CN109214050B - Polar region vertical line deviation non-singularity detailed calculation model and modeling method thereof - Google Patents
Polar region vertical line deviation non-singularity detailed calculation model and modeling method thereof Download PDFInfo
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Abstract
The invention discloses a detailed calculation model for deviation of polar region vertical lines without singularities and a modeling method thereof, and belongs to the field of geodetic measurement. The establishing process of the model comprises the following steps: based on the basic properties of the Legendre function, researching and obtaining a non-singularity calculation formula of the Legendre function on the first derivative of theta and a polynomial of a class, substituting the formula into a traditional calculation model of vertical line deviation, fully considering the conditions when m is equal to 0, 1 and other quantities, and finally establishing a detailed calculation model of the polar region vertical line deviation without theta singularity. The invention enables the vertical line deviation calculation model to be closer to the physical characteristics of a study object, has simple formula form and better stability and practicability, thoroughly solves the theta singular problem existing in the vertical line deviation calculation formula in the two-pole region, lays a theoretical foundation for calculating the vertical line deviation of the two-pole region and the nearby region, and can also calculate the vertical line deviation result of any region in the whole world by using the model provided by the invention.
Description
The technology belongs to the field:
the invention discloses a detailed calculation model for deviation of polar region vertical lines without singularities and a modeling method thereof, and belongs to the field of geodetic measurement.
The background technology is as follows:
in solving geophysical problems, the choice of the form of coordinates is often dependent on the geometry of the boundary surface of the object under investigation in order to bring the mathematical model closer to the physical properties of the object under investigation. Therefore, in the representation of the vertical deviation, the geodetic coordinates (r, θ, λ) are more preferable, which makes the calculation of the vertical deviation simpler and more practical.
The geocentric coordinate expression of the meridian component and the unitary component of the deviation of the vertical line on the ground is
Wherein, xi and eta respectively represent meridian components and mortise unitary components of vertical deviation; θ and λ represent the residual latitude and the geocentric longitude of the calculation point, respectively;representing a completely normalized earth disturbance gravity coefficient; n and m represent the order and order of the spherical harmonic coefficients, respectively; />Indicating a fully normalized associated Legendre function.
In formula (1)The recursive calculation formula of (2) is as follows
In the middle ofIs a kroneker symbol, when m=0,/is>When m is not equal to 0, ">
By substituting the expression (3) into the expression (1), a conventional calculation model of the vertical line deviation can be obtained.
As can be seen from formulas (1) - (3), in the two poles and the vicinity thereof, the calculation of the vertical deviation becomes infinite due to sinθ=0 or close to 0 on the denominator. However, the two poles of the earth and the vicinity thereof are common areas on the earth's surface, the calculated value of the deviation of the perpendicular line should be limited as with other earth's surfaces, and the magnitude of the difference is not too great in different areas. Therefore, the problem of theta singularity exists in the calculation of the deviation of the polar region vertical line. Along with the development of gravity measurement technology, it is more and more important to thoroughly solve the problem of theta singularity existing in polar region vertical line deviation calculation under the spherical coordinates in theory.
In order to solve the problem of theta singularity in polar region vertical line deviation calculation, the traditional method is mainly a rectangular coordinate representation method.
The basic principle of the rectangular coordinate representation method is that the vertical deviation is directly represented in the form of the rectangular coordinate of the earth center, so that the denominator does not contain a term equal to or close to 0, and the theta singularity problem existing in the calculation of the vertical deviation is avoided. But other problems are thereby introduced: the formula form and parameters are complex, and the stability and the accuracy of recursive computation are greatly affected. Therefore, the patent provides a detailed calculation model with no theta singularities for deviation of vertical lines of polar regions.
Disclosure of Invention
The purpose of the invention is that: a detailed calculation model without theta singularities of the vertical line deviation of the polar region is provided, namely, a first derivative of the Legendre function on theta and a non-singularity calculation formula of a class of polynomials are researched and obtained from the basic property of the Legendre function, and then the calculation formula is substituted into a traditional calculation model with the vertical line deviation, and meanwhile, the conditions when m is equal to 0, 1 and other quantities are fully considered, so that the detailed calculation model without theta singularities of the vertical line deviation of the polar region is finally established. The invention enables the vertical line deviation calculation model to be closer to the physical characteristics of a study object, has simple formula form and better stability and practicability, thoroughly solves the theta singular problem existing in the vertical line deviation calculation formula in the two-pole region, lays a theoretical foundation for calculating the vertical line deviation of the two-pole region and the nearby region, and can also calculate the vertical line deviation result of any region in the whole world by using the model provided by the invention.
The technical scheme of the invention is as follows: as shown in fig. 1: a modeling method of a detailed calculation model with no singularity on the deviation of a polar region vertical line comprises the following steps:
the first step: and calculating to obtain a non-singular result of the Legendre function on the first derivative of theta.
By usingRepresenting an accompanying Legendre function, where P n (x) Is an n-order Legendre polynomial, and x is replaced by cos theta to obtain
(4) The two ends derive theta to obtain
Because of the Legendre polynomial P n (x) Satisfies the following relationship
Combining the formula (5) and the formula (6) to obtain
Normalizing the Legendre function in the formula (7) to obtain a non-singular calculation formula of the Legendre function on the first derivative of theta, wherein the calculation formula is as follows:
and a second step of: calculating to obtain a class of polynomialsIs not a singular result of (c).
The Legendre function satisfies the following accompanying Legendre equation
Thereby can be obtained
Continuing to derive theta at two ends of the step (7) to obtain
Substituting the formula (7) into the formula (12) to obtain a calculation formula of the second derivative of the Legendre function
Substituting the formula (7) and the formula (13) into the formula (11) to obtain
According to the Legendre functionStandard forward push methods of (2) have
Thus, the first and second substrates are bonded together,
substituting the formula (16) and the formula (17) into the formula (14) to obtain
Normalizing the Legendre function in the formula (20) to obtain a polynomialThe calculation formula without singularity is as follows:
and a third step of: substituting the formula into a traditional calculation model of vertical line deviation, and fully considering the conditions when m is equal to 0, 1 and other quantities, and establishing a detailed calculation model of polar region vertical line deviation without theta singularities.
Substituting the formulas (9), (10) and (20) into the formulas (1) to (3) to obtain a detailed calculation model without theta singularities of vertical line deviation, wherein the detailed calculation model comprises the following components:
formulas (20) and (21) are detailed theta-free singularities calculation models of vertical line deviation in polar regions.
Compared with the prior art, the invention has the following advantages:
thoroughly solving the problem of theta singularity existing in the polar region vertical line deviation calculation formula;
the newly established detailed calculation model with no theta singularities for the vertical line deviation has the advantages of simple formula form and good stability and practicability;
the newly established detailed calculation model without theta singularities for the vertical line deviation lays a theoretical foundation for the calculation of the vertical line deviation between two poles and the areas nearby the two poles, and the model provided by the invention can be used for calculating the vertical line deviation result of any area of the world.
Description of the drawings:
basic flow for establishing detailed calculation model without theta singularities for vertical line deviation of polar region in figure 1
The difference between the measured data and the deviation of the vertical line calculated by the 360-order EGM2008 earth gravity field model in the embodiment of FIG. 2
The specific embodiment is as follows:
the detailed calculation model of the polar region vertical line deviation without theta singularity provided by the invention is used for calculating the ground vertical line deviation data of a certain domestic region by adopting a 360-order EGM2008 earth gravity field model, and comparing the ground vertical line deviation data with the measured value of the region, wherein the difference value of the result is shown in figure 2:
the difference statistics are shown in table 1:
table 1 vertical deviation calculated for 360 th order EGM2008 model and measured data difference result statistics (")
As can be seen from fig. 2, the meridian components are substantially-5 "to 25" and the unitary mortise component is-15 "to 10" except for a few points, which are the most of the difference ranges of the point values and the measured data.
As can be seen from table 1, the mean square error of the difference between the perpendicular deviation meridian component, the unitary mortise component and the measured data is approximately 5 "and 16", respectively.
Claims (1)
1. A modeling method of a detailed calculation model with no singularity on the deviation of polar region vertical lines,
the model is as follows:
in the formula, xi and eta respectively represent meridian components and mortise unitary components of vertical deviation; θ and λ represent the residual latitude and the geocentric longitude of the calculation point, respectively;representing a completely normalized earth disturbance gravity coefficient; n and m represent the order and order of the spherical harmonic coefficients, respectively;representing a fully normalized associative Legendre function;
the modeling method of the model is characterized by comprising the following steps of:
the first step: calculating to obtain a non-singular result of the Legendre function on the first derivative of theta;
by usingRepresenting an accompanying Legendre function, where P n (x) Is an n-order Legendre polynomial, and x is replaced by cos theta to obtain
The two ends derive theta to obtain
Because of the Legendre polynomial P n (x) Satisfies the following relationship
Combined with the formula to obtain
Normalizing the Legendre function in the formula to obtain a non-singular calculation formula of the Legendre function on the first derivative of theta, wherein the calculation formula is as follows:
and a second step of: calculating to obtain a class of polynomialsIs a non-singular result of (2);
the Legendre function satisfies the following accompanying Legendre equation
Thereby can be obtained
Continuing to derive theta at two ends to obtain
Substituting the formula into the formula to obtain the calculation formula of the second derivative of the Legendre function
Substituting the formula and the formula into the formula to obtain
According to the Legendre functionStandard forward push methods of (2) have
Thus, the first and second substrates are bonded together,
substituting the formula and the formula into the formula to obtain
Normalizing the Legendre function in the formula (20) to obtain a polynomialThe calculation formula without singularity is as follows:
and a third step of: substituting the formula into a traditional calculation model of vertical line deviation, and fully considering the conditions when m is equal to 0, 1 and other quantities, so as to establish a detailed calculation model of the polar region vertical line deviation without theta singularities;
and substituting the formulas (9), (10) and (20) into the formulas (1) to (3) to obtain the detailed calculation model without theta singularities with vertical deviation.
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| CN110727914B (en) * | 2019-09-30 | 2022-12-02 | 中国人民解放军战略支援部队信息工程大学 | Vector operation-based plumb line deviation single-point calculation method |
| CN111797360B (en) * | 2020-06-11 | 2024-03-26 | 南京信息工程大学 | Polynomial grid method for constructing sea area vertical line deviation model based on frequency domain characteristics |
| CN111829553A (en) * | 2020-06-18 | 2020-10-27 | 中国船舶重工集团公司第七0七研究所 | PC-104-based high-precision inertial navigation system disturbance gravity compensation method |
| CN111833208B (en) * | 2020-07-14 | 2022-03-11 | 宁夏大学 | Underground water reserve monitoring method and system based on vertical deviation disturbance |
| CN113885101B (en) * | 2021-09-28 | 2023-12-12 | 中国船舶重工集团公司第七0七研究所 | Method for constructing gravity gradient reference map based on ellipsoidal model |
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