A method and device for calculating earth gravity field, and storage medium

Abstract

The present invention discloses a method and device for calculating the earth's gravity field, and a storage medium, comprising: step S1, obtaining a first recursive term, a second recursive term, a first denormalized coefficient, and a second denormalized coefficient of a self-defined formula according to a spherical harmonic representation of the earth's gravity field and a trigonometric function representation of associated Legendre polynomials; step S2, obtaining a first recursive term, a second recursive term, a first denormalized coefficient, and a second denormalized coefficient of an improved recursive formula according to a recursive relationship in the associated Legendre polynomials; step S3, calculating a gravitational potential and a gravitational acceleration according to the first recursive term, the second recursive term, the first denormalized coefficient, and the second denormalized coefficient of the improved recursive formula.

Description

Technical Field

[0001] The present invention belongs to the technical field of earth gravity field, and in particular relates to an earth gravity field calculation method and device, and a storage medium. Background Art

[0002] The Earth's gravity field information is a basic information of natural science. It is widely used in modern science and technology and plays an important role in the development of modern space science and earth science. At present, the main Earth gravity field model is the spherical harmonic gravity field model, which can expand the Earth's gravity potential into a theoretically convergent full-order spherical harmonic series. Since the internal mass distribution of the Earth is unknown, the terrain coefficient cannot be calculated according to the definition and must be determined indirectly. In 1998, the United States published the EGM96 global gravity field model, which provided 360-order spherical harmonic gravity field coefficients. Subsequently, the EGM2008 global gravity field model published in 2008 increased the order to 2190-order spherical harmonic coefficients. The increase in the order of the coefficients of the spherical harmonic model has improved the calculation accuracy while also bringing a greater burden to the calculation. Summary of the invention

[0003] The technical problem to be solved by the present invention is to provide a method and device for calculating the earth's gravity field, and a storage medium.

[0004] To achieve the above object, the present invention adopts the following technical solution:

[0005] A method for calculating the earth's gravity field, comprising:

[0006] Step S1: Obtain the first recursive term of the self-defined formula based on the spherical harmonic representation of the earth's gravity field and the trigonometric function representation of the associated Legendre polynomials. , the second recursive term , the first denormalization coefficient and the second denormalization coefficient ;

[0007] Step S2: According to the recursive relationship in the associated Legendre polynomial, an improved recursive formula is obtained. , , and ;

[0008] Step S3: According to the improved recursive formula , , and , calculate the gravitational potential and gravitational acceleration.

[0009] Preferably, the gravitational potential is calculated The equation is:

[0010] ;

[0011] in, and is the order and degree of the spherical harmonic gravity field; is the standard gravitational constant; and are the mass of the Earth and the radius of the Earth respectively; , , and Both are improved recursive , , and .

[0012] Preferably, the equation for calculating the acceleration due to gravity is:

[0013] ;

[0014] in, , , , , , , , , , , Both are improved recursive , , , , , , , , , , , , , are the gravitational accelerations of corresponding orders respectively.

[0015] The present invention also provides an earth gravity field calculation device, comprising:

[0016] The first calculation module is used to obtain the first recursive term of the self-defined formula according to the spherical harmonic representation of the earth's gravity field and the trigonometric function representation of the associated Legendre polynomials. , the second recursive term , the first denormalization coefficient and the second denormalization coefficient ;

[0017] The second calculation module is used to obtain the improved recursive formula according to the recursive relationship in the associated Legendre polynomials. , , and ;

[0018] The second calculation module is used to improve the recursive formula , , and , calculate the gravitational potential and gravitational acceleration.

[0019] The present invention also provides a storage medium, on which a computer program is stored, and the computer program executes the method for calculating the earth's gravity field when running.

[0020] The present invention also provides a storage medium, on which a computer program is stored, and the computer program executes the method for calculating the earth's gravity field when running.

[0021] Aiming at the rapidly growing factorial terms in the coefficients appearing in the original recursive relationship, the present invention uses an improved spherical harmonic function recursive algorithm to accurately calculate the earth's gravitational potential and gravitational acceleration. Specifically, a recursive relationship is improved. Compared with Montenbruck's recursive relationship, the calculation of factorial terms in the calculation process is improved, and the rapidly growing factorial terms in the original method are improved, avoiding numerical problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on the provided drawings without paying creative work.

[0023] Figure 1 The figure is a flow chart of a method for calculating the earth's gravity field according to an embodiment of the present invention. DETAILED DESCRIPTION

[0024] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0025] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and understandable, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0026] Embodiment 1:

[0027] like Figure 1 As shown, an embodiment of the present invention provides a method for calculating the earth's gravity field, which uses a new recursive relationship to calculate the gravity potential and gravity acceleration of a spherical harmonic gravity field of any model and any order, including:

[0028] Step S1: Obtain the first recursive term of the self-defined formula based on the spherical harmonic representation of the earth's gravity field and the trigonometric function representation of the associated Legendre polynomials. , the second recursive term , the first denormalization coefficient and the second denormalization coefficient ;

[0029] Step S2: According to the recursive relationship in the associated Legendre polynomial, an improved recursive formula is obtained. , , and ;

[0030] Step S3: According to the improved recursive formula , , and , calculate the gravitational potential and gravitational acceleration.

[0031] As an implementation of an embodiment of the present invention, in step S1,

[0032] Establish equations for calculating gravitational potential and gravitational acceleration using spherical harmonics:

[0033] ;

[0034] in, is the gravitational potential; is the acceleration due to gravity; and is the order and degree of the spherical harmonic gravity field; is the standard gravitational constant; and are the mass of the Earth and the radius of the Earth respectively; Respectively represent the length, longitude, and latitude in spherical coordinates; is the potential coefficient of the first gravity field, is the potential coefficient of the second gravitational field.

[0035] Since the potential coefficient of the first gravitational field and the potential coefficient of the second gravity field This covers a range of ten or a dozen orders of magnitude, and even for a small model the first normalized coefficients are usually given. and the second normalization coefficient . and , and There are the following relationships:

[0036] ;

[0037] in, is the Kronecker notation, when hour, , in other cases .

[0038] It is a trigonometric representation of the associated Legendre polynomials, which transforms the longitude dependence of the angle function into a single recursive term. This allows the Earth potential and the resulting acceleration to be efficiently calculated in the satellite Cartesian coordinates (x, y, z).

[0039] Based on this definition, the longitude-dependent recursive term is introduced to remove the factorial operation in the recursive process. and Distinguish and customize it as the first recursive item and the second recursive term , which is expressed as follows:

[0040] ;

[0041] The factorial term in the formula is defined separately as , which is expressed as follows:

[0042] ;

[0043] Will and Combined with the factorial term in the above formula, we get the first new denormalization coefficient and the second new denormalization coefficient as follows:

[0044] ;

[0045] As an implementation of the embodiment of the present invention, in step S2,

[0046] Given the definition of associated Legendre polynomials:

[0047] ;

[0048] in, is the trigonometric form of Legendre polynomials, which are defined as follows:

[0049] ;

[0050] It has the following recursive properties:

[0051] ;

[0052] According to the above recursive relationship, the combination step S1 gives and The definition of can be deduced , There is the following recursive relationship:

[0053] ;

[0054] Montenbruck's recursive term and The recursive method is improved by introducing two new intermediate variables and , these two new intermediate quantities and The recursive relationship is as follows:

[0055] Since the Legendre polynomials above have a recursive property, and When , its definition contains Legendre polynomials, so we can deduce and The improved recursive formula is used to calculate different orders in actual calculations. and of and When , the improved recursive formula is used for calculation instead of the custom formula. The specific expression is as follows:

[0056] ;

[0057] make , the above equation can be simplified into the following form:

[0058] ;

[0059] Compared with Montenbruck's recursive relation, so far, the improved recursive calculation and The factorial terms with medium to high growth rates have been completely removed, and the improved recursive calculation and The coefficient of the term can be calculated by two multiplications and one division. To further improve the computational efficiency, the entire coefficient (including the R term) can be calculated in each and The value is calculated in advance, and only one multiplication is required for each subsequent calculation. Therefore, the calculation efficiency has been greatly improved.

[0060] Similarly, the first denormalized coefficient of the improved recursive formula is and the second denormalization coefficient The normalization coefficient number stored in the gravity field model file , The relationship is:

[0061] ;

[0062] Here, the factorial term is defined separately as , the expression is as follows:

[0063] ;

[0064] It has a recursive property, so it can be calculated as follows:

[0065] ;

[0066] ;

[0067] in, is the Kronecker notation, when = hour, =1, otherwise = 0. Using any of the three recursive formulas above, you can construct an "enhanced" array by traversing the normalization coefficients diagonally, vertically, or horizontally. Depending on the value range, you can use the following method to convert m Set to 1, and simplify the recurrence relationship by approximation and calculate , further reducing the number of calculation steps.

[0068] Improved for:

[0069] ;

[0070] As an implementation of the embodiment of the present invention, in step S3,

[0071] Calculate the gravitational potential by recursion relation The equation is:

[0072] ;

[0073] in, , , and Both are improved recursive , , and , which improves the variables in the recursive algorithm given by Montenbruck , , , .

[0074] This extra complexity comes from the inclusion of an extra factor in the denormalized gravity model coefficients, intended to simplify the calculation of the acceleration. If one wishes to maintain simplicity in the potential function, then this factor can be simply removed and middle.

[0075] Following the acceleration calculation method given by Montenbruck, a new recursive relationship is obtained to calculate the equation for gravitational acceleration:

[0076] ;

[0077] ;

[0078] in, , , , , , , , , , , Both are improved recursive , , , , , , , , , , , , , are the gravitational accelerations of corresponding orders respectively, and the total gravitational acceleration is the sum of the accelerations of each order.

[0079] The embodiments of the present invention have the following characteristics:

[0080] 1. The improved new recursive relationship of the standardized constant of the present invention is more concise and easier to operate.

[0081] 2. The present invention can calculate any order of any gravity field model, the numerical problems that may occur when calculating high-order gravity fields are solved, and the reliability of the calculation is improved.

[0082] 3. Compared with the previous recursive algorithm, the present invention improves the items that will be calculated multiple times in one integration step, reduces the complexity of the recursive relationship calculation, and improves the operation efficiency.

[0083] 4. The steps of calculating acceleration in the present invention can be simplified to use fewer multiplications and no division is required. For occasions where acceleration is frequently calculated, such as the single-step integration method of the Runge-Kutta series for satellite dynamics orbit integral prediction, the present invention can significantly reduce its computational complexity.

[0084] Embodiment 2:

[0085] The embodiment of the present invention further provides an earth gravity field calculation device, comprising:

[0086] The first calculation module is used to obtain the first recursive term of the self-defined formula according to the spherical harmonic representation of the earth's gravity field and the trigonometric function representation of the associated Legendre polynomials. , the second recursive term First denormalization coefficient and the second denormalization coefficient ;

[0087] The second calculation module is used to obtain the improved recursive formula according to the recursive relationship in the associated Legendre polynomials. , , and ;

[0088] The second calculation module is used to improve the recursive formula , , and , calculate the gravitational potential and gravitational acceleration.

[0089] Embodiment 3:

[0090] The present invention also provides a storage medium, on which a computer program is stored, and the computer program executes the method for calculating the earth's gravity field when running.

[0091] The embodiments described above are only descriptions of the preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.

Claims

1. A method for calculating the earth's gravity field, characterized in that: include: Step S1: According to the spherical harmonic representation of the earth's gravity field and the trigonometric function representation of the associated Legendre polynomials, the first recursive term V' of the self-defined formula is obtained. nm , the second recursive term W' nm , the first denormalization coefficient C' nm and the second denormalization coefficient S' nm ; Step S2: According to the recursive relationship in the associated Legendre polynomial, the improved recursive formula V' is obtained. nm , W' nm , C' nm and S' nm ; Step S3: V' according to the improved recursive formula nm , W' nm , C' nm and S' nm , calculate the gravitational potential and gravitational acceleration; In step S2, Where r represents the length in spherical coordinates, n and m are the order and degree of the spherical harmonic gravity field, R is the radius of the earth, make The above equation can be simplified into the following form: Improved recursive formula of the first denormalized coefficient C' nm and the second denormalization coefficient S' nm Normalization coefficients stored in the gravity field model file The relationship is: Here, the factorial term is defined separately as g nm , the expression is as follows: g nm It has a recursive property and is calculated using the following method: Among them, δ nm is the Kronecker notation. When n=m, δ nm =1, in other cases δ nm =0; Set m to 1 and calculate g by simplifying the recurrence relation by approximation nm , the improved g n1 for: g n1 ≈2 n log2n; In step S3, the equation for the gravitational potential V is calculated: Where G is the standard gravitational constant; M is the mass of the Earth; V' nm , W' nm , C' nm and S' nm All are improved recursive V' nm , W' nm , C' nm and S' nm ; The equation for calculating the acceleration due to gravity is: Among them, C′ n0 , V′ n+1，1 , C′ nm , V′ n+1，m+1 , V′ n+1，m-1 , S′ nm , W′ n+1，m+1 , W′ n+1，m-1 , W′ n+1，1 , W′ n+1，m+1 , W′ n+1，m-1、 V′ n+1，m+1 , V′ n+1，m-1 , V′ n+1，m , W′ n+1，m Both are improved recursive formulas of C′ n0 , V′ n+1，1、 C′ nm , V′ n+1，m+1 , V′ n+1,m-1 , S′ nm , W′ n+1,m+1 , W′ n+1，m-1 , W′ n+1,1 , W′ n+1,m+1 , W′ n+1,m-1 , V′ n+1,m+1 , V′ n+1,m-1 , V′ n+1,m , W′ n+1,m , are the gravitational accelerations of corresponding orders respectively.

2. An earth gravity field calculation device for implementing the earth gravity field calculation method according to claim 1, characterized in that: include: The first calculation module is used to obtain the first recursive term V' of the self-defined formula according to the spherical harmonic representation of the earth's gravity field and the trigonometric function representation of the associated Legendre polynomials. nm , the second recursive term W' nm , the first denormalization coefficient C' nm and the second denormalization coefficient S' nm ; The second calculation module is used to obtain the improved recursive formula V' according to the recursive relationship in the associated Legendre polynomial nm , W' nm , C' nm and S' nm ; The second calculation module is used to calculate V' according to the improved recursive formula nm , W' nm , C' nm and S' nm , calculate the gravitational potential and gravitational acceleration.

3. A storage medium, characterized in that: The storage medium stores a computer program, which executes the method for calculating the earth's gravity field as claimed in claim 1 when running.

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