CN111473775A - Sand and dust migration monitoring method - Google Patents

Sand and dust migration monitoring method Download PDF

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CN111473775A
CN111473775A CN202010251048.5A CN202010251048A CN111473775A CN 111473775 A CN111473775 A CN 111473775A CN 202010251048 A CN202010251048 A CN 202010251048A CN 111473775 A CN111473775 A CN 111473775A
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sand
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CN111473775B (en
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张艳
胡平
赵廷宁
王若水
吴海龙
孙永康
李晓霖
陈童
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Beijing Forestry University
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    • G01C11/00—Photogrammetry or videogrammetry, e.g. stereogrammetry; Photographic surveying
    • G01C11/02—Picture taking arrangements specially adapted for photogrammetry or photographic surveying, e.g. controlling overlapping of pictures
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    • G05D1/10—Simultaneous control of position or course in three dimensions
    • G05D1/101—Simultaneous control of position or course in three dimensions specially adapted for aircraft
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N15/00—Investigating characteristics of particles; Investigating permeability, pore-volume or surface-area of porous materials
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Abstract

The embodiment of the invention provides a sand and dust transport monitoring method, which comprises the following steps: carrying out unmanned aerial vehicle aerial survey on an area to be monitored at a first moment and a second moment respectively to obtain a first aerial image corresponding to the first moment and a second aerial image corresponding to the second moment; the first time and the second time are separated by a set time interval; and respectively extracting the sand and dust monitoring parameters of the first aerial image and the second aerial image, and comparing the sand and dust monitoring parameters of the first aerial image with the sand and dust monitoring parameters of the second aerial image to obtain the migration and change rule information of the area to be monitored in a set time period. According to the embodiment of the invention, the area to be monitored is monitored by adopting the unmanned aerial vehicle aerial survey technology, and the sand and dust monitoring parameters corresponding to aerial images at different moments are compared to obtain the sand and dust migration and change rule information of the area to be monitored in a set time period, so that sand and dust monitoring can be realized; compared with the prior art, the monitoring efficiency is obviously improved, and manpower and material resources are saved.

Description

Sand and dust migration monitoring method
Technical Field
The embodiment of the invention relates to the field of sand and dust monitoring, in particular to a sand and dust migration monitoring method.
Background
In order to effectively prevent and control sand and prevent and control the damage of wind erosion of soil, the movement law of wind erosion needs to be accurately known and mastered on the basis of wind erosion monitoring in the field, which is very important for guiding the selection of wind erosion prevention and control measures. Adopt the sand collector to monitor sand and dust usually among the prior art, nevertheless because the sand collector only can realize the sand and dust monitoring of limited height and scope, under the condition that needs monitor the sand district of large tracts of land, need adopt a large amount of sand collectors to arrange in great scope, arrange earlier stage and will consume a large amount of manpower and materials, influence monitoring efficiency.
Disclosure of Invention
In order to solve the above problems, embodiments of the present invention provide a sand transport monitoring method that overcomes or at least partially solves the above problems.
The embodiment of the invention provides a sand and dust transport monitoring method, which comprises the following steps: carrying out unmanned aerial vehicle aerial survey on an area to be monitored at a first moment and a second moment respectively to obtain a first aerial image corresponding to the first moment and a second aerial image corresponding to the second moment; the first time and the second time are separated by a set time interval; and respectively extracting the sand and dust monitoring parameters of the first aerial image and the second aerial image, and comparing the sand and dust monitoring parameters of the first aerial image with the sand and dust monitoring parameters of the second aerial image to obtain the migration and change rule information of the area to be monitored in a set time period.
According to the sand and dust migration monitoring method provided by the embodiment of the invention, the to-be-monitored area is monitored by adopting the unmanned aerial vehicle aerial survey technology, sand and dust monitoring parameters corresponding to aerial images at different moments are compared, and the migration and change rule information of the to-be-monitored area in a set time period is obtained, so that sand and dust monitoring can be realized; and compare in the mode of arranging the sand-collecting instrument among the prior art and show and improve monitoring efficiency, saved manpower and materials.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below. It is obvious that the drawings in the following description are only some embodiments of the invention, and that for a person skilled in the art, other drawings can be derived from these without inventive effort.
Fig. 1 is a schematic flow chart of a sand and dust transportation monitoring method according to an embodiment of the present invention;
fig. 2 is a schematic view of an aerial image of a sand dune area in the current 12 months of the year in the sand dust transportation monitoring method according to the embodiment of the present invention;
fig. 3 is a schematic view of an aerial image of a sand dune area in the next 1 month of the year in the sand dust migration monitoring method according to the embodiment of the present invention;
fig. 4 is a schematic diagram illustrating a position of a sand belt in the sand transportation monitoring method according to the embodiment of the present invention;
fig. 5 is a schematic diagram illustrating a sand dune wind erosion analysis in the sand dust migration monitoring method according to the embodiment of the present invention;
fig. 6 is a schematic view of a complete flow of a sand and dust transportation monitoring method according to an embodiment of the present invention;
fig. 7 is a schematic diagram of an arrangement position of a sand trap provided in the embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some embodiments, but not all embodiments, of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The embodiment of the invention provides a sand and dust transport monitoring method, and referring to fig. 1, the method comprises the following steps:
101, respectively carrying out unmanned aerial vehicle aerial survey on an area to be monitored at a first moment and a second moment to obtain a first aerial image corresponding to the first moment and a second aerial image corresponding to the second moment; the first time and the second time are separated by a set time interval.
The pilotless aircraft is an unmanned aircraft, abbreviated as "unmanned aerial vehicle" in english, and abbreviated as "UAV", and is an unmanned aircraft operated by a radio remote control device and a self-contained program control device, or is operated autonomously by an onboard computer, completely or intermittently. The sand dune area in the embodiment comprises a near-ground area and an overhead area thereof, the particle size of the near-ground area is larger, the unmanned aerial vehicle is adopted for aerial survey under the ground condition, and the non-rotary sand-collecting observation tower and/or the sand-collecting instrument is adopted under the water surface condition; the particle size of the upper region is small, and an unmanned aerial vehicle can be used for carrying a dust measuring device to measure the concentrations of TSP (Total Suspended Particulate), PM2.5 and PM10 of the sand dust with different heights. Wherein the particle size of the particles is measured using a laser particle sizer. According to the embodiment of the invention, an unmanned aerial vehicle aerial survey technology formed by combining an unmanned aerial vehicle and aerial photogrammetry is adopted to carry out aerial survey on the sand dune area in the area to be monitored, so that an aerial image of the sand dune area shot by the unmanned aerial vehicle can be obtained.
Specifically, although the above step 101 is described as performing aerial survey only twice, it is understood that the above step 101 may be repeated multiple times, so that aerial survey is performed every set time interval, and a series of aerial images separated by the set time interval can be obtained. In addition, the specific value of the set time period is not limited in the embodiment of the present invention, and the set time period may be one month, for example. The series of aerial images may include aerial images of each month from 1 to 12 months through a year of aerial survey. Fig. 2 shows an aerial image of 12 months of the year, i.e., the first moment, and fig. 3 shows an aerial image of 1 month of the next year, i.e., the second moment.
102, respectively extracting sand and dust monitoring parameters of the first aerial image and the second aerial image, and comparing the sand and dust monitoring parameters of the first aerial image with the sand and dust monitoring parameters of the second aerial image to obtain migration and change rule information of the area to be monitored in a set time period.
The dust monitoring parameter is a parameter capable of reflecting the dust condition. On one hand, each aerial image has corresponding dust monitoring parameters; on the other hand, aerial images obtained by aerial survey at different moments correspond to different moments; therefore, by comparing the sand and dust monitoring parameters of the first aerial image and the second aerial image, the migration and change rule information of the sand dune area in the set time period between the first moment and the second moment can be obtained, and the migration and change rule information is the sand and dust monitoring result.
According to the sand and dust migration monitoring method provided by the embodiment of the invention, the sand and dust area is monitored by adopting the unmanned aerial vehicle aerial survey technology, sand and dust monitoring parameters corresponding to aerial images at different moments are compared, and the migration and change rule information of the sand and dust area in a set time period is obtained, so that sand and dust monitoring can be realized; and compare in the mode of arranging the sand-collecting instrument among the prior art and show and improve monitoring efficiency, saved manpower and materials.
Based on the content of the above embodiment, as an optional embodiment, the dust and sand monitoring parameter includes elevation information of the area to be monitored; correspondingly, the step of obtaining the change rule information of the area to be monitored in the set time period by comparing the sand and dust monitoring parameter of the first aerial image with the sand and dust monitoring parameter of the second aerial image comprises the following steps: acquiring the difference of the altitude information of the area to be monitored in the first aerial image and the second aerial image; acquiring the difference of the sand dune volumes in the area to be monitored according to the difference of the elevation information; and calculating the sand and dust variation of the area to be monitored according to the difference of the sand dune volumes in the area to be monitored.
Specifically, the dust monitoring parameters include gradient information and a gradient factor in addition to elevation information of an area to be monitored. In other words, the elevation and the related information of the aerial images in the two aerial survey time periods are extracted, the gradient and the slope factor of the sand dune image map are extracted, and the migration and change rules of the sand dune area in the preset time period can be researched.
Based on the content of the above embodiment, as an optional embodiment, the dust and sand monitoring parameter of the first aerial image is compared with the dust and sand monitoring parameter of the second aerial image by the following formula, so as to obtain the change rule information of the area to be monitored in the set time period:
Q＝Δv*ρ；
Δv＝∫∫h(x,y)ds；
h(x,y)＝z1(x,y)-z2(x,y)；
wherein Q is the amount of change in dust and sand in the region to be monitored, Δ v is the difference between the dune volumes calculated based on the first aerial image and the second aerial image, ρ is the density of dust and sand, and dsH (x, y) is the difference between the elevation information of the first aerial image and the second aerial image taken by the same pixel, z1(x, y) is the elevation information of the first aerial image shot by the same pixel, z2And (x, y) is the elevation information of the second aerial image shot by the same pixel.
Based on the content of the foregoing embodiment, as an optional embodiment, a method for performing unmanned aerial vehicle aerial survey on an area to be monitored is provided, which includes, but is not limited to, the following steps: adopt unmanned aerial vehicle to carry out the aerial survey along main wind direction each sand ribbon in the sand hill region respectively. Referring to fig. 4, a plurality of sand belts, e.g., three as shown in fig. 4, may be included in the sand hill area. Usable unmanned aerial vehicle carries out the aerial survey to each sand ribbon respectively.
Before adopting unmanned aerial vehicle to carry out the aerial survey along main wind direction each sand ribbon in the sand hill region respectively, still include the following step: positioning a flying point of the unmanned aerial vehicle by an RTK carrier phase difference technology, and recording longitude and latitude information and elevation information of the flying point; the flying spot is marked by adopting an inserting manner, and the pricking point processing and the image control point arrangement processing are carried out on the flying spot through RTK.
The RTK (Real Time Kinematic) carrier phase difference technique can provide a three-dimensional positioning result of a station in a specified coordinate system in Real Time, and achieve centimeter-level accuracy. The pricking point treatment is that in a navigation photo, a needle is perpendicular to the front of the photo, and a small hole is pricked on the image of the object point obviously at the turning position of the ownership boundary line, namely the pricking point.
Specifically, the takeoff point of the unmanned aerial vehicle needs to be determined before aerial survey of the unmanned aerial vehicle. Firstly, selecting a flat sand ground as a flying point in a sand dune area; then accurately positioning the selected flying point of the unmanned aerial vehicle by using RTK (real-time kinematic), and recording longitude and latitude information and an elevation average value (namely elevation information) of the flying point of each sand belt; after the flying points of each sand belt are determined, each flying point can be marked by plugging thousand, so that the flying points of each aerial survey are the same; meanwhile, pricking points by using RTK, and then arranging image control points. After the processing is finished, the same flying points of the unmanned aerial vehicles with the same sand belts in each aerial survey can be ensured, and in addition, the flying points of the three sand belts can be respectively recorded as P1，P2And P3By the use of P1，P2And P3And calculating the average value of the dust monitoring parameters of the three groups of images at the first moment and the average value of the dust monitoring parameters of the three groups of images at the second moment by the three groups of aerial survey images.
And in the aerial survey of the unmanned aerial vehicle corresponding to the first moment and the aerial survey of the unmanned aerial vehicle corresponding to the second moment, the air lines aiming at the same sand belt are the same. In particular, in many aerial surveys, it is necessary to keep the origin and course of the same sand belt the same. Particularly, during aerial survey, the noon time period of calm weather is selected, so that the influence of external factors on aerial survey can be reduced as much as possible. And aerial photography is carried out on the sand belts of each sand dune by utilizing unmanned aerial vehicle route planning software, and the planned route map is stored, so that the same route corresponding to each sand belt is realized during aerial survey of the unmanned aerial vehicle every time.
Based on the content of the foregoing embodiment, as an optional embodiment, after obtaining the first aerial image corresponding to the first time and the second aerial image corresponding to the second time, the method further includes the following steps: and comparing the first aerial image with the second aerial image to obtain the sand dune wind erosion amount of the sand dune area in a set time period.
The wind erosion amount is the difference between the amount of surface material blown away by wind and the accumulation amount within a certain time, that is, the variation of the ground elevation. Because the aerial images can reflect the elevation of the sand dune area, the sand dune wind erosion amount of the sand dune area in a set period can be obtained through comparing the first aerial image with the second aerial image. For example, comparing the aerial image of fig. 2 with the aerial image of fig. 3, the analysis result of the sand dune wind erosion amount is shown in fig. 5. Further, in the case that the step 101 is repeated for a plurality of times, so that aerial survey is performed every set time interval and a series of aerial images separated by the set time interval are obtained, the aerial image obtained by the first aerial survey can be used as a base map, and the sand dune wind erosion amount can be obtained by comparing the later obtained aerial image with the base map. In addition, the elevation of each aerial image is extracted, and the change rule of the elevation of each aerial image in the time period can be counted.
Based on the content of the above embodiment, as an optional embodiment, the area to be monitored further includes a river channel area; correspondingly, the sand and dust movement monitoring method further comprises the following steps:
if the water surface width of the river channel area is smaller than a first preset threshold value, when the first moment and the second moment are in a non-icing period of the river channel area, building a multi-gradient non-rotary type sand collecting observation tower on two banks of the water surface of the river channel area along a main wind direction, and measuring the sand amount of the water inlet surface of the river channel area based on the multi-gradient non-rotary type sand collecting observation tower; arranging a set number of sand collectors on the ice surface of the river channel region along the main wind direction when the first time and the second time are in the icing period of the river channel region; collecting sand collecting samples in the primary sand collector at a first moment and a second moment respectively, and weighing the sand collecting samples to obtain the amount of the sand dust passing through the river; if the water surface width of the river channel area is larger than or equal to the first preset threshold value, when the river channel area is in an icing period, constructing multi-gradient non-rotary sand collection observation towers on two sides of the water surface of the river channel area along the main wind direction, arranging sand collection instruments on the ice surface of the river channel area at equal intervals, measuring the water inlet surface sand amount of the river channel area based on the multi-gradient non-rotary sand collection observation towers, and acquiring the river channel passing sand amount of the river channel area based on the sand collection instruments.
Specifically, because the amount of sand in the river course region is less and the sand bed thickness is thinner, consequently can not adopt the mode of unmanned aerial vehicle aerial survey to monitor sand and dust. Referring to fig. 6, when the water surface width of the river channel region is smaller than a first preset threshold, the water surface is narrow, and a multi-gradient non-rotary sand-collecting observation tower is used for measuring the sand amount entering the water surface of the river channel region in a non-icing period; during the icing period, a sand collector, such as a non-rotary multi-gradient sand collector, is used for measuring the sand and dust amount of the river channel passing through the river channel area. When the water surface width of the river channel area is larger than or equal to a first preset threshold value, the water surface is wide, a multi-gradient non-rotary sand collecting observation tower is used for measuring the sand amount entering the water surface of the river channel area in the icing period, and a sand collecting instrument is used for measuring the sand dust amount of the river channel passing through the river channel area.
When the sand-collecting instrument is used for measuring the sand and dust amount of the river channel passing through the river channel area, referring to fig. 7, firstly, a firmer ice surface can be selected on the ice surface of the river channel, two sand-collecting instruments (the sand-collecting instrument 1 and the sand-collecting instrument 3 are arranged on the left bank, and the sand-collecting instrument 2 and the sand-collecting instrument 4 are arranged on the right bank) are respectively arranged on the two banks of the river channel, and the sand-collecting directions are staggered in the northwest direction and the right west direction (the sand-collecting direction of the sand-collecting instrument can be specifically determined according to the movement direction of the sand and dust. The interval time for collecting the sand collecting samples in the sand collector can be consistent with the set time interval of aerial survey of the unmanned aerial vehicle. For example, when the set time period is one month, the sand collecting samples in the sand collector are collected once per month, and then the samples collected every month are weighed to obtain the amount of the river course sand; and (4) carrying out granularity measurement by using a laser granularity meter to obtain the granularity of the sand dust passing through the river channel. The amount of the sand and dust passing through the river channel and the granularity of the sand and dust passing through the river channel are the results of monitoring the sand and dust on the ice surface of the river channel. The non-rotary sand-collecting observation tower is composed of a multi-gradient sand collector, and the height of the observation tower is higher than that of a local desert sand dune, and is usually 8 to 10 m. The multiple gradients may be preset multiples of preset lengths, the preset lengths being one or more, and the preset multiple corresponding to each preset length being one or more. For example, the first preset length is 0.02, and the corresponding preset multiple is 1-24; the second preset length is 0.5, and the corresponding preset multiple is 1-3; the third preset length is 2, and the corresponding preset multiple is 1-5. The predetermined length is in centimeters.
On the basis of the above embodiment, in this embodiment, the inflow surface sand amount of the river channel area is measured based on the multi-gradient non-rotating sand-collecting observation tower by the following formula:
Wsurface of water＝(SUpwind direction-SDownwind direction)*L；
Wherein, WSurface of waterThe amount of sand entering the water surface of the river channel area, SUpwind directionSand wind input rate, S, determined for upwind direction to said non-rotating sand-collecting observation towerDownwind directionAnd the wind sand output rate measured for downwind the non-rotary sand-collecting observation tower is L, which is the water surface length of the river channel area.
On the basis of the above embodiment, in this embodiment, the amount of the river course dust passing through the river course area is obtained based on the sand trap by the following formula:
Figure BDA0002435495570000071
wherein, W'Surface of waterThe amount of the sand and dust passing through the river course in the river course area, n is the total number of the sand collecting instruments, Si+1And SiThe sand transport rates measured by the i +1 th sand collector and the i th sand collector respectively, LiThe distance between the (i + 1) th sand collector and the ith sand collector.
Finally, it should be noted that: the above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and the modifications or the substitutions do not make the essence of the corresponding technical solutions depart from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims (8)

1. A sand and dust movement monitoring method is characterized by comprising the following steps:
carrying out unmanned aerial vehicle aerial survey on an area to be monitored at a first moment and a second moment respectively to obtain a first aerial image corresponding to the first moment and a second aerial image corresponding to the second moment; the first time and the second time are separated by a set time interval;
and respectively extracting the dust and sand monitoring parameters of the first aerial image and the second aerial image, and comparing the dust and sand monitoring parameters of the first aerial image with the dust and sand monitoring parameters of the second aerial image to obtain the migration and change rule information of the area to be monitored in the set time period.
2. The method of claim 1, wherein the dust monitoring parameters comprise elevation information of the area to be monitored;
correspondingly, the step of obtaining the change rule information of the area to be monitored in the set time period by comparing the sand and dust monitoring parameter of the first aerial image with the sand and dust monitoring parameter of the second aerial image comprises the following steps:
acquiring the difference of the altitude information of the area to be monitored in the first aerial image and the second aerial image;
acquiring the difference of the sand dune volumes in the area to be monitored according to the difference of the elevation information;
and calculating the sand and dust variation of the area to be monitored according to the difference of the sand dune volumes in the area to be monitored.
3. The method according to claim 2, wherein the sand and dust monitoring parameters of the first aerial image are compared with the sand and dust monitoring parameters of the second aerial image by the following formula to obtain the change rule information of the area to be monitored in the set time period:
Q＝Δν*ρ；
Δv＝∫∫h(x,y)ds；
h(x,y)＝z1(x,y)-z2(x,y)；
wherein Q is the amount of change in dust and sand in the region to be monitored, Δ v is the difference between the dune volumes calculated based on the first aerial image and the second aerial image, ρ is the density of dust and sand, and dsH (x, y) is the difference between the elevation information of the first aerial image and the second aerial image taken by the same pixel, z1(x, y) is the elevation information of the first aerial image shot by the same pixel, z2And (x, y) is the elevation information of the second aerial image shot by the same pixel.
4. The method according to claim 1, wherein the step of performing drone aerial survey of the area to be monitored specifically comprises:
positioning a flying starting point of an unmanned aerial vehicle through a carrier phase difference technology RTK, and recording longitude and latitude information and elevation information of the flying starting point;
marking the flying spot by adopting a kilogrammes method, and carrying out pricking point processing and image control point arrangement processing on the flying spot by using RTK (real time kinematic);
respectively aerial surveying each sand belt of a sand dune area in the area to be monitored along a main wind direction by adopting an unmanned aerial vehicle;
and in the aerial survey of the unmanned aerial vehicle corresponding to the first moment and the aerial survey of the unmanned aerial vehicle corresponding to the second moment, the air lines of the sand belts are the same.
5. The method of claim 1, wherein after obtaining the first aerial image corresponding to the first time and the second aerial image corresponding to the second time, further comprising:
and comparing the first aerial image with the second aerial image to obtain the sand dune wind erosion amount of the area to be monitored in the set time period.
6. The method of any one of claims 1-5, wherein the area to be monitored further comprises a river area; correspondingly, the sand and dust movement monitoring method further comprises the following steps:
if the water surface width of the river channel area is smaller than a first preset threshold value, when the first moment and the second moment are in a non-icing period of the river channel area, building a multi-gradient non-rotary type sand collecting observation tower on two banks of the water surface of the river channel area along a main wind direction, and measuring the sand amount of the water inlet surface of the river channel area based on the multi-gradient non-rotary type sand collecting observation tower;
when the first time and the second time are in the icing period of the river channel area, arranging a set number of sand collectors on the ice surface of the river channel area along the main wind direction, collecting sand collecting samples in the sand collectors once at the first time and the second time respectively, and weighing the sand collecting samples to obtain the amount of the sand dust passing through the river channel of the river channel area;
if the water surface width of the river channel area is larger than or equal to the first preset threshold value, when the river channel area is in an icing period, constructing multi-gradient non-rotary sand collection observation towers on two sides of the water surface of the river channel area along the main wind direction, arranging sand collection instruments on the ice surface of the river channel area at equal intervals, measuring the water inlet surface sand amount of the river channel area based on the multi-gradient non-rotary sand collection observation towers, and acquiring the river channel passing sand amount of the river channel area based on the sand collection instruments.
7. The method of claim 6, wherein the amount of surface sand entering the river channel region is measured based on the multi-gradient non-rotating sand-catchment observation tower by the following formula:
Wsurface of water＝(SUpwind direction-SDownwind direction)*L；
Wherein, WSurface of waterThe amount of sand entering the water surface of the river channel area, SUpwind directionSand wind input rate, S, determined for a non-rotating sand-collecting observation tower in upwind directionDownwind directionAnd the wind sand output rate measured for the downwind non-rotary sand collecting observation tower is L, which is the water surface length of the river channel area.
8. The method of claim 6, wherein the amount of the river course dust passing through the river course area is obtained based on the sand trap by the following formula:
Figure FDA0002435495560000031
wherein, W'Surface of waterThe amount of the sand and dust passing through the river course in the river course area, n is the total number of the sand collecting instruments, Si+1And SiThe sand transport rates measured by the i +1 th sand collector and the i th sand collector respectively, LiThe distance between the (i + 1) th sand collector and the ith sand collector.
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