CN111597496B - Rational polynomial coefficient-based single satellite-borne remote sensing image target height calculation method - Google Patents

Rational polynomial coefficient-based single satellite-borne remote sensing image target height calculation method Download PDF

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CN111597496B
CN111597496B CN202010422096.6A CN202010422096A CN111597496B CN 111597496 B CN111597496 B CN 111597496B CN 202010422096 A CN202010422096 A CN 202010422096A CN 111597496 B CN111597496 B CN 111597496B
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王峰
胡玉新
尤红建
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Aerospace Information Research Institute of CAS
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Abstract

A single satellite-borne remote sensing image target height calculation method based on rational polynomial coefficients comprises the following steps: acquiring a bottom coordinate and a top coordinate of a target on a remote sensing image; determining the relation between the image plane coordinates of the original data and the three-dimensional coordinates of the ground through rational polynomial coefficients: expressing the target bottom coordinates in a rational polynomial coefficient mode; expressing the target top coordinate in a rational polynomial coefficient mode; directly solving the height of the top of the regularized target based on the characteristic that the longitude and latitude of the bottom of the target are consistent with the longitude and latitude of the top of the target; obtaining the absolute height of the top of the target by adopting the regularization formula inverse transformation; and (4) performing difference on the top height and the bottom height of the target to obtain the target height. The method does not need to use a space triangular relation between a satellite or the sun and the target, and has higher calculation precision; the target body is directly observed, shadow information of the target is not needed, and the calculation process is simple, convenient and efficient.

Description

Rational polynomial coefficient-based single satellite-borne remote sensing image target height calculation method
Technical Field
The invention relates to the technical field of satellite-borne remote sensing images, in particular to a rational polynomial coefficient-based single satellite-borne remote sensing image target height calculation method.
Background
With the rapid development of urban processes, more and more man-made buildings are provided. Unlike naturally occurring objects that take on a gradual change in shape, man-made objects generally have the characteristic of being vertically erected, such as buildings, monuments, towers, power transmission poles, and the like. For the vertical target height information, the difficulty of field manual measurement is high. The satellite-borne remote sensing image can be used for accurately, objectively and quickly acquiring the vertical target height information.
The existing target height calculation method based on a single satellite remote sensing image generally utilizes the target shadow length, the solar altitude angle and azimuth angle, and the satellite altitude angle and azimuth angle to calculate.
As shown in FIG. 1, H is the building height, and M is the corner point of the building roof; beta is the solar altitude; theta solar azimuth angle; alpha is the satellite altitude;
Figure BDA0002496015810000011
is the satellite azimuth; delta is the azimuth angle of intersection of the satellite and the sun; m' is the position of the shadow of the corner point M on the roof of the building on the image; m' is the imaging position of the building roof corner point M on the image; m' M ", L, is the distance between the corner point of the building roof and its shadow on the image. According to the geometric relationship of satellite imagingThe following formula is obtained:
Figure BDA0002496015810000012
in Δ M' M "N, the cosine theorem can be applied
M′M″2＝M′N2+M″N2-2M′N*M″N*cosδ
Namely, it is
L2＝H2ctan2β+H2ctan2α-2H2ctanβctanαcosδ
Thus, the height H of the building is
Figure BDA0002496015810000013
After the remote sensing image is subjected to orthorectification, the influence of the solar azimuth angle and the satellite azimuth angle is eliminated, namely the solar azimuth angle and the satellite azimuth angle are considered to be equal, the formula can be simplified and expressed as
H＝L/(ctanβ-ctanα)
Therefore, calculating the height of a building requires knowing the altitude, azimuth and distance of the building's roof corner point from its shadow on the image.
In the process of implementing the invention, the applicant finds that the existing single remote sensing image target height calculation technology has the following technical defects:
(1) the existing single remote sensing image height calculation method needs to know the satellite and the solar altitude angle when the remote sensing image is imaged, in the actual process, the two angles are difficult to obtain, and some satellite remote sensing products cannot provide the satellite altitude angle and the solar altitude angle at the image imaging moment;
(2) the target height is calculated by utilizing the trigonometric function relationship, the influence of the angle precision on the height calculation result precision is large, and the precision of the satellite and the solar altitude at the imaging moment given by a satellite remote sensing product cannot be guaranteed, so that a satisfactory height calculation result cannot be obtained by utilizing the method of the solar altitude and the satellite altitude.
Disclosure of Invention
In view of the above, the present invention provides a method for calculating a target height of a single satellite-borne remote sensing image based on rational polynomial coefficients, so as to partially solve at least one of the above technical problems.
In order to achieve the above object, as an aspect of the present invention, a method for calculating a target height of a single satellite-borne remote sensing image based on rational polynomial coefficients is provided, including the following steps:
acquiring a bottom coordinate and a top coordinate of a target on a remote sensing image;
determining the relation between the image plane coordinates of the original data and the three-dimensional coordinates of the ground through rational polynomial coefficients:
expressing the target bottom coordinate in a rational polynomial coefficient mode;
expressing the coordinates of the top of the target in a rational polynomial coefficient mode;
directly solving the height of the top of the regularized target based on the characteristic that the longitude and latitude of the bottom of the target are consistent with the longitude and latitude of the top of the target;
obtaining the absolute height of the top of the target by adopting the regularization formula inverse transformation;
and the target height is obtained by the difference between the top height and the bottom height of the target.
Determining the relationship between the image plane coordinates of the original data and the three-dimensional coordinates of the ground through rational polynomial coefficients, wherein the relationship is shown as the following formula:
Figure BDA0002496015810000031
NumL(P，L，H)＝a1+a2L+a3P+a4H+a5LP+a6LH+a7PH+a8L2+a9P2+a10H2+a11PLH+ a12L3+a13LP2+a14LH2+a15L2P+a16P3+a17PH2+a18L2H+a19P2H+a20H3
Nums(P，L，H)＝c1+c2L+c3P+c4H+c5LP+c6LH+c7PH+c8L2+c9P2+c10H2+c11PLH+ c12L3+c13LP2+c14LH2+c15L2P+c16P3+c17PH2+c18L2H+c19P2H+c20H3
DenL(P，L，H)＝b1+b2L+b3P+b4H+b5LP+b6LH+b7PH+b8L2+b9P2+b10H2+b11PLH+ b12L3+b13LP2+b14LH2+b15L2P+b16P3+b17PH2+b18L2H+b19P2H+b20H3
Dens(P，L，H)＝d1+d2L+d3P+d4H+d5LP+d6LH+d7PH+d8L2+d9P2+d10H2+d11PLH+ d12L3+d13LP2+d14LH2+d15L2P+d16P3+d17PH2+d18L2H+d19P2H+d20H3
wherein an, bn, cn, dn are parameters of rational polynomial coefficients, (P, L, H) are normalized ground coordinates, and (X, Y) are normalized image plane coordinates.
Wherein, the target bottom coordinate is expressed by a rational polynomial coefficient as follows:
Figure BDA0002496015810000032
wherein, X0，Y0Image plane coordinates of the bottom of the object, P, for regularization0，L0，H0Ground coordinates at the bottom of the regularized target, (P)0，L0) As its longitude and latitude coordinates.
And the object space ground absolute elevation at the bottom of the target is obtained by adopting elevation offset in rational polynomial coefficients or by carrying out interpolation by using DEM data and radar height measurement data.
Wherein, the target top coordinate is expressed by a rational polynomial coefficient as follows:
Figure BDA0002496015810000033
wherein, X1，Y1Image plane coordinates of the top of the object, P, for regularization1，L1，H1Ground coordinates of the top of the object being regularized.
Wherein, due to P1＝P0，L1＝L0Then only regularized target top height H in the formula1The specific calculation process for the unknown quantity is as follows:
the rational polynomial coefficients are converted into the following two formulas:
NumL(P0，L0，H1)-Y1DenL(P0，L0，H1)＝0
＝(a20-Y1b20)H1 3 +(a10+a14L0+a17P0-Y1(b10+b14L0+b17P0))H1 2 +(a4+a6L0+a7P0+a11P0L0+a18L0 2+a19P0 2-Y1(b4+b6L0+b7P0+b11P0L0+b18L0 2+b19P0 2))H1 +((a1+a2L0+a3P0+a5L0P0++a8L0 2+a9P0 2+a12L0 3+a13L0P0 2+a15L0 2P0+a16P0 3) -Y1(b1+b2L0+b3P0+b5L0P0+b8L0 2+b9P0 2+b12L0 3+b13L0P0 2+b15L0 2P0+b16P0 3))
Nums(P0，L0，H1)-X1Dens(P0，L0，H1)＝0
＝(c20-X1d20)H1 3 +(c10+c14L0+c17P0-X1(d10+d14L0+d17P0))H1 2 +(c4+c6L0+c7P0+c11P0L0+c18L0 2+c19P0 2-X1(d4+d6L0+d7P0+d11P0L0+d18L0 2+d19P0 2))H1 +((c1+c2L0+c3P0+c5L0P0+c8L0 2+c9P0 2+c12L0 3+c13L0P0 2+c15L0 2P0+c16P0 3) -X1(d1+d2L0+d3P0+d5L0P0+d8L0 2+d9P0 2+d12L0 3+d13L0P0 2+d15L0 2P0+d16P0 3))
subtracting the two formulas, and removing the cubic term to obtain an one-element quadratic equation about the height of the top of the regularized target, which is shown as follows:
s2H1 2+s1H1+s0＝0；
wherein:
s2＝(a10+a14L0+a17P0-Y1(b10+b14L0+b17P0))(c20-X1d20)-(c10+c14L0+c17P0-X1(d10+d14L0+d17P0))(a20-Y1b20)
s1＝(a4+a6L0+a7P0+a11P0L0+a18L0 2+a19P0 2-Y1(b4+b6L0+b7P0+b11P0L0+b18L0 2+b19P0 2))(c20-X1d20) -(c4+c6L0+c7P0+c11P0L0+c18L0 2+c19P0 2-X1(d4+d6L0+d7P0+d11P0L0+d18L0 2+d19P0 2))(a20-Y1b20)
s0＝(c1+c2L0+c3P0+c5L0P0+c8L0 2+c9P0 2+c12L0 3+c13L0P0 2+c15L0 2P0+c16P0 3 -X1(d1+d2L0+d3P0+d5L0P0+d8L0 2+d9P0 2+d12L0 3+d13L0P0 2+d15L0 2P0+d16P0 3))(a20-Y1b20)) -((a1+a2L0+a3P0+a5L0P0+a8L0 2+a9P0 2+a12L0 3+a13L0P0 2+a15L0 2P0+a16P03 -Y1(b1+b2L0+b3P0+b5L0P0+b8L0 2+b9P0 2+b12L0 3+b13L0P0 2+b15L0 2P0+b16P0 3))(c20-X1d20)
using a quadratic equation of unitySolving a formula to obtain a regularized target top height H1。
Based on the technical scheme, compared with the prior art, the single satellite-borne remote sensing image target height calculation method based on rational polynomial coefficients at least has one part of the following beneficial effects:
for a single remote sensing image, the target height can be directly calculated according to rational polynomial coefficient information attached to the image without using solar height angle or satellite height angle information. The method realizes the direct calculation of the target height by transforming the rational polynomial positioning model based on the consistent constraint of the positioning information of the bottom and the top of the target. The method does not need to use a space triangular relation between a satellite or the sun and the target, and has higher calculation precision; the target body is directly observed, shadow information of the target is not needed, and the calculation process is simple, convenient and efficient.
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FIG. 1 is a schematic diagram of a target altitude calculation process based on solar altitude and satellite altitude in the prior art;
FIG. 2 is a flow chart of a target height calculation method of a single satellite-borne remote sensing image based on rational polynomial coefficients according to the invention;
fig. 3 is a thumbnail of remote sensing images of beijing area shot by a 2016 year 12, month 9 and day high-resolution satellite in accordance with an embodiment of the present invention;
fig. 4 is a schematic diagram of coordinates of a real object according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings in combination with specific embodiments.
Aiming at the determination of the prior art scheme, the invention provides a single satellite-borne remote sensing image target height calculation method based on rational polynomial coefficients, which has the advantages that: 1) calculating the target height by using a rational polynomial coefficient carried by the remote sensing image without satellite height angle and solar height angle information at the imaging moment; 2) the longitude and latitude information of the bottom and the top of the target are the same and serve as constraint conditions, a single remote sensing image target height calculation method based on rational polynomial coefficients is constructed, and the target height can be directly calculated only by acquiring coordinates of the bottom and the top of the target in the calculation process.
The technical scheme of the invention is as follows:
1) obtaining the bottom coordinate (Sample) of the target on the level 1 remote sensing image0，Line0) And top coordinates (Sample)1，Line1) Wherein Sample0Line, column number of the bottom of the object on the image0Sample is the line number of the target bottom on the image1Line, column number of the bottom of the object on the image1Is the line number of the target bottom on the image.
2) The relation between the original data image plane coordinates (Line, column Sample) and the ground three-dimensional coordinates (longitude Lon, latitude Lat, Height) is determined by rational polynomial coefficients of the satellite-borne remote sensing image level 1 standard product:
Figure BDA0002496015810000061
NumL(P，L，H)＝a1+a2L+a3P+a4H+a5LP+a6LH+a7PH+a8L2+a9P2+a10H2+a11PLH+ a12L3+a13LP2+a14LH2+a15L2P+a16P3+a17PH2+a18L2H+a19P2H+a20H3
Num(P，L，H)＝c1+c2L+c3P+c4H+c5LP+c6LH+c7PH+c8L2+c9P2+c10H2+c11PLH+ c12L3+c13LP2+c14LH2+c15L2P+c16P3+c17PH2+c18L2H+c19P2H+c20H3
DenL(P，L，H)＝b1+b2L+b3P+b4H+b5LP+b6LH+b7PH+b8L2+b9P2+b10H2+b11PLH+ b1２L3+b13LP2+b14LH2+b15L2P+b16P3+b17PH2+b18L2H+b19P２H+b２0H3
Dens(P，L，H)＝d1+d２L+d3P+d4H+d5LP+d6LH+d7PH+d8L2+d9P2+d10H2+d11PLH+ d12L3+d13LP2+d14LH2+d15L2P+d16P3+d17PH2+d18L2H+d19P2H+d20H3wherein an, bn, cn, dn are parameters of rational polynomial coefficients, (P, L, H) are normalized ground coordinates, and (X, Y) are normalized image plane coordinates, and the relationship therebetween is shown as follows:
Figure BDA0002496015810000062
Figure BDA0002496015810000063
wherein LAT _ OFF, LAT _ SCALE, LONG _ OFF, LONG _ SCALE, HEIGHT _ OFF, HEIGHT _ SCALE are regularization parameters of ground coordinates. SAMPLE _ OFF, SAMPLE _ SCALE, LINE _ OFF, LINE _ SCALE are regularization parameters for the image coordinates.
3) Coordinate of the bottom of the target (Sample)0，Line0) Expressed by the form of rational polynomial coefficients:
Figure BDA0002496015810000064
wherein, X0，Y0For regularized target bottom image plane coordinates, P0，L0，H0Ground coordinates at the bottom of the regularized target.
For specifying image coordinates (Sample)0，Line0) The absolute height of the ground of the object space0For simplicity of calculation, height, which is the elevation offset in rational polynomials, is used0Then, the longitude and latitude coordinate (P) can be obtained0，L0)。
4) Coordinate of the top of the target (Sample)1，Line1) Expressed by the form of rational polynomial coefficients:
Figure BDA0002496015810000065
wherein, X1，Y1For regularized target top image plane coordinates, P1，L1，H1Ground coordinates of the top of the object being regularized.
Considering that the latitude and longitude coordinates of the bottom and the top of the target are the same, the following relationship exists: p is1＝P0，L1＝L0. Then only the regularized target top height H in the formula1The specific calculation process for the unknown quantity is as follows:
converting rational polynomial coefficients into the following two formulas:
NumL(P0，L0，H1)-Y1DenL(P0，L0，H1)＝0
＝(a20-Y1b20)H1 3 +(a10+a14L0+a17P0-Y1(b10+b14L0+b17P0))H1 2 +(a4+a6L0+a7P0+a11P0L0+a18L0 2+a19P0 2-Y1(b4+b6L0+b7P0+b11P0L0+b18L0 2+b19P0 2))H1 +((a1+a2L0+a3P0+a5L0P0++a8L0 2+a9P0 2+a1２L0 3+a13L0P0 2+a15L0 ２P0+a16P0 3) -Y1(b1+b2L0+b3P0+b5L0P0+b8L0 2+b9P0 2+b12L0 3+b13L0P0 2+b15L0 2P0+b16P0 3))
Nums(P0，L0，H1)-X1Dens(P0，L0，H1)＝0
＝(c20-X1d20)H1 3 +(c10+c14L0+c17P0-X1(d10+d14L0+d17P0))H1 2 +(c4+c6L0+c7P0+c11P0L0+c18L0 2+c19P0 2-X1(d4+d6L0+d7P0+d11P0L0+d18L0 2+d19P0 2))H1 +((c1+c2L0+c3P0+c5L0P0+c8L0 2+c9P0 2+c12L0 3+c13L0P0 2+c15L0 2P0+c16P0 3) -X1(d1+d2L0+d3P0+d5L0P0+d8L0 2+d9P0 2+d12L0 3+d13L0P0 2+d15L0 2P0+d16P0 3))
subtracting the two formulas, and removing cubic terms to obtain a quadratic equation of a first order about the regularization target top height, which is shown as follows:
s2H1 2+s1H1+s0＝0
wherein:
s2＝(a10+a14L0+a17P0-Y1(b10+b14L0+b17P0))(c20-X1d20)-(c10+c14L0+c17P0-X11(d10+d14L0+d17P0))(a20-Y1b20)
s1＝(a4+a6L0+a7P0+a11P0L0+a18L0 2+a19P0 2-Y1(b4+b6L0+b7P0+b11P0L0+b18L0 2+b19P0 2))(c20-X1d20) -(c4+c6L0+c7P0+c11P0L0+c18L0 2+c19P0 2-X1(d4+d6L0+d7P0+d11P0L0+d18L0 2+d19P0 2))(a20-Y1b20)
s0＝(c1+c2L0+c3P0+c5L0P0+c8L0 2+c9P0 2+c12L0 3+c13L0P0 2+c15L0 2P0+c16P0 3 -X1(d1+d2L0+d3P0+d5L0P0+d8L0 2+d9P0 2+d1２L0 3+d13L0P0 2+d15L0 2P0+d16P0 3))(a20-Y1b20)) -((a1+a2L0+a3P0+a5L0P0+a8L0 2+a9P0 2+a12L0 3+a13L0P0 2+a15L0 2P0+a16P0 3 -Y1(b1+b2L0+b3P0+b5L0P0+b8L0 2+b9P0 2+b1２L0 3+b13L0P0 2+b15L0 2P0+b16P0 3))(c20-X1d20)
thirdly, solving a formula by adopting a quadratic equation of a unit to obtain the regularized top height H of the target1. The calculation formula is as follows:
Figure BDA0002496015810000071
5) obtaining the absolute height of the top of the target by adopting the regularization formula inverse transformation1：
height1＝H1*HEIGHT_SCALE+HEIGHT_OFF
6) And the difference between the top height and the bottom height of the target is obtained to obtain the target height force.
h＝H1-H0
The flow chart of the technical scheme of the invention is shown in figure 2.
Furthermore, the above definitions of the methods are not limited to the specific embodiments mentioned in the examples, and those skilled in the art may make simple modifications or substitutions, for example:
in the step 3, the elevation offset in the rational polynomial coefficient is used as the object space ground absolute elevation of the target bottom, and the object space ground absolute elevation of the target bottom can also be obtained by interpolation of DEM data, radar height measurement data and the like. The method for calculating the top height of the target by firstly calculating the bottom longitude and latitude information of the target and using the bottom longitude and latitude information as the top longitude and latitude information of the target belongs to the scope of the invention.
The method for calculating the target height of the single satellite-borne remote sensing image based on rational polynomial coefficients according to the present invention will be described in detail with reference to the following examples. A pair of 2016, 12, 9, day and high-scoring second satellites is selected to shoot Beijing area remote sensing images, and the product number of the satellite is GF2_ PMS2_ E116.3_ N40.0_20161209_ L1A 0002174008-PAN2, as shown in FIG. 3.
For example, in a monument, the bottom and top information of the target is obtained by manual point selection, the bottom coordinates are (26479.5934, 18879.0860), and the top coordinates are (26495.6202, 18878.5689), as shown in fig. 4.
The height of the building obtained by calculation through the method is 37.81 meters, the target actual height is 37.94 meters, and the height error calculated by the method is only 0.13 meter.
The above-mentioned embodiments are intended to illustrate the objects, technical solutions and advantages of the present invention in further detail, and it should be understood that the above-mentioned embodiments are only exemplary embodiments of the present invention and are not intended to limit the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (2)

1. A single satellite-borne remote sensing image target height calculation method based on rational polynomial coefficients is characterized by comprising the following steps:
acquiring a bottom coordinate and a top coordinate of a target on a remote sensing image;
determining the relation between the image plane coordinates of the original data and the three-dimensional coordinates of the ground through rational polynomial coefficients:
expressing the target bottom coordinate in a rational polynomial coefficient mode;
expressing the coordinates of the top of the target in a rational polynomial coefficient mode;
directly solving the top height of the regularized target based on the characteristic that the bottom longitude and latitude and the top longitude and latitude of the target are consistent;
the height of the top of the target is different from the height of the bottom of the target to obtain the height of the target;
the relation between the image plane coordinates of the original data and the three-dimensional coordinates of the ground is determined through rational polynomial coefficients, and the relation is shown as the following formula:
Figure FDA0003640042390000011
NumL(P,L,H)＝a1+a2L+a3P+a4H+a5LP+a6LH+a7PH+a8L2+a9P2+a10H2+a11PLH+a12L3+a13LP2+a14LH2+a15L2P+a16P3+a17PH2+a18L2H+a19P2H+a20H3
Nums(P,L,H)＝c1+c2L+c3P+c4H+c5LP+c6LH+c7PH+c8L2+c9P2+c10H2+c11PLH+c12L3+c13LP2+c14LH2+c15L2P+c16P3+c17PH2+c18L2H+c19P2H+c20H3
DenL(P,L,H)＝b1+b2L+b3P+b4H+b5LP+b6LH+b7PH+b8L2+b9P2+b10H2+b11PLH+b12L3+b13LP2+b14LH2+b15L2P+b16P3+b17PH2+b18L2H+b19P2H+b20H3
Dens(P,L,H)＝d1+d2L+d3P+d4H+d5LP+d6LH+d7PH+d8L2+d9P2+d10H2+d11PLH+d12L3+d13LP2+d14LH2+d15L2P+d16P3+d17PH2+d18L2H+d19P2H+d20H3
wherein an, bn, cn and dn are parameters of rational polynomial coefficients, (P, L and H) are normalized ground coordinates, and (X and Y) are normalized image plane coordinates;
the target bottom coordinate is expressed by a rational polynomial coefficient, which is shown as the following formula:
Figure FDA0003640042390000012
wherein, X0,Y0Image plane coordinates of the bottom of the object, P, for regularization0,L0,H0Ground coordinates of the bottom of the target for regularization, (P)0,L0) The longitude and latitude coordinates are obtained;
the target top coordinate is expressed by a rational polynomial coefficient, which is shown as the following formula:
Figure FDA0003640042390000021
wherein, X1,Y1Image plane coordinates of the top of the object, P, for regularization1,L1,H1Ground coordinates of the top of the object being regularized.
2. The computing method of claim 1, wherein P is due to1＝P0,L1＝L0Then only regularized target top height H in the formula1The specific calculation process for the unknown quantity is as follows:
the rational polynomial coefficients are converted into the following two formulas:
NumL(P0,L0,H1)-Y1DenL(P0,L0,H1)＝0
＝(a20-Y1b20)H1 3+(a10+a14L0+a17P0-Y1(b10+b14L0+b17P0))H1 2+(a4+a6L0+a7P0+a11P0L0+a18L0 2+a19P0 2-Y1(b4+b6L0+b7P0+b11P0L0+b18L0 2+b19P0 2))H1+((a1+a2L0+a3P0+a5L0P0++a8L0 2+a9P0 2+a12L0 3+a13L0P0 2+a15L0 2P0+a16P0 3)-Y1(b1+b2L0+b3P0+b5L0P0+b8L0 2+b9P0 2+b12L0 3+b13L0P0 2+b15L0 2P0+b16P0 3))
Nums(P0,L0,H1)-X1Dens(P0,L0,H1)＝0
＝(c20-X1d20)H1 3+(c10+c14L0+c17P0-X1(d10+d14L0+d17P0))H1 2+(c4+c6L0+c7P0+c11P0L0+c18L0 2+c19P0 2-X1(d4+d6L0+d7P0+d11P0L0+d18L0 2+d19P0 2))H1+((c1+c2L0+c3P0+c5L0P0+c8L0 2+c9P0 2+c12L0 3+c13L0P0 2+c15L0 2P0+c16P0 3)-X1(d1+d2L0+d3P0+d5L0P0+d8L0 2+d9P0 2+d12L0 3+d13L0P0 2+d15L0 2P0+d16P0 3))
subtracting the two formulas, and removing the cubic term to obtain a quadratic equation of a unit about the regularization target top height, which is shown as follows:
Figure FDA0003640042390000022
wherein:
s2＝(a10+a14L0+a17P0-Y1(b10+b14L0+b17P0))(c20-X1d20)-(c10+c14L0+c17P0-X1(d10+d14L0+d17P0))(a20-Y1b20)
s1＝(a4+a6L0+a7P0+a11P0L0+a18L0 2+a19P0 2-Y1(b4+b6L0+b7P0+b11P0L0+b18L0 2+b19P0 2))(c20-X1d20)-(c4+c6L0+c7P0+c11P0L0+c18L0 2+c19P0 2-X1(d4+d6L0+d7P0+d11P0L0+d18L0 2+d19P0 2))(a20-Y1b20)
s0＝(c1+c2L0+c3P0+c5L0P0+c8L0 2+c9P0 2+c12L0 3+c13L0P0 2+c15L0 2P0+c16P0 3-X1(d1+d2L0+d3P0+d5L0P0+d8L0 2+d9P0 2+d12L0 3+d13L0P0 2+d15L0 2P0+d16P0 3))(a20-Y1b20))-((a1+a2L0+a3P0+a5L0P0+a8L0 2+a9P0 2+a12L0 3+a13L0P0 2+a15L0 2P0+a16P0 3-Y1(b1+b2L0+b3P0+b5L0P0+b8L0 2+b9P0 2+b12L0 3+b13L0P0 2+b15L0 2P0+b16P0 3))(c20-X1d20)
solving a formula by using a quadratic equation to obtain the regularized target top height H1。
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