CN112965126B - Method for calculating central area effect of eastern component of external disturbance gravity - Google Patents

Method for calculating central area effect of eastern component of external disturbance gravity Download PDF

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CN112965126B
CN112965126B CN202110179203.1A CN202110179203A CN112965126B CN 112965126 B CN112965126 B CN 112965126B CN 202110179203 A CN202110179203 A CN 202110179203A CN 112965126 B CN112965126 B CN 112965126B
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component
external disturbance
east
gravity
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CN112965126A (en
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邓凯亮
陈欣
黄谟涛
李凯锋
范龙
徐广袖
邹舸
张靓
王耿峰
郭忠磊
周德久
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92859 TROOPS PLA
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01V—GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
    • G01V7/00—Measuring gravitational fields or waves; Gravimetric prospecting or detecting
    • G01V7/02—Details
    • G01V7/06—Analysis or interpretation of gravimetric records
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
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    • G06F17/12—Simultaneous equations, e.g. systems of linear equations

Abstract

The invention relates to a method for calculating an eastern component central area effect of external disturbance gravity, which comprises the following technical characteristics: computing central region effect of eastern component of external disturbance gravity by using generalized Stokes formula
Figure DDA0002941671110000011
Generalized Stokes formula for extracting and calculating east direction component of external disturbance gravity
Figure DDA0002941671110000012
Main item F ofψ(r, ψ) and sin α; and (3) performing plane approximation conversion and Taylor series expansion on the main terms of the generalized Stokes formula under a polar coordinate system to obtain an external disturbance gravity east component central area effect calculation formula, and calculating the high-precision external disturbance gravity east component central area effect by using the formula. The method has reasonable design, adopts the generalized Stokes formula to calculate the effect of the central area of the east component of the external disturbance gravity, improves the calculation precision of the effect of the central area of the east component of the external disturbance gravity, and can be widely applied to the technical field of physical geodetic measurement.

Description

Method for calculating central area effect of eastern component of external disturbance gravity
Technical Field
The invention belongs to the technical field of physical geodetic measurement, relates to a technology for calculating an east component of an external disturbance gravity based on gravity anomaly, and particularly relates to a method for calculating a central area effect of the east component of the external disturbance gravity.
Background
The external disturbance gravity east component is an important component of earth gravity field approximation modeling research content, is one of main application targets for solving geodetic edge value problems, and has important application value in precise calculation of flight trajectories of aerospace vehicles and space science and technology research. The generalized Stokes formula for calculating the east component of the external disturbance gravity by using gravity anomaly is widely applied. In the actual calculation process, the theoretical distance from the projection point of the calculation point on the spherical surface and the central area of the adjacent area to the calculation point is close to zero, the grid data block does not act on the east component, the grid data block is generally deducted from the integral domain when the calculation is carried out, and meanwhile, the problem of integration singularity can be avoided. However, when the area of the grid data block is large and the gravity anomaly field around the calculation point changes more severely, this simple processing method also brings errors of milligal level to the calculation result. For the calculation of the east component of the gravity of the external disturbance with high precision, the influence quantity cannot be ignored. The invention provides a method for calculating the central area effect of an external disturbance gravity east component by using a generalized Stokes formula aiming at the problem that the central area effect cannot be ignored in the calculation of the high-precision external disturbance gravity east component, so as to calculate the high-precision external disturbance gravity east component central area effect.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, provides the method capable of calculating the central area effect of the east component of the external disturbance gravity with high precision, and improves the precision of calculating the central area effect of the east component of the external disturbance gravity.
The technical scheme adopted by the invention for solving the technical problem is as follows:
a method for calculating an east component central area effect of external disturbance gravity utilizes a generalized Stokes formula to calculate the east component central area effect of the external disturbance gravity
Figure BDA0002941671100000011
The method comprises the following steps:
step 1, extracting and calculating generalized Stokes formula of east-direction component of external disturbance gravity
Figure BDA0002941671100000012
Main item F ofψ(r, ψ) and sin α;
and 2, combining main terms of the generalized Stokes formula by adopting plane approximate conversion and Taylor series expansion under a polar coordinate system to obtain an external disturbance gravity east component central area effect calculation formula, and calculating the high-precision external disturbance gravity east component central area effect by using the formula.
Furthermore, the generalized Stokes formula in step 1
Figure BDA0002941671100000013
Main item F ofψThe calculation formula for (r, ψ) and sin α is:
Figure BDA0002941671100000021
in the formula (I), the compound is shown in the specification,
Figure BDA0002941671100000022
computing an east component of gravity for the external disturbance of the external space computation point; Δ g is the known observed gravity anomaly at the flow point on the sphere;
Figure BDA0002941671100000023
respectively calculating the geocentric diameter, latitude and longitude of the point;
Figure BDA0002941671100000024
the geocenter radial, latitude and longitude of the flow point are respectively, wherein R is the average radius of the earth ellipsoid; 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 BDA0002941671100000025
is to calculate the point-to-integral flowThe spatial distance between points; fψ(r, ψ) is an integral kernel function; α is the azimuth angle from the flow point to the computation point.
Moreover, the specific implementation method of the step 2 is as follows:
to formula
Figure BDA0002941671100000026
And performing plane approximation processing on the expressed integral kernel function by adopting a polar coordinate system (s, alpha):
Figure BDA0002941671100000027
the integral of the data block coinciding with the calculated point is written as:
Figure BDA0002941671100000028
in the formula, s0Is half the size of the data grid, s is 1 'x 1' when the data grid is0＝0.5'；
Spherical projection point R of gravity anomaly delta g at space calculation point PPThe process is expanded to taylor series:
Figure BDA0002941671100000029
in the formula, 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 BDA00029416711000000210
First order gradient of east component
Figure BDA00029416711000000211
Second order gradient of north component and east component mixture
Figure BDA00029416711000000212
Second order gradient of north component
Figure BDA00029416711000000213
Second order gradient of east component
Figure BDA00029416711000000214
Taking a data grid coincident with the calculation point as (i, j), and calculating the first-order gradient g of the east component of the disturbance gravity according to the following formulay：
Figure BDA00029416711000000215
In the formula (I), the compound is shown in the specification,
Figure BDA00029416711000000216
computing a point latitude for the data grid (i, j);
substituting the formula II into the formula I to obtain the central region effect of the east component of the external disturbance gravity
Figure BDA00029416711000000217
The calculation formula of (2) is as follows:
Figure BDA0002941671100000031
the invention has the advantages and positive effects that:
the invention provides a method for calculating the central area effect of an external disturbance gravity east component by using a generalized Stokes formula aiming at the problem that the central area effect cannot be ignored in the calculation of the high-precision external disturbance gravity east component, so as to calculate the high-precision external disturbance gravity east component central area effect. The method is reasonable in design, and aims at the problem that the central area effect cannot be ignored in high-precision calculation of the east component of the external disturbance gravity, the central area effect of the east component of the external disturbance gravity is calculated by adopting the generalized Stokes formula, so that the calculation precision of the central area effect of the east component of the external disturbance gravity is improved, and the method can be widely applied to the technical field of physical geodetic measurement.
Detailed Description
The present invention is further illustrated by the following specific examples, which are intended to be illustrative, not limiting and are not intended to limit the scope of the invention.
A method for calculating an east component central area effect of external disturbance gravity utilizes a generalized Stokes formula to calculate the east component central area effect of the external disturbance gravity
Figure BDA0002941671100000032
The method comprises the following steps:
step 1, extracting and calculating generalized Stokes formula of east-direction component of external disturbance gravity
Figure BDA0002941671100000033
Main item F ofψ(r, ψ) and sin α.
In this step, the generalized Stokes formula
Figure BDA0002941671100000034
Main item F ofψThe calculation formula for (r, ψ) and sin α is:
Figure BDA0002941671100000035
Figure BDA0002941671100000036
in the formula (I), the compound is shown in the specification,
Figure BDA0002941671100000037
computing an east component of gravity for the external disturbance of the external space computation point; Δ g is the known observed gravity anomaly at the flow point on the sphere;
Figure BDA0002941671100000038
respectively calculating the geocentric diameter, latitude and longitude of the point;
Figure BDA0002941671100000039
respectively the earth center radial direction of the flow point,Latitude and longitude, where R is the mean radius of the earth's ellipsoid; 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 BDA00029416711000000310
calculating the space distance between the point and the integral flow point; fψ(r, ψ) is an integral kernel function; α is the azimuth angle from the flow point to the computation point.
Considering that the space distance l between the calculation point and the integrated flow point is a small quantity compared with the average radius R of the earth ellipsoid in the central area of the projection point of the calculation point on the sphere and the adjacent area, the disturbance gravity east component integral kernel function F can be formedψ(r, ψ) is subjected to a simplification process, retaining only the first term which plays a dominant role therein:
Figure BDA0002941671100000041
and 2, combining main terms of the generalized Stokes formula by adopting plane approximate conversion and Taylor series expansion under a polar coordinate system to obtain an external disturbance gravity east component central area effect calculation formula, and calculating the high-precision external disturbance gravity east component central area effect by using the formula.
In this step, the radius of the grid data block coinciding with the calculation point is taken as psi00Since the resolution of currently available gravity observation data has reached a higher level, the corresponding data grid can typically reach 5 'x 5' or even less, for the formula
Figure BDA0002941671100000042
And performing plane approximation processing on the expressed integral kernel function by adopting a polar coordinate system (s, alpha):
Figure BDA0002941671100000043
the integral of the data block coinciding with the calculated point is written as:
Figure BDA0002941671100000044
in the formula, s0Is half the size of the data grid, s is 1 'x 1' when the data grid is0＝0.5'。
Spherical projection point R of gravity anomaly delta g at space calculation point PPThe process is expanded to taylor series:
Figure BDA0002941671100000045
in the formula, 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 east component
Figure BDA0002941671100000046
First order gradient of east component
Figure BDA0002941671100000047
East component and east component hybrid second order gradient
Figure BDA0002941671100000048
Second order gradient of east component
Figure BDA0002941671100000049
Second order gradient of east component
Figure BDA00029416711000000410
Taking a data grid coincident with the calculation point as (i, j), and calculating the first-order gradient g of the east component of the disturbance gravity according to the following formulay：
Figure BDA00029416711000000411
In the formula (I), the compound is shown in the specification,
Figure BDA00029416711000000412
computing a point latitude for the data grid (i, j);
substituting the formula II into the formula I to obtain the central region effect of the east component of the external disturbance gravity
Figure BDA0002941671100000051
The calculation formula of (2) is as follows:
Figure BDA0002941671100000052
the effect of the present invention is verified by a specific embodiment as follows:
the super high-order 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 1'× 1' grid gravity anomaly on the surface of the earth. In order to represent the test result, a Marina sea ditch with severe change of a gravity abnormal field is specially selected as a test area, and the specific coverage range is as follows: 6 ° × 6 ° (
Figure BDA0002941671100000053
N-16 degrees N; lambda is 142 DEG E-148 DEG E). Selecting ri＝R+hiR is 6371km, the effect of the central region of the gravity east component of the external disturbance at 5 altitude planes is calculated by using the algorithm (formula (7)) of the present invention, and the 5 altitudes are respectively taken as: h isi0km,0.1km,0.3km,1km,3 km. Table 1 gives the effect of the central region of the gravity east component of the external perturbation at 5 elevation planes.
TABLE 1 center area effect (unit: mGal) of the external disturbance gravity east component of 5 altitude surfaces calculated by the inventive algorithm
Figure BDA0002941671100000054
As can be seen from table 1, the effect of the central region of the east component of the externally perturbed gravity decreases with increasing altitude, being negligible at 3km altitude. At 0km, the maximum value of the central area effect of the externally disturbed gravity east component can reach 3.62mGal, the minimum value reaches-6.36 mGal, and the root mean square is 1.08mGal, which shows that the central area effect is very necessary for the calculation of the externally disturbed gravity east component with high precision requirement, and proves the necessity and the effectiveness of the algorithm.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various changes and modifications can be made without departing from the inventive concept, and these changes and modifications are all within the scope of the present invention.

Claims (2)

1. A method for calculating the effect of an eastern component central area of external disturbance gravity is characterized in that: computing central region effect of eastern component of external disturbance gravity by using generalized Stokes formula
Figure FDA0003472845440000011
The method comprises the following steps:
step 1, extracting and calculating generalized Stokes formula of east-direction component of external disturbance gravity
Figure FDA0003472845440000012
Main item F ofψ(r, ψ) and sin α;
step 2, performing plane approximation conversion and Taylor series expansion on the main terms of the generalized Stokes formula under a polar coordinate system to obtain an external disturbance gravity east component central area effect calculation formula, and calculating a high-precision external disturbance gravity east component central area effect by using the formula;
generalized Stokes formula in the step 1
Figure FDA0003472845440000013
Main item F ofψThe calculation formula for (r, ψ) and sin α is:
Figure FDA0003472845440000014
in the formula (I), the compound is shown in the specification,
Figure FDA0003472845440000015
computing an east component of gravity for the external disturbance of the external space computation point; Δ g is the known observed gravity anomaly at the flow point on the sphere; r is the sum of the total number of the carbon atoms,
Figure FDA0003472845440000016
lambda is the geocentric diameter, latitude and longitude of the calculation point respectively; r is the total number of the carbon atoms in the carbon fiber,
Figure FDA0003472845440000017
λ' is the geocentric diameter, latitude and longitude of the flow point respectively, wherein R is the average radius of the earth ellipsoid; 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 FDA0003472845440000018
calculating the space distance between the point and the integral flow point; fψ(r, ψ) is an integral kernel function; α is the azimuth angle from the flow point to the computation point.
2. The method for calculating the effect of the central area of the east component of the gravity of the external disturbance according to claim 1, wherein: the specific implementation method of the step 2 comprises the following steps:
to formula
Figure FDA0003472845440000019
And performing plane approximation processing on the expressed integral kernel function by adopting a polar coordinate system (s, alpha):
Figure FDA00034728454400000110
R2dσ≈sdsdα；
the integral of the data block coinciding with the calculated point is written as:
Figure FDA00034728454400000111
in the formula, s0Is half the size of the data grid, s is 1 'x 1' when the data grid is0＝0.5'；
Spherical projection point R of gravity anomaly delta g at space calculation point PPThe process is expanded to taylor series:
Figure FDA0003472845440000021
in the formula, 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 FDA0003472845440000022
First order gradient of east component
Figure FDA0003472845440000023
Second order gradient of north component and east component mixture
Figure FDA0003472845440000024
Second order gradient of north component
Figure FDA0003472845440000025
Second order gradient of east component
Figure FDA0003472845440000026
Taking a data grid coincident with the calculation point as (i, j), and calculating the first-order gradient g of the east component of the disturbance gravity according to the following formulay：
Figure FDA0003472845440000027
In the formula (I), the compound is shown in the specification,
Figure FDA0003472845440000028
computing a point latitude for the data grid (i, j);
will be represented by the formula 2Entering a formula I to obtain the central region effect of the east component of the external disturbance gravity
Figure FDA0003472845440000029
The calculation formula of (2) is as follows:
Figure FDA00034728454400000210
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