CN112836373A - Method for calculating external gravity anomaly central region effect based on Poisson theory - Google Patents

Method for calculating external gravity anomaly central region effect based on Poisson theory Download PDF

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CN112836373A
CN112836373A CN202110171230.4A CN202110171230A CN112836373A CN 112836373 A CN112836373 A CN 112836373A CN 202110171230 A CN202110171230 A CN 202110171230A CN 112836373 A CN112836373 A CN 112836373A
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gravity anomaly
gravity
point
order gradient
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邓凯亮
陈欣
黄谟涛
陈鑫舟
陈轶
吴超
范龙
张博
张瑞瑞
郭忠磊
高飞
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CHINESE PEOPLE'S LIBERATION ARMY 92859 TROOPS
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Abstract

The invention relates to a method for calculating an external gravity anomaly central area effect based on a Poisson theory, which solves the problem that the high-precision external gravity anomaly can not ignore the central area effect, combines plane approximate conversion and Taylor series expansion under a polar coordinate system, calculates the external gravity anomaly central area effect based on the Poisson theory, can effectively perform high-precision calculation on the external gravity anomaly central area effect, and simultaneously performs numerical verification on a calculation result of the method by using simulated standard field data established by a global position model, thereby proving the necessity and effectiveness of the method, and being widely applied to the technical field of geophysics measurement.

Description

Method for calculating external gravity anomaly central region effect based on Poisson theory
Technical Field
The invention belongs to the field of geographic measurement, and particularly relates to a method for calculating an external gravity anomaly central area effect based on a Poisson theory.
Background
The external gravity anomaly is a key element for providing gravity anomaly field compensation for precisely determining flight trajectories of spacecrafts such as carrier rockets, artificial satellites, spacecrafts, missile weapons and space shuttles, and is also essential basic information for providing comparison reference for carrying out quality evaluation on gravity measurement data in the near-earth space. The Poisson theory is a basic mathematical model for calculating the external gravity anomaly, and in the actual calculation process, the theoretical distance from a projection point of a calculation point on a spherical surface and a central area of an adjacent area to the calculation point is close to zero, so that the grid data block can cause a serious singularity problem; a common approach is to subtract it directly from the integral domain or treat it as a constant value to avoid singular effects. However, when the area of the grid data block is large and the variation of the gravity anomaly field around the calculation point is severe, the processing method also brings large errors to the calculation result. For the external gravity anomaly calculation required with high accuracy, the amount of such influence cannot be ignored. But at present, a high-precision calculation method for external gravity anomaly does not exist.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a method for calculating the effect of the abnormal central area of the external gravity based on the Poisson theory, which can effectively calculate the effect of the abnormal central area of the external gravity with high precision.
The technical problem to be solved by the invention is realized by adopting the following technical scheme:
a method for calculating the effect of an external gravity abnormal central area based on Poisson theory comprises the following steps:
step 1, calculating an external gravity anomaly delta g based on a Poisson theory;
step 2, performing plane approximation processing on an integral kernel function in the external gravity anomaly delta g by adopting a polar coordinate system (s, alpha);
step 3, converting the gravity anomaly delta gqExpanding the spherical projection point Rp of the space calculation point P into a Taylor series;
step 4, calculating a second-order gradient g of the gravity abnormal north component according to the data grids (i, j) coincident with the calculation pointsxxAnd gravity anomaly east component second order gradient gyy；
Step 5, carrying out second-order gradient g on the gravity abnormal north componentxxAnd gravity anomaly east component second order gradient gyySubstituted gravity anomaly Δ gqExpanding Taylor series at the spherical projection point Rp of the space calculation point P to obtain the effect delta g of the external gravity anomaly central area0。
Moreover, the specific implementation method of the step 1 is as follows:
Figure BDA0002938979860000021
Figure BDA0002938979860000022
wherein Δ g is a calculation point
Figure BDA0002938979860000023
Gravity anomaly; Δ gqAs flow points on a spherical surface
Figure BDA0002938979860000024
Known observed gravity anomalies; r is the average radius of the earth ellipsoid; r is the centroid radial of the calculation point;
Figure BDA0002938979860000025
to calculate the latitude and longitude of a point；
Figure BDA0002938979860000026
Latitude and longitude as flow points; sigma is a unit spherical surface; d sigma is the area element of the unit sphere; psi is the spherical angular distance between the calculated point and the flow point;
Figure BDA0002938979860000027
l is the spatial distance between the calculation point and the flow point; k (r, ψ) is the integral kernel function.
Moreover, the specific implementation method of the step 2 is as follows:
Figure BDA0002938979860000028
Figure BDA0002938979860000029
Figure BDA00029389798600000210
R2dσ≈sdsdα
the integral of the data block coinciding with the calculation point is:
Figure BDA00029389798600000211
wherein h is the height of the calculation point from the earth surface, and h is R-R; Δ g0Is an external gravity anomaly central zone effect; s0Half the size of the data grid.
Moreover, the specific implementation method of step 3 is as follows:
Figure BDA00029389798600000212
wherein the x-axis points to true north; the y-axis is east; northbound distance x ═ scos α; eastDistance y is ssin α; first order gradient of north component
Figure BDA00029389798600000213
First order gradient of east component
Figure BDA00029389798600000214
Second order gradient of north component and east component mixture
Figure BDA00029389798600000215
Second order gradient of north component
Figure BDA00029389798600000216
Second order gradient of east component
Figure BDA00029389798600000217
Moreover, the specific implementation method of the step 4 is as follows:
Figure BDA0002938979860000031
moreover, the specific implementation method of step 5 is as follows:
Figure BDA0002938979860000032
the invention has the advantages and positive effects that:
aiming at the problem that the central area effect cannot be ignored due to high-precision external gravity anomaly is calculated, plane approximate conversion and Taylor series expansion under a polar coordinate system are jointly adopted, the central area effect of the external gravity anomaly is calculated based on the Poisson theory, the high-precision calculation of the central area effect of the external gravity anomaly can be effectively carried out, meanwhile, numerical verification is carried out on the calculation result of the method through simulation standard field data established by using a global position field position model, the necessity and the effectiveness of the method are proved, and the method can be widely applied to the technical field of geodetic measurement.
Detailed Description
The present invention will be described in further detail with reference to examples.
A method for calculating the effect of an external gravity abnormal central area based on Poisson theory comprises the following steps:
step 1, calculating an external gravity anomaly delta g based on a Poisson theory:
Figure BDA0002938979860000033
Figure BDA0002938979860000034
wherein Δ g is a calculation point
Figure BDA0002938979860000035
Gravity anomaly; Δ gqAs flow points on a spherical surface
Figure BDA0002938979860000036
Known observed gravity anomalies; r is the average radius of the earth ellipsoid; r is the centroid radial of the calculation point;
Figure BDA0002938979860000037
to calculate the latitude and longitude of the point;
Figure BDA0002938979860000038
latitude and longitude as flow points; sigma is a unit spherical surface; d sigma is the area element of the unit sphere; psi is the spherical angular distance between the calculated point and the flow point;
Figure BDA0002938979860000039
l is the spatial distance between the calculation point and the flow point; k (r, ψ) is the integral kernel function.
Step 2, because the central area for calculating the external gravity anomaly is the projection point of the calculation point on the spherical surface and the adjacent area thereof, the radius of the grid data block which is coincident with the calculation point is taken as psi00Calculating dot and product in the central areaThe space distance l between the shunt points is a small quantity compared with the average radius R of the earth ellipsoid, the resolution of currently used gravity observation data reaches a higher level, the corresponding data grid can generally reach 5 'multiplied by 5' or even smaller, and a polar coordinate system (s, alpha) is adopted to carry out plane approximation processing on an integral kernel function:
Figure BDA0002938979860000041
Figure BDA0002938979860000042
Figure BDA0002938979860000043
R2dσ≈sdsdα
the integral of the data block coinciding with the calculation point is:
Figure BDA0002938979860000044
wherein h is the height of the calculation point from the earth surface, and h is R-R; Δ g0Is an external gravity anomaly central zone effect; s0Half the size of the data grid.
Step 3, converting the gravity anomaly delta gqExpanding into a Taylor series at the spherical projection point Rp of the spatial calculation point P:
Figure BDA0002938979860000045
wherein the x-axis points to true north; the y-axis is east; northbound distance x ═ scos α; east distance y ═ ssin α; first order gradient of north component
Figure BDA0002938979860000046
First order gradient of east component
Figure BDA0002938979860000047
Second order gradient of north component and east component mixture
Figure BDA0002938979860000048
Second order gradient of north component
Figure BDA0002938979860000049
Second order gradient of east component
Figure BDA00029389798600000410
Step 4, calculating a second-order gradient g of the gravity abnormal north component according to the data grids (i, j) coincident with the calculation pointsxxAnd gravity anomaly east component second order gradient gyy：
Figure BDA00029389798600000411
Step 5, carrying out second-order gradient g on the gravity abnormal north componentxxAnd gravity anomaly east component second order gradient gyySubstituted gravity anomaly Δ gqExpanding Taylor series at the spherical projection point Rp of the space calculation point P to obtain the effect delta g of the external gravity anomaly central area0：
Figure BDA0002938979860000051
According to the method for calculating the effect of the external gravity anomaly central area based on the Poisson theory, a global gravity field position model EGM2008 is used as a reference standard field for numerical calculation and inspection and is used for simulating and generating the observation quantity of the 5 'x 5' grid gravity anomaly on the earth surface, meanwhile, in order to reflect the representativeness of an inspection result, a Marina sea ditch with severe change of a gravity anomaly field is specially selected as a test area, and the specific coverage range is as follows: 6 ° × 6 ° (
Figure BDA0002938979860000052
Lambda is 142 DEG E-148 DEG E). Selecting ri＝R+hi，R＝6371km。
The invention is adopted to calculate the effect delta g of the central area of the external gravity anomaly0The method calculates the external gravity anomaly central region effect of 5 height surfaces, wherein the 5 heights are respectively taken as: 2km,4km,6km,8km,10 km. As shown in table 1, the external gravity anomaly central zone effect is given at 5 elevation planes.
TABLE 1 external gravity anomaly Central zone Effect (units: mGal) for 5 height planes calculated by the present invention
Figure BDA0002938979860000053
As can be seen from table 1, as the height increases, the external gravity anomaly central zone effect decreases; at 2km, the maximum value of the effect of the central area of the external gravity anomaly can reach 1.10mGal, the minimum value can reach-1.90 mGal, and the root mean square is 0.24mGal, which shows that the effect of the central area is very necessary for the external gravity anomaly calculation with high precision requirement, and proves the necessity and effectiveness of the invention.
It should be emphasized that the embodiments described herein are illustrative rather than restrictive, and thus the present invention is not limited to the embodiments described in the detailed description, but also includes other embodiments that can be derived from the technical solutions of the present invention by those skilled in the art.

Claims (6)

1. A method for calculating the effect of an external gravity abnormal central area based on Poisson theory is characterized in that: the method comprises the following steps:
step 1, calculating an external gravity anomaly delta g based on a Poisson theory;
step 2, performing plane approximation processing on an integral kernel function in the external gravity anomaly delta g by adopting a polar coordinate system (s, alpha);
step 3, converting the gravity anomaly delta gqExpanding the spherical projection point Rp of the space calculation point P into a Taylor series;
step 4, calculating a second-order gradient g of the gravity abnormal north component according to the data grids (i, j) coincident with the calculation pointsxxAnd gravity anomaly east component second order gradient gyy；
Step 5, carrying out second-order gradient g on the gravity abnormal north componentxxAnd gravity anomaly east component second order gradient gyySubstituted gravity anomaly Δ gqExpanding Taylor series at the spherical projection point Rp of the space calculation point P to obtain the effect delta g of the external gravity anomaly central area0。
2. The method for calculating the effect of the abnormal central zone of the external gravity based on the Poisson theory as claimed in claim 1, wherein: the specific implementation method of the step 1 comprises the following steps:
Figure FDA0002938979850000011
Figure FDA0002938979850000012
wherein Δ g is a calculation point
Figure FDA0002938979850000013
Gravity anomaly; Δ gqAs flow points on a spherical surface
Figure FDA0002938979850000014
Known observed gravity anomalies; r is the average radius of the earth ellipsoid; r is the centroid radial of the calculation point;
Figure FDA0002938979850000015
to calculate the latitude and longitude of the point;
Figure FDA0002938979850000016
latitude and longitude as flow points; sigma is a unit spherical surface; d sigma is the area element of unit sphere(ii) a Psi is the spherical angular distance between the calculated point and the flow point;
Figure FDA0002938979850000017
l is the spatial distance between the calculation point and the flow point; k (r, ψ) is the integral kernel function.
3. The method for calculating the effect of the abnormal central zone of the external gravity based on the Poisson theory as claimed in claim 1 or 2, wherein: the specific implementation method of the step 2 comprises the following steps:
Figure FDA0002938979850000018
Figure FDA0002938979850000019
Figure FDA00029389798500000110
R2dσ≈sdsdα
the integral of the data block coinciding with the calculation point is:
Figure FDA0002938979850000021
wherein h is the height of the calculation point from the earth surface, and h is R-R; Δ g0Is an external gravity anomaly central zone effect; s0Half the size of the data grid.
4. The method for calculating the effect of the abnormal central zone of the external gravity based on the Poisson theory as claimed in claim 3, wherein: the specific implementation method of the step 3 is as follows:
Figure FDA0002938979850000022
wherein the x-axis points to true north; the y-axis is east; northbound distance x ═ scos α; east distance y ═ ssin α; first order gradient of north component
Figure FDA0002938979850000023
First order gradient of east component
Figure FDA0002938979850000024
Second order gradient of north component and east component mixture
Figure FDA0002938979850000025
Second order gradient of north component
Figure FDA0002938979850000026
Second order gradient of east component
Figure FDA0002938979850000027
5. The method of claim 4, wherein the computing of the effect of the abnormal central region of external gravity is based on Poisson's theory, and comprises: the specific implementation method of the step 4 comprises the following steps:
Figure FDA0002938979850000028
6. the method of claim 5, wherein the computing of the effect of the abnormal central region of external gravity is based on Poisson's theory, and comprises: the specific implementation method of the step 5 is as follows:
Figure FDA0002938979850000029
CN202110171230.4A 2021-02-08 2021-02-08 Method for calculating external gravity anomaly central region effect based on Poisson theory Pending CN112836373A (en)

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CN103399350A (en) * 2013-07-29 2013-11-20 中国人民解放军国防科学技术大学 Airborne gravity downward continuation method based on integral iteration algorithm
CN108416082A (en) * 2018-01-19 2018-08-17 中国人民解放军92859部队 A kind of marine site pour point external disturbance gravitation horizontal component without unusual computational methods
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