is calculated the real length from the beginning of the lower hypsometric level of the arc that belongs to the circle where each point of measurement belongs, up to the point of measurement.

For the calculation will be needed the coordinates Xa, Ya of each point of measurement in the cross-section, as well as the azimuth F that is calculated in Step 5, as well as the R2 distance from the center of the circle up to the point of measurement that is calculated in Step 3.

The calculation of the length of the arc is the following:

L4= PI x R2 x JF.

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STEP 7. Development of points of measurement in the cross-section.

The development of points of measurement in the cross-section will be used for each point of measurement that exists in the cross-section in order to attribute the final form of the total development. It consequently, will be calculated the two-dimensional transformation in the development of the cross-section.

For the calculation will be needed the followings:

The Z for each point as it was calculated in Step 4.

The theoretical values of the radius of the arcs of the circles that precede the arc where each point of measurement belongs. That is to say, from the top circle to the next circles, are given the values Rk 1 , Rk2, Rk3....

It will be supposed also that are given as known the azimuths of the cross-section. That is to say, from the top circle will be supposed are given the azimuth values Azkl, Azk2,

Azk3....from the under hypsometric level of each arc that precedes the arc where the point of measurement belongs. This azimuth is given by the geometric characteristics of the theoretical cross-sections of each study. That is to say, beginning from the top circle, it will be in effect the followings: For the Rk] the Azkl , for the Rk2 the AzK2, for the Rk3 the Azk3....

For the arc that belongs to the top circle will result from the following calculation:

Ll= PI x Rkl x Azkl

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For the right arc of the second circle will result the following calculation:

L2= PI x Rk2 x (Azk2-Azkl)

200

For the right arc of the third circle will result the following calculation:

L3= PI x Rk3 x (Azk3-Azk2)....

200

In order to calculate the total length of the parallel arc up to the point of measurement in the cross-section it will be: Lm= L1+L2+L3+L4... The L4 is the length of arc that was calculated in Step 6.

STEP 8. Three-dimensional coordinates transformation for the attribution of mapping of cylindrical development of the existing surface of a tunnel.

It is calculated the transformation X, Y, Z for each point in the surface of a tunnel so that becomes import in the final calculating attribution of mapping.

As known will be needed the followings:

To X0, Y0 of axis. According to Step 1 it is the vertical projection of each point to the axis.

The Z of each point where it is calculated according to Step 4.

The length of development Lm of each point that is calculated according to Step 7.

The absolute XI, Yl of point as became the reception from the total station in the existing surface of a tunnel.

Considering the development concerning the azimuth, is directed rightly and absolute the mapping according to the direction of axis of a tunnel. In order to calculate the absolute azimuth AZl from each point of measurement up to the axis, it will result from the following calculation:

AZ1= X1 - X0

Yl - Y0

From this value will be calculated the arc of the tangent and the finding of correct quadrant of the circle.

The calculation that will be used in order to become the transformation for the X will be: X of (cylindrical development) =Y0+Lm (always with positive sign) x SIN (AZl)

The calculation that will be used in order to become the transformation for the Y will be: Y of (cylindrical development) =Y0+Lm (always with positive sign) x COS (AZl).

The method of mapping and control of surfaces of tunnels in projective development that can be realized with the help of a computer is characterized by calculation X of (cylindrical development), Y of (cylindrical development). The Z comes from the calculation of Step 4. The transformation of the existing cylindrical surface of a tunnel in the three-dimensional projective development, allows the reading of information of the situation of the existing surface of tunnels to all length and width, in a single drawing of projective mapping. The total comparison results concerning the existing surface are reported in each theoretical cross- section of tunnels.

2. Method of mapping and control of surfaces of tunnels in projective development, which can be realized with the help of a computer, is characterized by the steps 1-8 according to claim 1.

According to claim 1 will be supposed initially are followed the steps of algorithm 1-8, in order to be calculated the mapping of surfaces of tunnels in projective development, so as to be realized the calculation of the excavations limit. The calculation of the excavations limits are realized in order to be revealed the regions of projective mappings, which are beyond the theoretical excavation limits that defined by an initial study. By determining the excavation limits, are remodeled the Z of measured points that exceeding the excavation limits.

Namely, it will be calculated the net value of dispute arising in relation to the excavation limits, and the current situation of the surfaces of projective mapping, as calculated in accordance with claim 1. Therefore for each measured point will be calculated the net value that resulting from the subtraction between Z of points that was calculated in claim 1 step 8, and the defined (M) value of excavation limit.

It refers in cases where the initial study does not allow existing overbreaks on the tunnel's surfaces that are above a permissible limit (M) of excavations.

As a limit (M) is defined, the maximum vertical distance between the theoretical cross section and each measured point.

Therefore, where the value of Z for each point that has been calculated in accordance of claim 1 step 4, is greater than the defined value (M), then those points will be isolated.

The purpose of the isolation of points with Z>M is the subtraction of their existing Z as calculated in step 4 of claim 1 , with the excavation limit (M).

Namely, for each measured point with Z>M, then the value of Z will be remodeled to Z-M. Therefore, for each point with Z>M will result in the net difference between the current situation of surfaces, and the limit (M) of excavation's provided by the initial study.

The value (M) of the excavation limit can be defined each time by the initial study, or the contractor of the tunnel manufacture.

The results of the new Z of points arising from the difference between Z-M, are used to update the projective mapping, in order to display the areas characterized by the exceeded of excavation limit (M).

In order to place greater emphasis to the regions which characterized by the exceeding of the excavation limits, then it will be used following acceptance:

For each point with Z <M then the value of Z will be zero.

In the final analysis, it is indicated a surface of re-mapping that displays flat regions where points are within the excavation limits and abnormalities where the Z of points exceeds the excavation limit (M).

The calculation of the excavation limits is used during the rehabilitation of surfaces of tunnels before the final investment, to map the minimum values of the regions that should be overlaid with gunite, since they exceed the outbreak limit (M).

The initial study or the contractor can define the outbreak limit (M).

3. Method of mapping and control of surfaces of tunnels in projective development that can be realized with the help of computer, and is characterized by the steps 1 -8 according to claim 1.

Relation of claim 1 steps 1-8 so that is calculated the merging of cross-sections of the primary surveys with explosions that were realized in the past.

Aim is to realize calculation of forecast of future cross-sections of excavations with explosions.

The result of calculations will present the estimate about how it could be a future cross- section if are followed similar cases of explosions of the past.

The conditions in order to calculate the projective mapping of the topographic primary surveys of excavations will be according to Step 1-8 claim 1.

The way in order to achieve the forecast of excavations is the identification of previous cross- sections that has been realized by the primary surveys after each explosion.

1) The cross-sections that will be identified should have same geology.

2) It has been used for each cross-section the same quantity of explosives.

3) Was realized the same length of perforation. Provided that the responsible engineers of manufacture fix the conditions above, then will be calculated future cross-sections. That is to say, it will be calculated virtual primary surveys, and will create cross-sections in the real space of a tunnel.

Requested elements for the calculation of forecast of future excavations with explosions:

The requested elements that will need for the calculation are based on the steps 1-8 of claim 1. For the cross-sections of excavation with explosives where they have been grouped according to the criteria above, and for each point that has been surveyed in the past on the cross- sections, will be supposed that are realized the calculations according to Steps 1-5, of claim 1 : Step 1. Calculation in the two-dimensional space (X, Y of cross-section).

Step 2. Grouping of points of measurement in the cross-section.

Step 3. Distance from the center of the circle until the point of measurement.

Step 4. Calculation of Z of point of measurement.

Step 5. Calculation of azimuth of cross-section.

A) Afterwards should be calculated the mean of the radius of the forecasting points. That is to say, the mean of the distances from the centers of the circles where the points belong, up to the points of the survey. The survey points that will be added, belongs to different cross- sections. Afterwards for each radius Rl , R2, R3, R4...of the surveyed points that belong to different cross-section, it will be RA= (Rl+R2+R3+R4)/4. Consequently if the theoretical radius where the points of the different cross-sections belong, is RP, then Z of point of forecast = RA-RP.

B) Calculation of the azimuthal determination.

The azimuthal determination fixes the ranges of azimuths where the averages of distances RA of paragraph (A), will be included. The azimuthal ranges can be fixed from the engineer of manufacture. The azimuthal ranges are fixed by the centers of each circle with equally spaced azimuthal values. In consequence, the different arcs where a theoretical cross-section is constituted.

C) Development of arcs in the cross-section, according to claim 1 Step 6,

is calculated real length L4 from the beginning of the lower hypsometric level of the arc that belong to the circle where each point of forecast included, up to the point of forecast.

D) Development of points of measurement in the cross-section according to claim 1 Step 7. The development of points of measurement in the cross-section will be used for each point of forecast of excavations that exists in the cross-section in order to attribute the final form of the total development. It consequently, will be calculated the two-dimensional transformation development of cross-section. It will be realized replacement of L4 where it was calculated in the Step of 7 claim 1 with the L4 that is calculated in the paragraph (c).

E) Calculation of the forecasting coordinates of the excavations in the two-dimensional space of cross-section.

As known, it will be supposed that exists the followings:

The XC, YC of the center of the circle where each point of forecast of excavation belongs. The RA that was calculated in paragraph (A).

Azimuth Fl for each point of the different cross-sections of excavation with explosions, where has been calculated according to claim 1 Step 5.

In order to calculate the forecasting coordinates in the two-dimensional space of cross-section Xa, Ya for the point, will be according to with the following calculation:

Xa= XC+RA*SI (Fl)

Ya= YC+ RA*COS (Fl). F) Calculation of the absolute coordinates XI, Yl of the virtual points of forecast of excavations:

Are calculated the absolute coordinates in the particular region where has been selected by the responsible engineers of the project, so that the steps of the calculations of the three- dimensional development of the forecasting future explosions can be realized. In the substance, the coordinates XL Yl will be the points where according to claim 1, they are given by the topographic primary surveys in the interior of a tunnel. In the case of merging of cross-sections with explosions, those coordinates don't exist. However, they will be calculated. The forecasting coordinates belongs in a region where has not been realized yet excavation of a tunnel.

Thus as known it will be supposed to exist the following elements according to claim 1 : 1 ) Axis coordinates X0, Y0, to the position where it will be selected by the responsible engineers to calculate the forecast of excavations. The X0, Y0 will be the vertical projection of the forecasting points, up to the axis.

2) The Xa of cross-section. That is to say, the distance of the forecasting point from the axis, that was calculated in the paragraph (E).

3) The absolute azimuth AZ1 from the axis up to the future points of forecast of excavations. In order to calculate the absolute azimuth AZ1, will be realized the calculation according to claim 1 Step 8. The difference is that is not calculated the azimuth of points up to the axis, but is calculated the azimuth from the axis up to the points of forecast of excavations.

Then it will be supposed it is fixed the following rules:

For each point of forecast of excavations that has a negative sign, that is to say, is found left from the axis according to the serial flow of the tunnel alignment, the azimuth of each points up to the axis will be AZl+300grads. In the opposite case, that is to say if the point is found right from the axis then will be: AZl+lOOgrads.

The calculation that will be used for the absolute coordinates XI, Yl of the virtual points of forecast of excavations is the following:

XI (absolute) = X0+Xa*SIN (AZ1)

Yl (absolute) = Y0+Xa*COS (AZ1)

In order to calculate the absolute altitude of each point of forecast of excavations, it will need to know the value of the vertical projection of each point to the longitudinal profile of the road construction. That is to say, for the point that is requested the altitude of the road construction vertical in the axis in which it belongs, according to claim 1 Step 1. Then the sum of the Ya and the altitude of the road construction that is calculated in paragraph (E), will also be the absolute Z of each point of forecast of excavations.

G) From the moment that was realized the calculation of the absolute coordinates of the virtual points of forecast of cross-section, will be used the mathematic calculation according to claim 1 in order to calculate the transformation of the crowd of the topographic virtual points from the cylindrical absolute system, to the three-dimensional cylindrical projective development. Thus, they will be used according to claim 1 the following steps:

Step 7 the total length Lm from the beginning of axis, up to each future point in the cross- section.

Where XI, Yl, are the absolute coordinates of the virtual points. Where X0, Y0, are the coordinates of the axis.

The calculation in order to realize the transformation according to claim 1 Step 8 for the X, will be:

X of (cylindrical development) =X0+Lm (always with positive sign) x SIN (AZ1)

The calculation in order to realize the transformation according to claim 1 Step 8 for the Y, will be:

Y of (cylindrical development) =X0+Lm (always with positive sign) x COS (AZ1)

The Z of point according to paragraph (A) will be the mean of the distance of the points from the theoretical value of their radius. The way of calculation that is reported above is linear.

It is linear because the final surface of mapping that will be created by the forecasting cross- section, will run from the forehead of excavation, and will present the same Z for each forecasting point in the all length of the future perforation.

The not linear method can be achieved with the following way:

If it has been realized primary survey of explosions in the past where according to the above criteria they are grouped depending on the geology of ground and the quantity of explosives that was used, then it will be become the merging of cross-sections according to the order that has been realized each primary survey.

It consequently, will be calculated the merging of cross-sections with the following way: The first cross-section of the explosion A with the first cross-section of the explosion B. The second cross-section of the explosion A with the second cross-section of the explosion B. The third cross-section of the explosion A with the third cross-section of explosion B, etc.... It thus is created a not linear future forecast of mapping of excavation, which has important resemblances from the erratic overbreaks of explosions of the past. It thus is achieved more analytical forecasting.

The precision of forecast will be better each time when it will be realized new explosion. 4. Method of mapping and control of surfaces of tunnels in projective development that can be realized with the help of a computer, and is characterized by the steps 1-8 according to claim 1, and the calculation of merging of cross-sections of excavation that will result according to claim 3.

Relation of claim 1 steps 1-8 in order to be calculated the statistical analysis of the primary surveys with explosions that were realized in the past. It will be realized a sequence of a new phase of calculations based on the results as it is reported in claim 3.

Aim is to determine again the estimate of the future forecasts of excavations with explosions, by decreasing the average of the quantities of explosives that were used in the past.

In this way are decreased with geometric progress the cases of big overbreaks that could result in the surface of a tunnel from the future explosions.

A) The conditions in order to it realize the calculation, are the followings:

The cross-sections that will be identified have same geology.

It has been used the same type of perforation.

It has been calculated the merging of future cross-sections according to claim 3.

B) According to the calculation of forecast of excavations that was realized in claim 3 paragraph (A), was calculated the mean of the RA of the radius of the points that belong in different cross-sections according to the criteria of claim 3.

The next step is to be given the values of the quantities of explosives El, E2, E3... that were used for each cross-section.

Then as known, it will be supposed that exists followings:

The mean of the radius of the points RA that was calculated in claim 3 paragraph (A).

The mean of the quantity of explosives that came from different cross-sections. The calculation that will result will be the following:

E= E1+E2+E3...

3

C) Calculation of the new estimate of the mean of the radius RA of the points, according to claim 3 paragraphs (A).

It will be supposed it is given as known:

1) The theoretical value of radius R of the circle where the points of the different cross- sections belong.

2) The mean of Z of the points that were calculated in claim 3 paragraph (A). 3) The azimuthal ranges that will be requested by the engineer of manufacture, according to claim 3 paragraph (B).

4) The mean of the radius RA of the points that were calculated in claim 3 paragraph (A).

5) The quantity of explosives EG, that it will be defined by the engineer of manufacture, in order to realize the future statistical forecast.

It thus results in the following calculation:

R2 = EG*RA. In this way was calculated again the radius of the

E

points in the particular azimuthal range, in accordance with the defined explosive quantities EG.

Thus, the new Z of the points that will result will be the difference between the R2 and the theoretical value of the radius R of the circle where the points of the different cross-sections belong. That is to say, Z2 = R2-R.

The difference between the initial Z of the points that were calculated from the mean of the radius and the Z2 is: DZ=Z-Z2.

To the next step will be become the subtraction DZ of the radius of each point of cross- section where was realized the cross section merging.

The same steps from the until now explanation of the statistical analysis of forecast of excavations, will be followed for the points of cross-section that belong to any circle. Each azimuthal range that is referred in claim 3 paragraph B, includes autonomous behavior of calculation according to the statistical criteria above. That is to say, the DZ of points that was calculated above will be proportional to the mean of the radius of the points of forecast of excavations that belong to each azimuthal range. The engineers of manufacture will define the azimuthal ranges.

In order to realize the calculation of the new points of statistical analysis that results from this claim, will be realized the steps of calculations of the algorithm of claim 3, paragraphs (A) until (G).

The difference with the algorithm of claim 3, will be the replacement of value RA that was calculated in claim 3 paragraph (A), with the value R2 that is calculated in claim 4.

5. The method of mapping and control surfaces in projective development of tunnels that can be realized with the help of a computer, characterized by the claims 1,2,3,4.

Relation of claims 1 ,2,3,4, with claim 5, where any topographical software of a terrain model can visualize the results of their calculations.

The steps followed are:

A). According to claim 1 , is calculated the mapping of each measurement point in the three- dimensional projective development.

The coordinates from the calculated mapping can be imported into software that supports terrain model design, in order to generate three-dimensional triangles between the points X, Y, Z of the projective development, calculated in accordance with claim 1 in step 8.

Therefore, the development of the three-dimensional projection of the measured points obtained in accordance with claim 1, contributes to the smooth construction of three- dimensional triangles and contour lines.

The contour lines generated between the points, are depended from the calculation of Z of points as referred in claim 1 step 4.

The positions of the generated triangles are depended from the calculation of the points X, Y, as referred in claim 1 step 8.

Therefore, the results of the coordinates of points in the three-dimensional projective mapping as calculated in accordance with the claim 1 step 8, can be imported into topographic software, which environment supports the design of three-dimensional triangles and contour lines. B) . According to claim \ , is calculated the mapping of surfaces of tunnels in projective development, in order calculate the excavation limit as referred in claim 2.

The results of the updated mapping that reveals the excavation limits can be visualized from the three-dimensional triangles and contour lines, as referred in paragraph A.

C) . According to claim 1 , is calculated the mapping of surfaces of tunnels in projective development. Consequently in accordance to claim 3 is calculated the merging of the cross sections and the forecasting of excavations used blasting's, based on the finite topographical measurements that were realized.

The results of the updated mapping of the excavation forecasting, can be visualized in three- dimensional triangles and contour lines, as referred in paragraph A.

D) . According to claim 1 , is calculated the mapping of the surfaces of tunnels in projective development so that according to claim 3, to calculate the merging of the cross sections and the forecasting of excavations used blasting. The excavation forecasting is based on the finite topographical measurements that were realized according to claim 3.

Consequently, in accordance with claim 4 is calculated the forecasting's of the excavations used blasting, in order to estimate the reduction of quantity of explosives that were used in the past, so that to prevent future outbreaks on the surfaces of the tunnels.

The results of the updated projection mapping that reveals the excavation forecasting after the calculation of the reduction of explosives, can be visualized in three-dimensional triangles and contour lines, as referred in paragraph A.