MXPA98010460A - Sampling and reconstruction of propagating wavefields - Google Patents

Abstract

LAS SE¥ALES ADQUIRIDAS QUE REPRESENTAN MUESTRAS DE CAMPO ONDULATORIO DE PROPAGACION SON PROCESADOS USANDO METODOS CON VARIABLES ESLABONADAS EN EL ESPACIO Y EN EL TIEMPO PARA FORMAR IMAGENES TENIENDO MEJOR DEFINICION Y RESOLUCION QUE LAS QUE SE PUEDEN OBTENER POR METODOS COMUNES Y CORRIENTES LAS FRECUENCIAS A LAS QUE SE TIENE ACCESO INCLUYEN AQUELLAS PREVISTAS ASI COMO ALGUNAS APARENTEMENTE MAS ALLA DE LAS CONSIDERADAS FACTIBLES DE ACUERDO CON LOS CRITERIOS NYQUIST ASI COMO LAS PRESENTES EN LA FUENTE DEL CAMPO ONDULATORIO. LAS SE¥ALES Y CARACTERISTICAS FISICAS PUEDEN OBTENERSE, POR EJEMPLO EN TOMOGRAFIA (EXPLORACIONES CT MEDICAS) UBICACION POR ECO (RADAR O SONAR) O PRESENTACIáN DE IMAGENES SISMICAS. LAS SE¥ALES SON PROCESADAS MANTENIENDOSE INTERRELACIáN DE LAS VARIABLES ESLABONANDAS DE TIEMPO Y ESPACIO, EN VEZ DE SEGUIR LAS TECNICAS ACTUALES ACEPTADAS POR CONVENCIáN LAS CUALES TRATAN LAS VARIABLES TEMPORALES Y ESPACIALES INDEPENDIENTEMENTE. LA DEFINCION Y RESOLUCIáN DE IMAGENES SON OBTENIDAS MAS ALLA DE LAS QUE NORMALMENTE CONSIDERADAS POSIBLES CON TECNICAS ANTERIORES MEDIANTE ENTENDIMIENTO E INTERPRETACIáN CORRECTA DEL CONTENIDO DE LA INFORMACIáN DE UN CAMPO ONDULATORIO MUESTREANDO LA VELOCIDAD DE ADQUISICION DE DATOS PUEDE AUMENTARSE GRANDEMENTE AL MISMO TIEMPO QUE TAMBIEN SE REDUCEN LOS COSTOS DEV ADQUISICION PUESTO QUE LOS METODOS DESCRITOS REQUIEREN MENOS DATOS Y EMPLEAN TALES DATOS SEGUN DISPONIBLES DE MANERA MAS EFICAZ.

Description

TITLE: SAMPLING AND RECONSTRUCTION OF PROPAGATION OBSTRUCTIVE FIELDS INVENTOR: NORMAN S. NEIDELL BACKGROUND OF THE INVENTION: 1. SCOPE OF THE INVENTION: The present invention relates to the acquisition and processing of signals representing samples of undulatory propagation fields to form images that have better definition and resolution than that can be obtained by traditional methods. 2. DESCRIPTION OF THE PREVIOUS TECHNIQUE: The measurement, observation, recovery or recordings of samples of a wave propagation field to derive information covers a wide range of applications. Wave fields of propagation can arise naturally or can be initiated intentionally by various means, but in all cases they behave according to the Wave Equation well known in some way. Since the Wave Equation describes a wave field completely as a function of time and spatial variations whether this wave field is reflected, directly received or both, there is information available in some way necessary to decipher, locate by echo, image, navigate or apply any desired reconstruction operation to achieve the desired use. It may be necessary to have complementary knowledge to complete these tasks, such as codes, times or reference positions and propagation parameters, but with these at hand the calculations are also well known and widely used. The Clear Wave Equation is one of the fundamental equations of classical physics. The solutions of this equation by analytical and numerical means have been treated as a continuous activity for many years. Even today, with the availability of massive computational power, solutions can remain a difficult task due to the "linkage" or interaction between all the variables. A function or velocity field is in fact the escalating mechanism that formally joins the time dimension with the components of space. Solving the wave equation in discrete approximation directly or working from some mode of solution by numerical means has been considered especially difficult if all the variables involved are treated simultaneously. Therefore, many previous approaches to the problems involving the Wave Equation have intentionally separated the variables of time and space to achieve the computational advantage. Mechanisms such as the Fourier transformation were used to achieve variable separation and allow the treatment of the wave field itself one frequency component at a time. Other approaches treat the Wave Equation itself in a computational mesh that has an effective sampling mesh as used to sample the wave field. Therefore, according to what is known, the traditional practice as it prevails today, has been to consider the variables of time and space one by one in terms of their content of information about the wave field. Such a view apparently limits the resolution of images achievable in terms of the properties of the source illumination and the wave field sampling according to the Nyquist theory as applied to the individual variables. SUMMARY OF THE INVENTION Briefly, with the present invention it has been discovered that the coupled treatment of wave field variables can have full access to the structure of the wave field information. Therefore, the image sampling of the wave field reconstruction may be finer than the wave field sampling itself. For applications such as tomography (CAT comput- ed medical scans), echo localization, seismic imaging, to mention a few representative, linked variable solutions allow definition and resolution beyond what is normally considered as possible that arises from the unnecessarily imposing limits for the theory of single variable sampling. CorrespondinglyIn fact, a thicker sampling or acquisition of less data is needed to achieve any level of definition or resolution than normal guidelines for single variable sampling theory can teach. The practical consequences of using variable-coupled Wave Equation solutions means that the speed of data acquisition will increase and costs will decrease with better results obtained at a lower price. The implications for the elaboration of medical images, navigation images, subsurface seismic, in fact any application of wave field reconstruction are very direct and immediate. BRIEF DESCRIPTION OF THE DRAWINGS: Figure 1 is a two-dimensional schematic diagram of a wave field function and certain of its spatial and temporal characteristics. Figure 2 is a two-dimensional schematic diagram showing minimum travel time paths and straight line approximations for a real source to a particular resolution element and to a designated receiver element of the antenna function. Figure 3 is a simplified three-dimensional schematic diagram in which a real source is at the origin of a Cartesian coordinate system and a designated receiver element of the antenna function lies in the z = 0. plane. rays straight to a particular resolution element. Figure 4 is a schematic diagram showing the point of all resolution elements having a total travel time common to the actual source and the particular receiving element of the antenna function for a constant velocity propagation means.

Figures 5 A, 5B and 5C are schematic diagrams of the interrelation between the various concepts of undulatory field image processing and the sampling of the variables. Figure 6 is a schematic diagram relating to the present invention used in relation to positioning and location of objects and navigation. Figure 7 is a two-dimensional schematic diagram illustrating the present invention used in connection with a study of seismic reflection of the subsurface. Figure 8 shows a diagram indicating the computing and data storage aspects of the preferred embodiment for a three-dimensional general case. Figure 9A shows a two-dimensional terrestrial model from which the simulated data used for illustration of this invention is computed. The detail of the model is indicated in enlarged form at the bottom of the Figure. Figure 9B shows the Huygens image processing (through the modified KirchhofF migration) of the simulated seismic data from the model in Figure 8A with and without the superimposed model. Figure 9C shows the Huygens image processing (through the modified Kircnhoff migration) of the simulated seismic data decimated with and without the superimposed model. Only ten percent of the original data has been used. The decimated recordings preserved for the calculations are indicated by the inverted black triangles. Figure 10 is a two-dimensional schematic diagram showing in a layered earth model the arrival times for symmetric source-sink pairs around a common surface point can be used to estimate a velocity function using a hyperbolic approximation . Figure 11 A shows a profile of acquired and imaged seismic data with 2 millisecond time sampling using a commercially available KirchhofF migration. Figure 1 1 B shows the profile as described in Figure 1 1 A as it presents its image using a KirchhofF migration according to the present invention. Figure 11C compares the amplitude spectra of the images of Figures 1 1 A and 1 IB at the designated common location. Figure 1 ID shows an enlarged view of a part of the image of Figure 11B. Figure 11E shows an analogous part of an image as in Figure 1 ID using the same initial data, but sampled at 1 millisecond, using the modified KirchhofF migration according to the present invention.

Figure 11F contrasts the amplitude spectra for the profile of Figures 1 1 A to HE, inclusive, as their images are presented, using the modified KirchhofF migration at 2 milliseconds and 1 millisecond, respectively, for an image below the location of the CDP (Common Depth Point) 2100 from 0.4 to 0.8 seconds. Figure 11G contrasts the amplitude spectra as they are in Figure 11F for displaying their images below the location of the CDP 2150 from 0.8 to 1.2 seconds. Figure 1 1H contrasts the amplitude spectra as shown in Figure 1F for the presentation of their images below the location of the CDP 2200 from 1.1 to 1.5 seconds.

Figure 1 II contrasts the amplitude spectra as shown in Figure 1 1H for the presentation of their images below the location of the CDP 2120 from 1.1 to 1.5 seconds.

DESCRIPTION OF THE PREFERRED EMBODIMENT: 1. Glossary of Terms: For ease of reference, the terms defined for use in the description of the present invention are set forth below. As will be evident to those skilled in the art, the definitions incorporate both current standard meanings as well as meanings by extension as they become necessary. They include: Acoustic Impedance - A material characteristic, the density-velocity product that gives rise to the reflection of wave fields acoustic propagators with respect to contrasts in this feature. See also impedance. Condition "aliasing" - A term in signal theory that describes the corruption of a discretely sampled signal that is due to the presence of frequency content in that signal before discrete sampling beyond the Nyquist limit of that sampling Amplitude - The function of intensity of space and time that describes a wave fields of propagation. Antenna Function - A mathematical description of locations that are to be used to sample a wave propagation field. . . Apparent Frequency - The Fourier transform component of a wave field image variable. Since the wave field sampling variables are not linearly related to the homonymous variables in the image space, they are related by the inherent linkage in the wave propagation, frequencies can be found outside of those considered to be properly sampled or achievable from the wave field illumination properties and the sampling operation. All the frequencies that describe the image, either in space or time, should be considered apparent if they are referred to the field of undulatory field sampling and its independent variables. Apparent Font - A Huygens or virtual font that is also a grid cell and an image or resolution display element.

Apparent Wave Number - See apparent frequency. Fourier Transformation - Linear integral formulas that consist of complex exponential functions which convert the timed space functions into their frequency domain equivalents. These are sums that consist of harmonic trigonometric functions in their discrete approximation. Frequency - A component of the Fourier decomposition together with a variable often used in signal theory to assess the information content. Fresnel Zone - As this term is used in the presentation of seismic images, the first Fresnel Zone is defined in optics for the dominant frequency that corresponds to that region of illumination that has constructive interference from which reflections arise and on which averaging occurs. The Fresnel Zones will increase in size at the same time as the dominant frequency decreases, while the speed of propagation is reduced, and also when the propagation time increases. Grid Cell - A region defined in a multidimensional space of size related to the sampling of the variables that define the space itself and that is also an element of resolution or image presentation. Huygens Principle - The concept to describe a wave propagation field in which each point on a wavefront is considered as a new and independent apparent source. This principle extends in the present invention to include wave fields that propagate in a material whose image has to be presented. Presentation of Images Huygens type or Extended Presentation of Huygens Images - Wave field reconstruction methods with linked variables for the presentation of virtual images or resolution elements that represent an extension of the Huygens Principle according to this present invention. Source Huygens - A virtual or apparent source consisting of an element of image presentation or resolution with a grid cell in a propagation medium. Such use also represents an extension of the Huygens Principle in accordance with the present invention. Image - A defined location or mapping of wave field amplitude values over grid cells, pixels, resolution elements or voxels that constitute selective wave field reconstruction. Presentation of image - The activity of forming an image. Impedance - A material property of the propagation medium whose changes affect the wave propagation field in all directions. Image Display Element - A grid cell or resolution element that can also be an apparent or virtual Huygens source within a medium in propagation.

Intermediate Grid - A work grid. Interpolation - A mechanical procedure to estimate values of a function other than where it is sampled based on averages of nearby samples, weighted only according to the proximity to the sampled values. KirchhofF method - Reconstruction of wave field or image presentation methods that consist of calculations on definable surfaces, usually simple conic sections or surfaces that have an axis of revolution. Commonly used in seismic reflection processing but not related to the KirchhofF integral analytical methods as practiced in optics and other fields. The Solution Methods of Linked Variables of wavefield reconstruction or image presentation that is derived from an appropriate Wave Equation or solution in which all the variables are treated simultaneously and are not separated to simplify the computational demands. Master Grid - The final arrangement of the pixels or grid cells in which the amplitude values are added to form an image. The sizes of the elements are adequate according to the sampling theory to represent the image. This term is most commonly used when the particular image presentation method requires the use of an intermediate or work grid. Modified Presentation KirchhofF or Modified Migration Kirchhoff - The reconstruction of undulatory field or image presentation according to the present invention using the modified image presentation KirchhofF in seismic applications and other applications in which simple speed functions define calculations on ellipsoids or hyperboloids of revolution.

Limit Nyquist - A term in signal theory that describes the limiting frequency beyond which it is no longer possible to faithfully represent discrete regular samples of a function of a variable. For discrete sampling with regular intervals, the Nyquist limit is the reciprocal of twice the sampling interval. Pixel - A grid cell in two-dimensional space. Means of Propagation - The environment in which real sources, undulatory fields in propagation and antenna functions exist, which is characterized by properties that affect the wave propagation field. Wave propagation field - A disturbance that has variant amplitude with space and time within a propagating medium initiated by a real source. Real Source - The causative element of a wave propagation field. Element Receiver - An elementary component of an antenna function in which samples from a propagation wave field can be received or recorded.

Reference Location - Position in a designated means of propagation designated for a specific purpose. Normally the location of a real signal or a receiver element of an antenna function. Resolution Element - A grid cell, pixel, voxel, or component of an image. It can also be an apparent Huygens or virtual source. Sampling - The activity of collecting a series of values that represent a whole; in this case, a series of values of amplitudes over space and time that represent a wave propagation field. Sampling Interval - The spacing between discrete samples of a variable. A concept of particular importance in signal theory. Seismic Wave - A wavelet as defined in a seismic reflection application. Source - A real source unless it is noted as an apparent source of Huygens or virtual.

Time - A variable independent of all wave equations. See also travel time. Travel Time - The travel time from one designated item to another whose meaning varies with the application of the case in question. The actual wave field travel trajectories consist of all the spatial variables and corresponding velocity functions, therefore the relation between travel time with any distance or spatial variable in the original wave field sampling can be variable and quite complex. Simple rescalations of spatial image variables using velocity functions are typically only unsuitable for traveling time to distance transformations and can lead to paradoxical results in terms of frequency content if relationships are considered as discrete linear systems.

Speed - A material property of a propagation medium that describes the travel speed for a wave propagation field in that medium. Speed Function - An expression of the material property of a propagation medium as a function of spatial and / or spatial and temporal variables. Virtual Source - An Apparent Source or Huygens. Also a grid cell or resolution element. Voxel - A grid cell or resolution element with three dimensions and rectangular geometry. Wave Equation - A mathematical structure that describes a wave propagation field within a propagation medium as a function of spatial and temporal variables. The actual source parameters and material parameters of the propagation medium are necessary data for such a description.

Wave field - A general description of an energy that travels through a medium in propagation. Reconstruction of wave field - The development of a wave propagation field from a sample in terms of amplitudes as spatial or temporal variables. The images constitute a selective reconstruction of the wave field. Wavefront - The amplitude values for a wave propagation field for a constant time value. Ripple - A history of amplitude time as recorded at the level of a receiving element that describes the wave field of propagation of short duration and impulsive character that has zero average value for elastic perturbations. Work Grid - An intermediate array of pixels or grid cells in which the amplitudes are added as a step in the presentation of wave field image or wave field reconstruction. The size of the elements is suitable for the representation of the image according to the criteria of the sampling theory 2. Sampling of Propagation Wave Fields: One can consider a wave field W (x, and, t) continuous in space and time that propagates in a two-dimensional space described by variables x and y in variable time t. Now, one can consider that the propagation velocity function V is everywhere known and the undulatory field geometry will be considered to be known at all times t as well. An antenna function is now introduced together with some function A (x, y) that will be used to sample the wave field W (x, y, t). A (x, y) can be continuous or discontinuous, of a single value, of multiple values, finite or infinite and / or sampled in a regular or irregular manner, but can not equal the wave field at all time t. Consider now a wavefront of the wave field at time t, having a constant amplitude value W ,. One can assume that the antenna function A (x, y) intercepts Wj at point A (Xjyj) donates more than one pair x, and can apply - but not all x values, and since this has previously been eliminated by the condition of absence of parallel equalization. Suppose now that the antenna function A (x, y) samples the wave field W (x, y, t) discretely in time with interval? T, and it is the case that tj constitutes such a time sample. It has become normal practice to believe that the content of variable frequency in time of W (x, y, t), the undulatory field sampled discretely in time for fixed x and y is limited to the bandwidth between zero and 1/2? T , being called this high frequency frequency limit Nyquist. Similar considerations can be used for a spatial variable to define the analogous spatial frequency Nyquist (or wave number).

Here it can be shown that such frequency limits are actually perceived and not really valid. The unpredictability that dients the collective thinking of the people derives in large part from those computational methods that efficiently treat the Wave Equation by decoupling the spatial and temporal variables. Such a procedure encourages the application of signal theory concepts to the variables independently without considering the effect of their linkage on the informational content. To appreciate the way to evade the apparent limitation - here with respect to the frequency as it is related to the variable time t for illustrative purposes, we consider two successive values of wave field amplitude sampled in time taken at the points x, - e yi. In particular, these are W (xJ, yJ, ^) and WC ^ y ^ +? T) respectively. The usual assumption is then that no wave field amplitude value is sampled between t, and t, +? T and thus a frequency limit Nyquist is applicable. However, it is clear that at least some undulatory field amplitude values as occur between the times tj and tj + t will necessarily appear at other times in other parts along with the antenna function A (x, y) - which is it for x? Xj, and? and ,, but at different times due to the dissimilarity between the antenna function A (x, y) and the wave field W (x, y, t). However, since the geometries are known for both functions, the effective sampling of the wave field W (x, y, t), as observed in the discrete time variable t is at a finer rate than indicated by? T. However, to obtain such information, one should not treat spatial variables independently of their coupling to time values. That is, those operators that face the wave field sampled in time as defined require the ability to explicitly recognize, for example, that the amplitude value W (xk, yk, tj) also represents W (xJ, ypt1 + n? t) in which n is a fractional value greater than 0 but less than 1. Until now this treatise has been made in fairly general terms except for the limitation of having only two spatial variables. To illustrate the benefits and the practical nature of this invention, a preferred embodiment for the case of a reflected wave field that requires intensive computation will be described below, which further reduces the efforts required to implement the solution of linked variables and present image of the medium of propagation with high resolution A special case illustrated below in Figure 1 illustrates this principle that occupies us quite simply In Figure 1, we observe a wave field function with constant amplitude wavefronts that are linear in x and y, and at any time t that affects a linear antenna function where y = o (on the x axis). A wavefront of the wave field corresponding to t = t, is shown as creating an angle? with the antenna function. The wavefront is also shown for t, +? T, The wave field W (x, y, t), as sampled in both (x¡O) and in (x¡ -? T, O) consists of samples of recorded time acquired synchronously at intervals of? t. If? X is in fact a discrete sampling interval at x for the antenna function, then no wave field amplitude at increments other than integral multiples of? T will be sampled if and only if? X0 = V? T / sine? where we are assuming also propagation with constant velocity V. It should be clear that continuous sampling along the antenna function or sampling increments less than? x0 will detect undulatory field amplitudes other than in the multiple integrals of? t. Therefore, for such a case, the operations of spatial-temporal variables developed from the Wave Equation will effectively treat samples in time less than? T due to the coupling of coordinates. For the case shown, when? x = l / 2? x0, the effective time sampling would in fact be precisely half of that calculated using signal theory in the normal way for time sampling using? t. It should be clear that the more complex wave fields and / or antenna geometries with or without discrete spatial sampling will quickly result in irregular sampling of wave field amplitudes in the time variable. Such displays are not easily handled by signal theory, unless this is done on a case-by-case basis. Alternatively, the spatial-temporal coupled formalisms derived from the appropriate wave equation gives us the possibility to fully utilize the content of the sampled undulatory field information by reconstructing the wave field in each variable beyond the Nyquist limits as normally defined for the variables individual when they are treated separately, provided that the image sampling is adequate to represent such information. Notwithstanding the foregoing, the described procedures incorporate sensitivities to various practical considerations. Certain limitations will be introduced by the precision with which wave field geometries and antenna functions are known. Also, the specification of the speed function and any related errors involved will play an important role with respect to the intended application. In fact, the real sources that develop the wave field should also be considered. Returning to Figure 1, a further important observation can still be made. As noted above, if? X for a discrete sampling of the antenna function is greater than? X0 while time samples other than increments of? T will occur, they will be inadequate in their number or better "density" to represent frequencies of the time variable up to the Nyquist limit corresponding to? t. For applications in which the recovery of such frequencies is required, two alternative options can be taken individually or some combination. Again it will be noted that for any of the described procedures, a coordinated use of calculations involving both the space variable and the time variable is necessary to achieve the sampling objective. By keeping the antenna function fixed, you can record different wave fields with the same required information. For example, can such wave fields be generated with different values? (or geometries, etc.) if it is shown that this is practical. The effective density of time samples can be increased to an appropriate level or to an even greater number, by then amalgamating the various undulatory field displays by means of Wave-space Equation operations. As an alternative, we can use more than one antenna function for a single wave field. Changing the sampling as well as the geometry qualifies as different antenna functions in this context. Of course, both procedures can be applied in some combination as appropriate for different applications. Using both methods in combination has been a fairly normal technique for the seismic reflection method within the scope of certain restrictions. With the present invention, it has been found that undulatory field sampling in the manner described herein, can access the undulatory field information structure beyond the limits normally considered possible.

A specific example that applies the mentioned principles as well as one that is about to be presented would be educational but before being addressed, it is important to understand the nature of a wave field reconstruction procedure that is derived from the Wave Equation as distinguished from other methods which are frequently applied to wave fields and which also use multiple variables. There are also other important dimensions of undulatory field information content that are not recognized, and these should be introduced to complete an appropriate background treatise. 3. Reconstruction of the Ondulatory Field of Linked Variables: The Wave Equation in its various incorporations: acoustic, elastic, electromagnetic, etc. It describes undulatory fields in terms of their space and time coordinates and includes those conditions that govern the particular circumstance. The most effective wavefield reconstructions are developed from the Wave Equation itself or from one of its more general solutions. More typical formulations based on the Wave Equation consist of discrete approximations that are then convenient to use using digital computers. Those procedures that start with general solutions to the Wave Equation are usually discrete, and again try to take advantage of the resources of modern computing. To complete and in order to provide additional knowledge, it is useful to introduce the Wave Equation itself at this time. Since the case studies to be presented have to do with the seismic reflection method, a simple form of the Wave Equation will be presented using Cartesian coordinates x, y, z: d2 d2f ^ f 1 32f 2 2 dxd dy dz The quantities named in the above equation are easily identified with x, y, z and t, with the usual spatial and temporal coordinates. Speed V functions may vary with all previous independent coordinates. In this case, f describes the propagating disturbance and represents the wave displacement (dilation, pressure, rotation, etc.). Since both space and time variables are involved, calculations even using modern computer systems can be particularly burdensome. Therefore, it has become quite common practice to decouple the variables of space and time in some way as we noted earlier. This is usually done by applying some point of a Fourier transformation in which the solution to the Wave Equation is applied by treating the frequency components one by one. However, it has been found that this widely accepted approach has resulted in a significant loss of information. On the contrary, with the present invention, one can sample and reconstruct wave propagation fields using methods based on the Wave Equation involving e | simultaneous treatment of spatial and temporal variables. However, the additional arithmetic load of using it is one aspect of achieving the additional information that is sought. Another aspect is the applicant's discovery that the resolution of the image is limited only by the sampling of the wave fields and not by the sampling of individual variables or parameters that describe the real source.

All the sampling of the wave fields is done in the variable time t as well as the spatial variables. However, most image presentation applications are desired as functions of only the spatial variable. It is clear then that sampling over time should represent, ultimately, distance sampling for the satisfactory presentation of images. While speeds seem to be a simple way to rescue time as distance, the relationship is not simple. In fact, a wave field value or amplitude at any time is quite ambiguous in its physical significance. It can be derived from any number of different locations, totally or partially from several of these.Many of the numerical solutions of discrete approximations of the Wave Equation are in fact linked variable calculations. Such approaches fail to achieve the full information potential of the undulatory field sampling because the preassigned variable increments are too thick to represent full resolution. Such selection of the computational grid is, of course, made on the basis of essentially conventional sampling theory criteria which, according to the present invention, have been recognized as being inappropriate. Then the grid of the wavefield sampled and the grid of the image, although they do use variables that have the same names, are not, in fact, related in a simple way. In this case, the physics incorporated by the Wave Equation together with the velocity function - which, for some applications, can vary in time - constitutes a type of transformation that may be non-analytical. For this reason, comparisons of undulatory field sampling and image sampling do not have the same meaning as one would presume for a simple linear system. From this it follows that the sampling of a wave field in an antenna function may appear to have specified increases in spatial and temporal variables, but the actual sampling of the propagation medium may be quite different. That is, when the travel trajectories corresponding to such sampling are accommodated, also recognizing the origins of the wave field, the density of the spatial sampling can be significantly greater than that indicated by parameters of the antenna function and the increase of time sampling. This notion will be further expanded when considering sampling in the reconstruction of the wave field. However, there are also other calculations frequently applied to problems related to wave field that involve the joint use of independent variables. One of these is, for example, interpolation. It is important to appreciate that the operations based on the Wave Equation as required, are easily differentiated from interpolation and other operations of that nature. Taking a concrete case, a velocity function must be provided and the travel trajectories considered as related to Wave Equation variables in an analytical or non-analytical manner, while no physical factors comparable to some are required for interpolation or any other these other calculations. Typical scaling operations for interpolation procedures only involve weights based on proximity considerations with respect to the samples. 4. Relation of the Real and Virtual Sources: The wave fields emerge from real sources and propagate in appropriate media. We continue to adapt a point of view regarding the reconstruction of undulatory fields that are different from the more usual approaches, but that unifies most of the relatives. In fact, philosophy, as it will be applied, owes much to Huygens' philosophy that every element of a wave field of propagation can be considered as a new source in itself. The reconstructions of wave fields can be used to present the image of the wave field itself, including sources as well as the propagation medium. Most applications form images of the source or medium, and show that they can be related in a very simple way. For the treatise on our points, it will be useful to call the sources by the term "real" sources. Also, the categories in which wavefield reconstruction applications are normally placed will be reformulated as a consequence of this reformed perspective. At this point, one must change its consideration to the means of propagation. If the medium is uniform and homogeneous as well as non-dispersive and non-attenuative, then its only property of consequence can be the velocity of propagation of the wave field. In case any property of the medium shows variation, then the description or presentation of the image of such property within the medium at some designated resolution and accuracy can be recognized as a task of undulatory field reconstruction. These are used ondulatory field reconstruction classes that seek to present images of real sources whose wave fields are considered to propagate more or less undisturbed in a particular medium. Such approaches include, to name a few, Astronomy, Navigation and Passive Sonar Detection. The detail applied in the reconstruction can vary from simple measurement of a distance or travel time, to a refined image of the source that identifies specific properties corresponding to it. Huygens is credited with the development of the notion by which a wave propagation field can be synthesized by treating each of its elements as an independent radiating source and noting the effective propagation envelope for such sources. Then one can extend the significance of the Huygens principle by considering the wave field as propagating within some material. If now one considers an element of a single resolution within the medium that one seeks to present in image, then for any real source wherever it is placed in position, this element can be considered in the spirit of Huygens as a new source, but now of virtual or apparent nature. You could even call it a Huygens source with respect to its enlarged view. All these names can be considered interchangeably. One could appreciate right away that the virtual source differs significantly from a real source in the sense that it does not radiate a wave field unless a real source is also present.In fact, its effective properties are directionally dependent and are derived from properties of the real source and its proximity to that real source. A wave field from an apparent source can be sampled by an antenna function as if it were a real source. In any wave field reconstruction procedure we must also accommodate the existence of the real font as well, which is usually quite frank. It is interesting to note that if the real source and the antenna function tend to be on one side of the resolution element, we could name such an application by the term "reflection" technique. In the same way, we could refer to the approach as the "transmission" method. or maybe "tomographic" application. All these applications that we consider different, are a continuity of wave field reconstruction implementations to present the image of an apparent source. Only the presentation of the image of a real font is clearly different. When we present an image of a real source, what you see is what you get - so to speak. That is, the size and shape, the signal character in time and space are all that any or, even the best presentation of images, can recover. Of course, the previously indicated considerations concerning wave field sampling continue to apply to whatever application. All limits in time or spatial frequency inherent to the real source will completely govern what is achievable as an image. This fact has become a guiding paradigm for essentially all the methods of presentation of images and reconstruction of undulatory fields. On the other hand, it is interesting to see that the characteristics inherent to the real source do not correspondingly limit how we can present the image of the real source. Although this statement may seem wonderful at first glance, its degree of precision is easily demonstrated with an intuitive argument. Let us suppose that there is only a single different resolution element in a propagation medium that is at the same time homogeneous or ideal. It should always have some effect on a wave propagation field, whatever its properties. Detecting such an effect can be difficult, but it is always only a question of "numbers". One can overcome noise and small size by using pure statistics if one wishes so. In fact, there are interesting options available to achieve such a task. As discussed above, one can employ more and better antenna functions as well as many sources. If one is determined, the available technology should allow one to achieve any goal of presenting images in terms of accuracy and resolution for the majority of cases. Hence, the reconstructions of undulatory fields that present images of real sources offer perhaps the most powerful technology of presentation of images developed so far thanks to the flexibility in implementation, as well as the nature of the results that can be achieved. It is important to note that in the detection of a small effect or resolution element, the sampling of the variables, if any, should be capable of such representation according to the normal criteria of the sampling theory. From another point of view, what we have described is also not wonderful in the sense that the presentation of holographic images using monochromatic radiation of the electromagnetic spectrum produces images that have different spatial frequency and time content than the source illumination . Of course, this technology completely exploits again the properties of the wave field to achieve such objectives. 5. Virtual Sources of Presentation of Images Presentation of Images Huygens One must recognize that by adopting the point of view of presentation of virtual sources, some basis for directional dependence of the undulatory effective field becomes quite apparent. For example, in the case where the virtual source is between the real source and the elements of an antenna function, the recorded wave field will, of course, represent an interference between the wave field from the real source and the Huygens contribution. from the apparent source. It will also be apparent from this that all methods of reconstructing the wave field for the presentation of the image of a virtual source must accommodate two reference locations. The most direct of such approaches would, of course, be to consider reference locations to the real source and to a particular resolution element that acts as a virtual source. The requirement for two reference locations leads to undulatory field reconstruction approaches as applied variously to define surfaces that are conical cuts, typically ellipsoid and hyperboloid figures of revolution. However, before illustrating such cases, one should note that undulatory field reconstruction methods involving the use of surfaces for computational reference are typically called KirchhofF methods in many applications (optics, exploratory seismology, etc.). Initial work by KirchhofF showed that an integral on a closed surface that encompasses real sources could completely represent the physics of propagating wave fields. Therefore, the methods of presenting images using surfaces derive their terminology in the same way that the KirchhofF methods do, by analogy. A more appropriate nomenclature in the same tradition would be to call them Huygens methods, although this approach extends Huygens' ideas as indicated above. Returning now to formulate the representative reconstruction of the wave field in a more elementary case, one can consider only a real source and a single resolution element that acts as a virtual source. The antenna function should now be included and in particular should consider the detection of wave fields in a location of such function. Figure 2 shows a general case where the key elements are viewed in a propagation medium having varying properties so that the minimum time ray paths shown are not represented as straight lines. Straight lines are also represented in order to recognize them as approximations.

Note from Figure 2 that the total travel time consists of a sum of two contributions - from the actual source to the apparent source and from the apparent source to a particular detection point of the antenna function. This becomes an important consideration in the matter of establishing a preferred calculation for the reconstruction of wave fields.

A Cartesian coordinate system on the right is illustrated in Figure 2 as well. The coordinates for key elements are presented with the subscripts s and r denoting the real source and the determined receptor element, respectively. The total travel time, as approximated by the straight-line ray trajectories, is simply a sum that includes two terms in the following way: (y, - y) - - z Y v? _ '(*. { yc - y) ¿* (• z) where Vs and Vr describe a certain "effective speed" for each of the two parts of the total travel trajectory. It is not necessary to place specific restrictions on the antenna function either in terms of continuity or geometry. However, one can easily conceive a simple statistical-based approach to present the image of a virtual source that one can also consider a Huygens source. , it is required that one have a reasonable estimate of travel times for each element of the antenna function for each real source used to illuminate any virtual source of interest. There may or may not be many real sources in the same way that we can employ a certain multiplicity of antenna functions. Then, one can identify all the amplitude values at least partially corresponding to that specific virtual source by means of the arrival or travel times and add them all. This sum in the zero-order direction then represents the image we are looking for by selective reinforcement of the desired response and cancellation of other contributions. Of course, as a practical matter, there must be some "normalization" that takes into account the number or scope of the addition operation.

The procedure just pointed out does not necessarily imply surfaces in any general implementation. It is also evident that the final achievable resolution - that is, the quantity and size of virtual sources whose images will be presented, will depend on the number of independent samples of the acquired wave field that can act to describe the propagation medium. Reliability of virtual source images will increase as the relationship between independent wavefield samples and the number of virtual sources increases. This relationship is a measure of data redundancy. Having large volumes of data is the procedure that is most frequently applied to the recovery of small amplitudes or to detect undulatory fields in environments with a high level of noise. The simple surfaces for the addition operation can be defined for special cases in which the locations of real sources and receiving elements within the antenna function are restricted in some way. For example, if we place all the real sources on a surface in a plane and we take the antenna functions as some sampling of this same plane, then the sum we have sought to form will be governed by travel times (straight line ray trajectories ) according to the expression: - Jx 2 * and 2. z2 * - < ] (xt - x) 2 Xyr - y) 2 - z2 3 r where the actual source is now the origin of coordinates (see Figure 3).

If now Vs is considered equal to Vr and the resolution element or virtual source is also found vertically below the origin of coordinates, then the expression of the travel time becomes:? 77 and The presentation of the image by the sum operation constrained in this way using an antenna function more or less regularly sampled on the designated plane, defines a sampled spacetime surface that is a hyperboloid of revolution. It is also a method used for the presentation of seismic images known as the KirchhofF Diffraction Stacking.

It is interesting that the undulatory field reconstruction approach just described is essentially unlimited in the sense that x, y and z can arbitrarily increase to any value. This "open" nature is a strong reason that leads one to look for some alternative method that does not share this particularity and that has some additional advantages. Again in seismic applications, and also with appropriate restrictions, this would be considered another KirchhofF method but in fact we note that this terminology is quite imprecise. The methods that are being considered should be more appropriately called Huygens-type approaches for reasons explained above and still remain with the traditions of the industry. Let us suppose once again that the travel times from any real source to a designated element of resolution are known, in the same way as are the travel times of such an element to any element of the antenna function. A different approximate procedure of wave field reconstruction for the presentation of the image of the apparent source representing the desired resolution element can be designed starting, as before, with a given virtual source and the actual source. Now one can take the actual source together with a certain receiver element of the antenna function. Continuing with this treatise, one should now consult Figure 4. For Figure 4, in order to simplify the drawing, the propagation medium surrounding the real source and receiving element is taken as a constant as well as all other relevant material properties. For any fixed total trip time value, any resolution element or virtual source that could contribute to the detected wavefield amplitude will have a location that is along an ellipsoid of revolution as shown. This follows from the geometric definition of such a surface, if now the real source and a given receptor element are taken as being in the two focal points. This surface is closed and thus it is clearly limited in the range of variables that must be considered Obviously once the substantial changes of parameters are included, the counterpart surface would have another form of greater complexity. It also becomes evident that complex antenna functions can even suppress any description of the collective positions of virtual sources that can contribute to the amplitude of the undulatory field detected that constitutes a surface of any kind. However, the image presentation approach as applied, distributes the amplitude detected in each defined resolution element that has come. If a sufficient number of real sources and / or receiving elements are used in such a manner and distributed in a similar manner, then the image of the propagation material can be presented at any desired resolution by the reinforcement of the valid information.

One should note that computational normalizations can be easily designed for this approach and these could accommodate scattering of wave fields since the total travel time is common for each detected amplitude that is subjected to the distribution process. The speed function is also known. In this way, the image formed can have even greater fidelity.

Note that the number of resolution elements over which the detected amplitudes are to be distributed depends only on the number of independent samples that are available. The factors that limit resolution are, of course, the accuracy of the speed function and travel times, signal / noise levels, characteristics of real sources, as well as similar practical issues. In the preferred computation, the method that has just been considered will be applied, but also allowing a different speed be applied for the travel of the wave field from the real source to the Huygens source or resolution element and from the resolution element to the receiving element. The actual computation adds the travel times in one direction with respect to the travel time components of the real source and the receiving element in order to develop travel times in two directions for the resolution elements as functions of couples consisting of real source and receiving element. It is then a simple matter of substituting amplitudes as detected and recorded for the given pair that corresponds to such travel times in both directions. A detailed expression of such calculation will follow in the context of the illustration using real data. It is important to recognize that the described methods, outside of the preferred calculation, they are commonly applied in many applications involving wavefield reconstruction and image presentation. The essence of this invention is the appreciation that both the undulatory field sampling and the achievable resolution of image presentation for virtual sources are not limited or related as is implied in traditional paradigms. All the considerations arising from this invention make it evident that the acquired data for traditional methods of presentation of images have significant information potential beyond what can be achieved by means of usual procedures. 6. The Sampling Functions in the Reconstruction of the Wave Field: These methods offered by the presentation of Huygens type images of virtual sources are in fact reconstructions of the wave field with linked variables since the travel times are related to the spatial variables They also represent duly the function that the sampling performs. Since several important concepts are interacting for all wavefield reconstructions, an intuitive guide of the differentiations made can be offered here, which will be followed later by the practical calculation illustrating advantages offered by the present invention. Figure 5 represents in diagrammatic form three alternate philosophies of the presentation of the wave field image that clarify the differences in which emphasis will be placed. In Figure 5A, the sampling principles are applied independently to real sources, the antenna function, the decoupled reconstruction of the wave field as well as the image of the wave field. The resultant undulatory field images even for virtual sources are then estimated to be subject to effects such as "aliasing" condition, limitations based on frequencies of real sources and other properties, as well as Nyquist limits to resolution. This is perhaps the most usual way for the application of presenting the current wave image.

In Figure 5B, the reconstruction of the wave field uses a linked-variable approach, but now the sampling considerations are again applied inadequately and unnecessarily as in Figure 5A. It is believed that the limitations, as perceived for Figure 5A, are applicable - they are not. This becomes a more common circumstance in various wave field image display approaches. Only Figure 5C indicates a correct function for sampling consideration. The resolution limits of the wave field image for the virtual sources depend only on the number of independent samples that are available, as well as the uncertainty in the system (positions, speed, etc.) and, of course, the signal levels. noise. The designated sampling that defines the resolution of the undulatory field image should be adequate to support the objective resolution and at the same time should have the adequate support of wave field sampling information. Since recorded samples of the wave field are taken through the travel time variable, it is also common practice to image using this same variable. Having a speed function makes it possible to achieve the presentation of images using only spatial variables. It is almost always considered that there is some error when it is a field of speeds that is estimated or even when it is measured, that is why many people consider that it is preferable to present the image using the travel time variable. It is especially important to Note that an undulatory field image that uses the time variable is actually properly described by distance variables, the travel times in this case represent some effective and continuously varying distance rescaling only with respect to travel trajectories. The importance of the opinion just expressed is highlighted if one seeks to use the Fourier domain to assess the content of information as is often done in signal theory. If, for example, a real source has a temporal frequency content in a reliable band, which represents the low and high limits, then the sampling of the wave field in the corresponding time variable is typically spaced using such knowledge to frame the necessary frequencies according to the Nyquist theory. In fact, the reachable resolution of the sampling of the undulatory field taken in its entirety can allow a much finer sampling. As indicated, the image is formed in consideration of the spatial variables noting the actual trip of the energy in some approximation during the reconstruction with linked variables of the wave field. A calculation of the travel time of a spatial variable may then be finer than the sampling of the travel time of the original undulatory field sampling. Therefore, this becomes possible in the evaluation of frequency content of the time variable using a time variable to detect frequencies that appear to be outside the band surrounded by the real source. Such frequency is an apparent frequency, however, it has significance in physical terms.

An apparent temporal frequency outside the width band of the real source becomes, to some extent, an enlarged resolution of the undulatory field image on which it is considered achievable using Nyquist cptepos and the original undulatory field sampling Since the relationship between the travel time and the spatial variable can be quite complex, you can only establish specific meanings treating each case one by one Note once again that in the various treaties here the term frequency is often used interchangeably with the term wave number . The context in each circumstance must make the meaning clear In this respect and for reasons that have analogous terms already mentioned, we will also find the apparent wave numbers again outside those considered reasonable using Nyquist principles applied individually to the independent variables of the undulatory field sampled 7. An Image Presentation Application in Seismic Reflection: To demonstrate what has been described, one can perform the wave field reconstruction using linked variables to present the image of the propagation medium - the earth, in terms of acoustic impedance below a profile of seismic soundings Although this is a two-dimensional application involving a three-dimensional subsurface, the generality will not be lost. One can present images with resolution beyond the Nyquist temporal frequency limit for the acquired data and reach the apparent temporal frequencies, one should not reasonably expect to be present with significance in the actual source according to the current prevailing paradigms. At the same time, we will show a preferred incorporation to perform such a calculation that easily extends to three dimensions. The second Huygens type image presentation method will be implemented as described earlier (and illustrated for simplified conditions in Figure 4 as ellipsoid of revolution) This preferred calculation also consists of an algorithm or computational "trick" that can be explained quite easily in terms of perhaps another more familiar application of image presentation of the wave field - Navigation using two base stations Figure 6 shows a vehicle that is sought to be placed in a plane with two base stations, A and B, transmitting discernible electromagnetic signals at known speed in the air for the vehicle to receive them. reference times are available to start signals at stations A and B, will the vehicle measure the travel times of T? and You from the two base stations Using the signal speed and a good map, the circles of appropriate radii can be drawn with respect to each base station Its intersection on the water in Figure 6 locates the vehicle in relation to the base stations All this , as shown, is, of course, simple triangulation Suppose that now the base station A transmits a signal, but the base station B receives an echo or reflection from our vehicle to place it in position The reflection has a signal in the course of time which represents amplitudes but there is only one measurement - what is insufficient to locate the vehicle It can only be said that the vehicle is somewhere along an ellipse over water that has base stations A and B as focal points . In case of using many analogous pairs of base stations to A and B, one would place this vehicle in position through intersections of ellipses similar to triangulation But now using ellipses instead of circles * 1 m To develop an image of the vehicle instead of just its Location for the navigation procedures that are considered, it is only necessary to distribute amplitude recordings as done on the circles or ellipses (See Figure 6). One could superimpose a grid over the surface as also shown in Figure 6 and this act of amplitude distribution could simply consist of writing a story as a function of the amplitude time of each "grid cell" or "pixel" closest to each other. the determined curve that is being used. The grid is also indicated in Figure 6. If the amplitude measurements are added arithmetically in the grid locations, one would position and at the same time present the vehicle image by noting the definitive amplitudes that appear in the various grid cells. The size of the grid cells that are to be chosen are now related to the positioning of the vehicle and also to the presentation of its image. There must be sufficiency of measurement trajectories across the grid so that the true image (correctly placed in position) is distinguished from the amplitudes as they are distributed elsewhere and this is a fundamental requirement of the invention. As well, the resolution of the image will depend on the size of the selected grid cell and all the factors identified above that play a role in such a process. Before examining the application of seismic reflection, one must again consider the case of navigation involving the ellipse. Calculations using ellipses usually require significantly more effort than those using circuits. Also, one can add a substantial level of complexity to the situation if the travel times of base stations A and B respectively to the vehicle require different speeds. For such cases, what had been an ellipse for equal speed functions would happen to be some other curve that depends on the different speeds. The necessary calculations now adopt extraordinary requirements which imposes a greater burden even on the largest and fastest computers currently available. Even when there are large and fast computers available, one is always interested in computational efficiencies. The situation of seismic reflection that is indicated by the simplistic scheme of Figure 7, has many things in common with Figure 6. The plane of Figure 7, unlike the one in Figure 6, is in a vertical position in instead of horizontal. A real source that acts as a counterpart of the base station A transmits a wave field. One seeks to present the counterpart image of the particular resolution elements corresponding to the vehicle of Figure 6. One knows that these resolution elements are also virtual sources? apparent or Huygens sources, and these have been predetermined in size and number to establish the image. In making such a selection, one must be sure that the amplitude distribution operation to be performed provides adequate inputs in each grid cell. This is done so that the assumed statistical benefits are applied and form resolved images in the correct locations. Now in this seismic application it is possible and even probable that the travel times from the resolution element shown to the real source and a specific receiving element of an antenna function make the journey at different speeds. One would wish to have some appropriate means that has computational efficiency to achieve the Huygens distribution operation while at the same time recognizing in some way the different speed functions that must be applied. a * One can take the position of the real source and use its velocity function to calculate circles that represent only the travel time in one direction. Correspondingly, one can also take the position of the receiving element and use its different speed function to again calculate different circles of travel time in one direction. Referring one to Figure 6, which was first discussed for the navigation application, note that the illustrated circles actually indicate travel times in one direction in the same way that is being suggested at this time for the seismic case. Therefore, as previously proposed, one can consider this same figure but representing a vertical plane as it is. representing again the case of the seismic reflection images as well as the navigation application. As indicated, one can propose a deviation from the direct methods of presentation of images described above. All these had in common the operation of adding amplitudes to the grid cells. Instead, one can add travel times in one direction to the grid cells shown, For any source-receiver pair (A and B in this case), aggregation in any grid cell that also constitutes an element of resolution in this extended scheme of presentation of images Huygens would evidently be of travel time in two directions since this incorporates both travel time paths between A and B. However, the operations of presentation of images require that the grid cells contain amplitude values instead of travel times, but getting them is now a fairly simple matter. Considering a receiver element at point B and the actual source at A, the recording is in fact a story as a function of the time of amplitudes at B in travel times in two directions. For pair A and B that we are dealing with, one can go to the grid cells that have travel times in two directions of the previous operation and substitute the counterpart of recorded amplitudes corresponding to the travel times in two directions. If the velocities at points A and B as applied to the determined element of resolution (vehicle) are equal, it is evident that the distribution of amplitudes, as achieved, according to such a method, would be along an ellipse through the grid cells as seen in Figure 6. Since the calculation described only consists of a given pair of real source and receiving element, it becomes evident that one will need a "master grid" in which the final image would be developed as well as an "execution grid" in which the results would be calculated for each given pair of real source and receiving element. In turn, each of the results of the execution grid would be transferred to the master grid by addition. Each grid, whatever its use, must have the capacity according to the principles of sampling theory to represent the image as specified. The additional efficiency in this preferred computation is derived from the fact that it is necessary to calculate only one time the circles of a travel time in one direction centered on each real source or receiver element, and only when the time velocity function associated with the location of particular terrain is different For the operation of summation of travel time in one direction, one simply superimposes by adding two grids of travel time in a direction of values appropriately spaced for the given real source and receiving element and, makes the substitutions to amplitudes using the recording for that particular couple of the receiving entity. The amplitude values thus derived are then transferred from the execution grid to the master grid again by addition operation It would be particularly instructive to explain the previous computation by means of a diagram since other figures thus used to help the treatise including Figure 6 emphasize the physical situation. Figure 8 seeks to clarify the calculation of the preferred embodiment for a complete three-dimensional circumstance including storage requirements. Nine numbered elements are noted in the schematic flow diagram of Figure 8. The propagation medium or rather a part thereof on which to present image is illustrated as Element 81. Element 82, by means of its components (82A, 82B and 82C), represents the real sources, the receiving elements and their physical locations as stored for computational purposes. Note that it is important to recognize which receiving elements are active for any particular real source. Element 83 is not part of computing but expresses the generation of wave fields propagating in the medium that will form the basis of the presentation of images. The master grid of Element 89 is the storage area of the computer on which the final image will be formed. Its resolution in spatial variables and time sampling is determined according to the principles of this invention, both being independent of considerations of real source frequency content and sampling theory as applied individually to the independent variables. Note that a normalization operation is part of the final calculation and Element 89 will be discussed again, but later in the order of the role it performs in the computation. However, it became necessary to introduce Element 89 from the beginning to establish the intervals of sampling that will be used later while the calculations proceed. The recorded wave field values for each receiving element for each real source are accessible to the computation through Element 84. In Element 85, the velocity field of the propagation medium is approximated by average speeds taken to apply as travel time functions only on centralized spherical wave fronts in each real source and in each receiving element. These average speeds may initially be available only at certain locations of real source or receiving element, but using interpolation as necessary, such information may be made available for each such location. The average speeds, as required, can be calculated from the measured or calculated speeds of data available as appropriate to the particular application. For the study of seismic reflection data, which will be noted as a practical demonstration, such a procedure is described in greater detail. The grid of travel times in a direction corresponding to Element 86 uses the same discrete sampling as that of the master grid of Element 89. These grid values each centralized around a real source location or receiver element where the velocity average as a function of time is different i b- Each pair of real source-receiver element corresponding to a recording of a story as a function of time of wave field values is stored as a function of time by means of Element 84 At the same time, two corresponding squares of times in one direction displaced spatially representing the two locations, they are added to the working grid of Element 87 to develop a time grid in two directions of the same real-receiver source pair. In this way, the lateral change approach is made velocity in the propagation matepal by using different average velocity functions in the real source and receiver element locations In the nucleus of such an approach one can also find the use of spherical wave fronts in one direction time.

Since the Element 84 catalog relates the total travel times or in two directions with the amplitudes of undulatoped fields, these can be substituted in the execution grid of Element 87 to develop Element 88, since it is well known that the amplitudes of Wave field decreases with time due to travel time, a normalization factor must be applied to adjust the amplitude effect of travel time. Average speeds can be incorporated in such correction as are available in Element 85, therefore the stripes connection between Elements 85 and 86 The final image must be contained in Element 89 where the contents of the execution grid 88 must be, in turn, added once it is placed in correct position with respect to the locations of real source and receiving element Normalization in this step is based on the amount of contributions added in each cell of quadri or pixel so that the image represents a matepal property and not the sampling density. A prior requirement for the effective presentation of the image of a propagation medium, as indicated, is some reasonable knowledge of the velocity function. normally represents impedance variations which, by definition, include the propagation velocity. For example, the acoustic impedance for the case of acoustic waves is only a product of density and velocity. If velocity is an impedance component, it seems that knowing a function of speed in advance of the presentation of image almost implies knowing the answer before facing the problem Of course, this is not the case So that from there it is clear why the application of seismic reflection, one can firstly tie the model data where speeds are known, and then proceed to situations in which this consideration should be applied that implied to speed determination 8. Model Data Set Seismic data were simulated for a model shown as a depth function in Figure 9A A numerical approximation to the elastic wave equation was used with the source and receiver element positions taken On the surface of this two-dimensional configuration The positions along the surface have been noted at 100-foot intervals. This model is characterized by two boundary boundaries that have 45-degree dips and vain abrupt "passages" of the receiver. For this study, focus on two irm of the horizontal reflectors separated in depth by 500 feet, each having a series of five indentations or "wells" that are also shown in enlarged format at the foot of the figure. Each such well in The reflectors are basically square having a depth that matches their width. A constant speed of 5 000 feet / second was supplied for the calculation and the relatively wide band seismic wave applied to mark reflections had a dominant and central frequency of 80 Hz. A complete dataset, as generated, consisted of a row of 401 receiving elements recorded in time for each of the 40 shots (real sources) Shots were spaced in increments of 100 feet along the surface starting at the marked location The receiving elements were spaced at distances of 10 feet starting at the same distance from the actual source Time sampling was done in increments of 2 milliseconds For such parameters, the indentation or the smallest well (10 feet x 10 feet) has two time samples that describe its depth and two spatial samples if one presents images using the CDP methodology. The CDP image presentation method also discovers Yilmaz, 1987 (Geophysics Research, Data Processing No 2, Tulsa, published by the Society of Exploration Geophysicists) pages 45-47 and develops the spatial sampling of the image at half the intervals of the receiving elements. Therefore, also the smallest well would be seen with two samples in their spatial dimensions as indicated. According to the Nyquist criterion, the smallest well is sampled at approximately, at least, at its dominant spatial and temporal frequencies Figure 9B shows the presentation of seismic image of a part of the deep reflector with the indentations characterized in the lower part of Figure 9A The Huygens image presentation, as described, has been applied Seismic specialists would say that an operation of "migration before stacking type" has been applied Kirchhof? " The superimposed model in the migrated image is illustrated in the lower part of Figure 9B. Up to the 10-foot well itself is easily identified. The display format seen for both Figures 9B and 9C is a standard used for known seismic image presentation. as "variable ripple trace area (wiggle trace)" A time-dependent story of amplitude values in each spatial sample is drawn in graph vertically below its reference location as a curve around a zero level through the own position of the spatial sample The positive amplitude values oscillate to the right and are filled to the zero level with black coloration This display is therefore "unbalanced" since the amplitude oscillations receive more visual emphasis than the black coloration Of course, in the development of terrestrial images, one expects a greater quantity of positive amplitudes that represent post reflections. as the acoustic impedances (both speeds and densities as well as the components that make them up) increase systematically with depth • c The set of basic recorded data was then decimated literally in terms of the number of recorded receiver elements. Nine out of ten original records were deleted replacing the amplitudes recorded by zeros. In total, only ten percent of the data that was originally recorded has been preserved. with effective spacing of sources or shots as well as 100-foot spacing of the receiving elements. A 50-foot CDP spacing follows and again following the guidance of the Nyquist sampling theory, it is possible to reason that in general, the sampling was too thick for present image of any of the wells except the largest of these that is 50 feet in size. At the top of Figure 9C, black inverted triangular locations mark the locations where CDP image traces are available - in 50-foot increments. spaced locations taken by themselves would stop deciphering the po Those representing the objective of image presentation Interpolation can not recover the missing information adequately and processing methods such as the variable decoupled display of migration images using the Fouper Transformations again could not produce satisfactory resolution of such input data set However, the reconstruction of the wave field with linkable vapables as applied with the spatial sampling of 5 feet and the temporal sampling of 2 milliseconds has developed an image quite comparable with the original computation shown in Fig. 9B. In this case, the appearance of the smaller indentation or well for decimated data clearly shows access to apparent spatial frequencies beyond those normally considered recoverable from effective spatial sampling Such frequencies (or numbers of waves) are real with respect to deciphering the subsurface image space, but apparent in terms of its relation to the spginai sampling of the undulatory field, especially seen from a conventional point of view with respect to the individual vapables Also, the drastic reductions of data used with relatively small detectors of the final image quality have clearly important implications for practical applications in terms of data acquisition cost 9. Real Seismic Data Set A demonstration of the wave field reconstruction (or "pre-stacking migration as the seismic specialists call it) in accordance with this invention requires that now The issue of obtaining appropriate velocity functions is therefore considered. It is therefore worthwhile to consider this matter with some generality before examining the results of the presentation of seismographic images of field data. One can measure the velocity in seawater as a Depth function (relative to salt quality and temperature) before subsequently presenting images of submarines in this same medium as in the case of sonar For the application of reflection seismic, the CDP method developed by Mayne (1962, horizontal data acquisition techniques with common reflection points, Geophygicg (Geophysics), 27, 927-938) easily allows a "velocity" calculation based on a synthetic aperture method and using a simplified model of the variant velocity field in the subsurface Figure 10 that emerges from the work by Mayne shows for a two-dimensional case, as shown, that the last of a series of reflectors, if recorded as indicated, has a seismic expression whose description incorporates an "averaged" speed that is easily estimated - the stacking speed. This stacking speed is determined as a parameter when approaching arrival times while varying with the separation of the real source and the receiving element for the particular reflector using a hyperbolic curve. Therefore, usually a part of a velocity field is available although it may not be very accurate since it also depends on the applicability of the subsurface model to the actual situation.

Using a determined speed field as indicated, we can, for the case of any resolution element, calculate a travel time to any real source position as well as to any part of the antenna function. It is also necessary to make a few additional comments about of the velocity field. The propagation velocities not only control the timing of the wave field but also the geometrical shape. The changes in velocity cause the ray trajectories to bend in accordance with the well-known Sneli Law. The equivalent effect in terms of wave field involves changes in its curvature. It then becomes evident that any error or uncertainty in knowing the velocity field limits inevitably the resolution of the image. The method of solution by wave equation with linked variables that aims to perform the presentation of image in a propagation medium must appropriately allow a change in wavefield geometry, but it will be subject to limitations in resolution achievable thanks to the speed field. A simple analogy concerning the situation would be to look at the scene through glasses with incorrect recipe The improvement of the recipe - equivalent to correcting the speed field - would necessarily improve the image Then it follows that the quality of the image can become a criterion to refine the velocity field iteratively That is, an initial velocity function can be estimated and applied with updates or corrections made that in turn improve the focus or sharpness of the image. The computation of undulatory field image presentation described above. using travel times in one direction makes such computing very practical. Turning now to the field seismic data, a profile is considered to originate from data originally acquired outside of Louisiana, United States in 1985. The actual source is an array of variably sized air cannons whose precise sign or effective seismic wavelet is unknown . It was rated by the manufacturer as producing significant frequency content between 10 and 90 Hz. A total of 120 receiving elements - in effect "grouped" closely spaced hydrophones that composed each element ~ were spaced at 82.5 feet intervals, and dragged to their positions as appropriate on the seafloor. Digital recording was applied using an interval of 2 milliseconds between samples, but not until after having submitted the data to high-cut filter while they were in analog form. In this way, frequencies greater than 128 Hz were attenuated to a significantly greater degree with the increase in frequency. m The data were acquired with reference to sea level and the sound velocities in salt water were known with acceptable accuracy. However, it was necessary to reshape the pulse that describes the wave field of propagation with time. This was achieved in fact by a statistical means. Operations such as these that are essentially normal are well documented in texts such as Yilmaz, op. cit., which can be consulted with respect to necessary detail. Speed analyzes based on CDP [CDP gathers] captures, and in the manner discussed above, were collected in quarter-mile intervals to develop the necessary velocity function. However, for the part of the treated profile practically no change or gradient was observed in the velocity field. Figure 11 A is a view of the particular seismic line as processed using a traditional two-dimensional migration method prior to KirchhofF type stacking commercially available for final image presentation. It was also based on a wave equation solution with linked variables. The corresponding Huygens type image presentation made in accordance with the present invention for these same basic data is presented in Figure IB. Such image presentation has been called modified KirchhofF migration and the result has been presented with the same spatial and temporal sampling intervals (41.25 feet and 2 milliseconds, respectively). As expected, the two deployments are quite comparable to each other, particularly for this case where the lateral velocity load is minimal. Figures 11 A and 1 IB use common formats and currents of seismic image display deployments as described above for the study of the model data. It will be apparent that the field data has a background noise level that was not presented in the model data (see Figures 9B and 9C). For the scale of deployment offered by Figures 1 1 A and 1 1 B, the comparison can not be easily made in detail. Therefore, in the similar positions indicated in both Figures, the amplitude spectra are calculated and these can be studied in Figure 1 1C. A common standardization method was applied and it is evident that the modified KirchhofF migration (Extended Huygens Image Display) of Figure 1 I B has a somewhat higher frequency content. This result is by no means definitive or of special significance since the 2 millisecond computation has only limited capacity to represent frequencies higher than 125 Hz, particularly in view of the filtering as applied to the original data. The modified KirchhofF migration or the Huygens type computation was nevertheless repeated, using a sampling interval time of 1 millisecond for the final presentation of the image. Now the comparison that must be made is between the 1 millisecond image presentation and the 2 millisecond one using the same Huygens procedure. One can look closely at comparable time history displays in the seismic format as described above as well as in terms of frequency content using amplitude spectra.

It is important to recognize that the evaluation of the results from a study of real data always incorporates a higher degree of uncertainty than a computation of a model study which is only due to the presence of noise about which it was commented. In this case, the detected presence of higher frequencies (125 Hz and more) could be attributed to the amplification of the small signal content present in the source signal or to the system noise. Therefore, the appropriate arguments must be made to support any conclusion that has been reached.

Then, you should be aware and well aware of contributions out of plane since this is a two-dimensional cut derived from a three-dimensional earth. In this case, such effects have been judged to not interfere with the results or conclusions at any level of significance.

There is an even more subtle element with respect to the image presentation of higher frequencies that have geological origin which should also be considered. First, one must accept that the highest frequencies in the temporal variable correspond to thinner units on earth. Thinner units usually cover less surface area than thicker units. Therefore, as the wave fronts of the propagation wave field grow with the increase of time (or depth), the spatial extent of an inhomogeneity must also increase to have a discernible coherent effect in the wave field. The consideration of the first Fresnel zones (analogous to their definition in optics) and their relationship is an alternative way of thinking about this effect, and comments Yilmaz in his work already cited. Therefore, given a limited accuracy of the velocity function and other practical issues such as system noise, etc., one would expect to see a decreasing ability to present the image of the highest frequencies on the ground with recorded travel time or the corresponding reflector depth. For this reason, comparisons to be made as indicated here would be observed using frequency analysis, but also at different travel times. The magnified displays of Figures 1 I D and 1 1 E clearly show the presence of a much higher frequency content in the version presented in the image with temporal sampling of 1 millisecond compared to the image presentation of 2 mii-seconds. These higher frequency contributions must be considered real for two reasons. First, they show the lateral continuity of spatial sample to sample and this continuity is congruent with the geological features as indicated by the presentation of images at lower frequencies (2 millisecond image presentation). The amplification of noise must not present this same degree of spatial coherence. Likewise, the spatial continuity derived from the blurry image, the speed function error drought, would show elliptical continuity instead of following geology, since the geological pattern was a particularity of the waveform image presentation method according to applied. Comparisons of the frequency domain of the amplitude spectra (Fig. 1 1F to 1 II inclusive) show the presence of the highest frequencies in the presentations of 1 millisecond images compared to the presentations of images of 2 milliseconds. Also as expected, the analysis of the shallowest part of the data (0.4 to 0.8 second) shows the highest concentration of the highest frequency components. It is very unlikely that frequencies three times higher than the nominal high cut-off point, as designated by the manufacturer of the actual seismic source (air cannon array), could be attributed to residual signal amplification -? especially in view of the analog filtering also thus applied It is only reasonable to conclude that such signal content is derived from the temporal spatial variable linkage inherent in the presentation of Huygens type images of the earth as virtual sources in each grid cell. These apparent frequencies that are probably beyond the effective bandwidth of the source suggest that the spatial resolution of the presentation of the seismic image is significantly greater than what the Nyquist criteria allow as applied to the original wave field sampling in consideration of the individual variables. ThusThis example, although two-dimensional, illustrates the practical nature of this invention. The frequencies, although apparent, were accessed by means of the presentation of image of reconstruction of wave fields that were both beyond those considered as representable using the Nyquist criteria (for the acquisition of data of 2 milliseconds) and beyond those present in the real font. The disclosure and description that precede the invention are illustrative and explanatory thereof. Various changes in the techniques, processing parameters, acquisition parameters and similar aspects, as well as in the details of the illustrated embodiments can be made without deviating from the spirit of the invention.

Claims (1)

-V Relationship of Claims: 1. A virtual source image presentation method that composes the steps of: illuminating a medium with a wave propagation field that has a certain spectrum of frequencies from a real source; sampling of the wave propagation field; performing the reconstruction of the wave field in the virtual source location to form the image, using at least one apparent frequency component that is not present in the wave field of illumination from the real source; . The method of Claim 1, wherein said step of performing wave field reconstruction uses apparent frequency components within the frequency spectrum of said actual source; The method of Claim 1, wherein said step of performing wave field reconstruction uses apparent frequency components within a discrete sampling of the frequency spectrum of said actual source; The method of Claim 1, wherein said apparent frequency component is greater than any comparable frequency component measured with respect to the actual source. The method of Claim 1, wherein said apparent frequency component is smaller than any comparable frequency component measured with respect to the actual source. The method of Claim 1, wherein said wave propagation field is of an ultrasonic character. . The method of Claim 1, wherein said wave propagation field is electromagnetic in character. . The method of Claim 1, wherein said wave propagation field is acoustic in character. . A method of presenting an image of a source from a wave propagation field that comprises the steps of: performing a sampling that is discrete, at least in one variable of the wave field; perform a reconstruction with linked variables of the wave field. m assign resolution during reconstruction with linked variables of the variable having apparent frequency content outside the Nyquist limits for said sampling; forming an output log of the results of said reconstruction step with linked variables, The method of Claim 9, wherein said source is a real source. The method of Claim 9, wherein said source is a virtual source. The method of Claim 9, wherein said step of performing the wave field reconstruction uses apparent frequency components within the frequency spectrum of said actual source. The method of Claim 9, wherein said step of performing the wave field reconstruction uses components of the apparent frequency within a discrete sampling of the frequency spectrum of said real source. The method of Claim 9, wherein said wave propagation field is of an ultrasonic character. The method of Claim 9, wherein said wave propagation field is electromagnetic in character. The method of Claim 9, wherein said wave propagation field is acoustic in character. A method of presenting the image of a medium as a result of its illumination by a wave propagation field, which consists of the steps of: sampling the wave propagation field with a known antenna function; said discrete sampling step being in at least one variable; Perform the reconstruction of the wave field in a location in the middle, performing the steps of: recovery of at least one apparent frequency corresponding to said output variable outside the band defined by criteria of the sampling theory; m use of coupled space-time solutions of the Wave Equation based on velocities of the wave propagation field in the middle. The method of Claim 17, wherein said step of performing wave field reconstruction uses apparent frequency components within the frequency spectrum of said actual source. The method of Claim 17, wherein said step of performing wave field reconstruction uses apparent frequency components within the discrete sampling of the frequency spectrum of said real source. The method of Claim 17, wherein said apparent frequency component recovered is greater than any significant frequency component measured for illumination. The method of Claim 17, wherein said registered apparent frequency component is less than any significant frequency component measured for illumination. The method of Claim 17, wherein said wave propagation field is of an ultrasonic character, The method of Claim 17, wherein said wave propagation field is electromagnetic in character. The method of Claim 17, wherein said wave propagation field is acoustic in character. A method of data processing indicative of physical objects in a zone of interest, composed of the following steps: cause the energy to travel at a speed that can vary as a function of both time and space during such a trip through the area of interest; form a recording by sampling the energy received after traveling through the area of interest variably in time; process the received energy register to obtain a representation of objects present in the area of interest that has resolution without being restricted according to any frequency limitation of the discrete sampling of the recording in space or in time derived from the sampling theory as applied to each one of the variables, performing the steps of:

1. m divide the record into several energy arrival times for a range of possible locations in the field of interest, allocate a range of postulated objects locations in the field of interest for energy at a given energy arrival time based on the speed of travel over the area from a power source, repeat each step of assigning for each of the energy arrival times with respect to each energy source and the locations of postulated objects; form an output record that indicates as locations of real objects those locations of postulated objects that exhibit probity according to the temporal and spatial link processing variables for the energy sources and the sampled records. The method of Claim 25, wherein said energy source is located outside the area of interest and wherein said additional processing step has unrestricted resolution by the frequency content of the energy in space or time. The method of Claim 25, wherein said wave propagation field is of ultrasonic character. The method of Claim 25, wherein said wave propagation field is electromagnetic in character. The method of Claim 25, wherein said wave propagation field is acoustic in character. A virtual source image presentation method that composes the steps of: illuminating a wave propagation field coming from a real source; sampling of the wave propagation field using one or more receiving elements; perform reconstruction with linked variables in the location of the virtual source using at least one apparent frequency that is not present in the wave field of illumination coming from the real source, said step being constituted by the step of: determining travel times in a address relative to each real source and each receiver element corresponding to each speed function; add to the virtual source the travel times in an address corresponding to each pair of real source and receiving element; replace a recorded amplitude with respect to the travel time in two directions for each virtual source corresponding to that recorded at each moment for said pairs of real source and receiving element. repeat and replace all real sources and all pairs of real source and receiving element; add said substituted amplitudes; and form an exit record to present this virtual source. The method of Claim 29, wherein said designated resolution is not constrained by the frequency content of any of said real sources or property of the undulatory field sampling required by the Nyquist sampling criteria as applied to each of the variables. The method of Claim 30, further comprising the steps of: normalizing the recorded value of the wave field to compensate for the action of scattering the wave field by the distance traveled; and normalize the results of said addition step based on the number of summed values. The method of Claim 30, wherein said wave propagation field is of an ultrasonic character. The method of Claim 30, wherein said wave propagation field is electromagnetic in character. The method of Claim 30, wherein said wave propagation field is acoustic in nature. The method of Claim 30, wherein said step of performing wave field reconstruction uses apparent frequency components within the frequency spectrum of said actual source. The method of Claim 30, wherein said step of performing wave field reconstruction uses apparent frequency components within the discrete sampling of the frequency spectrum of said actual source. A method of presenting the image of a virtual source that is composed of the steps of: illuminating a medium with a wave propagation field from at least one real source; and sampling said wave field using one or more receiving elements, said sampling being discrete in at least one variable; perform the reconstruction with rolling variables of the wave field using said sampling at the location of the virtual source at least one apparent frequency component for said discrete variable outside those allowed by the sampling theory as applied to each of the variables, being said step comprising the step of: determining travel times in one direction for each real source and each receiving element to each virtual source using a speed function; add to the virtual sources the travel times in one direction for each pair of real source and receiving element; replacing by a recorded value of the wave field the travel time in two directions for each virtual source corresponding to the one recorded at each moment for said pair of real source and receiving element. repeat said substitutions for all virtual sources and all pairs of real source and receiving element; add all of said substituted amplitudes; and form an exit record to present this virtual source. The method of Claim 38, further comprising the steps of: normalizing the recorded value of the wave field to compensate for the scattering action of the wave field by the distance traveled; and - • normalizes the results of said addition step based on the number of summed values. The method of Claim 38, wherein said wave propagation field is of ultrasonic character. The method of Claim 38, wherein said wave propagation field is electromagnetic in character. The method of Claim 38, wherein said wave propagation field is acoustic in nature. The method of Claim 38, wherein said step of performing wave field reconstruction uses apparent frequency components within the frequency spectrum of said actual source. The method of Claim 38, wherein said step of performing wave field reconstruction uses apparent frequency components within a discrete sampling of the frequency spectrum of said actual source. A method of presenting the image of a virtual source in a propagation medium, of a function of known speed, through which the energy travels from, at least, a real source to a multiplicity of receiving elements, by crawled reconstruction of wave field, which is composed of the steps of: determining travel times in one direction of each real source and each receiving element in pairs to the virtual source based on a known speed function; calculating said travel times in one direction using centralized spherical wave fronts in each real source and receiving element based on a speed function appropriate to each; add the determined travel times in one direction for each of the real source and recipient element pairs for each virtual source; replace by a measured amplitude, a travel time in two directions in each virtual source location corresponding to that registered at each moment for the real source pair and the receiving element; repeating each substitution step for each of the real source pairs and the receiving element; add the substituted amplitudes in each virtual source location; and forming an output record of the results of said addition step in order to present the image of the virtual source. The method of Claim 45, wherein said wave propagation field is of an ultrasonic character. The method of Claim 45, wherein said wave propagation field is electromagnetic in character. The method of Claim 45, wherein said wave propagation field is acoustic in nature. 4 The method of Claim 45, further comprising the steps of: normalizing the recorded value of the wave field to compensate for the scattering action of the wave field by the distance traveled; and normalizing the results of said addition step based on the number of summed values. The method of Claim 45, which further includes the step of: sampling the energy traveling through the medium. The method of Claim 50, wherein said step of performing wavefield reconstruction utilizes reconstruction of apparent frequency components within the frequency spectrum of said actual source. The method of Claim 50, wherein said step of performing wave field reconstruction uses apparent frequency components within a discrete sampling of the frequency spectrum of said actual source. The method of Claim 50, wherein said image contains at least one apparent frequency component greater than any significant frequency component measured for the actual source. The method of Claim 50, wherein said image contains at least one apparent frequency component smaller than any significant frequency component measured for the actual source. The method of Claim 50, wherein said image contains at least one apparent frequency component outside the Nyquist sampling limits as applied to the individual variables.