Ellipsoidal harmonic-based full tensor magnetic gradient model computation method without singularity

Abstract

The application is a kind of singular-free full tensor magnetic gradient model calculation method based on ellipsoidal harmonic function, which establishes an accurate full tensor magnetic gradient model to provide an effective natural coordinate system for auxiliary navigation. The application obtains full tensor magnetic gradient by calculating the second order derivative of the magnetic potential equation based on ellipsoidal harmonic function expansion. Firstly, the second type of semi-normalized Legendre function in the gradient expression is reorganized, and the first and second order derivatives of the first and second type of semi-normalized Legendre are calculated. In order to meet the singularity-free, the full tensor magnetic gradient Vyy, Vxy, Vxz containing sine function in the denominator is reconstructed. Finally, the singular-free full tensor magnetic gradient modeling based on ellipsoidal harmonic function is realized. The application can solve the technical problem that the full tensor gradient cannot be constructed by using the measured data. The ellipsoidal harmonic method is more consistent with the real shape of the earth, which can avoid the technical bottleneck problem of function divergence when calculating the polar region position due to the assumption that the earth is a complete sphere, and can provide strong support for auxiliary navigation.

Description

Technical Field

[0001] This invention belongs to the field of Earth magnetic gradient measurement, specifically a method for calculating a singularity-free full tensor magnetic gradient model based on ellipsoidal harmonic functions. Background Technology

[0002] The full tensor magnetic gradient has significant application value in the field of navigation aids. The geomagnetic field is the physical field of the Earth's space environment, and its physical information is inherent, passive information of the Earth. The full tensor magnetic gradient of any point in near-Earth space corresponds to the latitude, longitude, and altitude of that point, providing a natural coordinate system for navigation. Therefore, establishing a full tensor magnetic gradient model is of great practical significance for the field of navigation aids.

[0003] Currently, large-scale aeromagnetic full-tensor magnetic gradient measurements have not been widely implemented, and the fundamental conditions for constructing a high-precision magnetic gradient benchmark database using measured full-tensor geomagnetic gradient data remain incomplete. Most methods for converting upper-air satellite magnetic field models into full-tensor magnetic gradient models employ spherical harmonic function expansions. However, these methods assume the Earth is a perfect sphere, leading to poor convergence when calculating the location in the outer space of the Brillouin sphere. Furthermore, when calculating pole locations, the sine function in the denominator of the full-tensor magnetic gradient is zero, resulting in singularities. Therefore, using an ellipsoidal harmonic model can better represent the spatial distribution of the full-tensor geomagnetic gradient and exhibits good convergence, but this method requires more solution coefficients and also suffers from the singularity problem. Summary of the Invention

[0004] The technical problem to be solved by this invention is to provide a method for calculating the non-singularity full tensor magnetic gradient model based on ellipsoidal harmonic functions, thereby solving the problem of singularities in gradient calculation at pole locations.

[0005] This invention is implemented as follows:

[0006] A method for calculating a singularity-free full tensor magnetic gradient model based on ellipsoidal harmonics, the method comprising:

[0007] Calculate the first derivative of the magnetic field potential function based on the ellipsoidal harmonic function expansion at any point outside the Earth in the ellipsoidal coordinate system to obtain the ellipsoidal harmonic function expression of the magnetic vector. Then calculate the derivative of each component of the ellipsoidal harmonic function expression to obtain the expression of each component of the full tensor magnetic gradient.

[0008] Based on the expressions for each component of the full tensor magnetic gradient, a singular-free full tensor magnetic gradient map constructed using ellipsoidal harmonics is plotted by inputting latitude, longitude, and altitude on a global or regional scale.

[0009] Furthermore, the magnetic potential field V(μ,δ,λ) at any point outside the Earth is calculated using formula (1) in the ellipsoidal coordinate system:

[0010]

[0011] In the ellipsoidal coordinate system, μ is the semi-minor axis of the confocal ellipsoid, λ is the longitude, and δ is the colatitude; R = 6371.2 km is the conventional geomagnetic reference radius; i 2 =-1; N is the maximum truncation order of the model expansion; Let represent the Gaussian ellipsoidal harmonic coefficients, and n and m represent the order and degree of the ellipsoidal harmonic coefficients, respectively; where and These are the first-class semi-normalized Legendre functions and the second-class semi-normalized Legendre functions, respectively.

[0012] Furthermore, the components V of the full tensor magnetic gradient are calculated using formulas (2)-(7). xx V yy V zz V xy V xz V yz :

[0013]

[0014]

[0015]

[0016]

[0017]

[0018]

[0019] in E is the linear eccentricity; in the expression... These are the renormalized second-type semi-normalized Legendre function and its first and second derivatives, respectively.

[0020] Furthermore, the renormalized second-type semi-normalized Legendre function is calculated as follows:

[0021]

[0022] After calculating and rearranging formula (8), the final renormalized second-type semi-normalized Legendre function is:

[0023] In the formula, k is the integer exponent of the hypergeometric series;

[0024] Calculate the first derivative of the seminormalized Legendre function of the second kind. for:

[0025]

[0026] Second derivative of the seminormalized Legendre function of the second kind for:

[0027]

[0028] Furthermore, the first type of semi-normalized Legendre function

[0029]

[0030] in The Kroneker symbol represents...

[0031]

[0032] The first-order derivative of the seminormalized Legendre of the first kind is:

[0033]

[0034] The second derivative of the first-kind seminormalized Legendre is:

[0035]

[0036] Replace the magnetic gradient component V of the full tensor according to formulas (16)-(18). yy The full tensor magnetic gradient component V xy The full tensor magnetic gradient component V yz In, there exist singular terms in:

[0037]

[0038]

[0039]

[0040] Furthermore, the Gaussian ellipsoidal harmonic coefficients are obtained through the spherical harmonic coefficients. The transformed result includes the following: The transformation relationship is as follows:

[0041]

[0042] in Represents the ellipsoidal harmonic coefficients; Let represent the spherical harmonic coefficients; S is the integer part of nm / 2; 0 ≤ p ≤ S; where g and h have the same conversion relationship, which can be simplified as:

[0043]

[0044]

[0045]

[0046] Compared with the prior art, the beneficial effects of this invention are as follows:

[0047] This invention, based on a global magnetic field model, utilizes a singularity-free full tensor magnetic gradient model based on ellipsoidal harmonics to calculate the full tensor magnetic gradient at arbitrary latitude, longitude, and altitude. This method effectively avoids singular values ​​at pole locations during calculation. Attached Figure Description

[0048] Figure 1 This is a model diagram of Vxx at an altitude of 0.2km according to the present invention;

[0049] Figure 2 This is a model diagram of Vyy at an altitude of 0.2km for the present invention;

[0050] Figure 3 This is a model diagram of Vzz at an altitude of 0.2km according to the present invention;

[0051] Figure 4 This is a model diagram of Vxy at an altitude of 0.2km for this invention;

[0052] Figure 5 This is a model diagram of Vxz at an altitude of 0.2km according to the present invention;

[0053] Figure 6 This is a model diagram of Vyz at an altitude of 0.2km according to the present invention. Detailed Implementation

[0054] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0055] The total tensor magnetic gradient on the Earth's surface varies with location for two main reasons: (1) the Earth is not a regular sphere, but an ellipsoid, and the surface is uneven; (2) the density distribution of the crust and its surrounding material is uneven.

[0056] The full tensor magnetic gradient is the second derivative of magnetic potential and is more sensitive to shallow materials near the Earth's surface than to magnetic fields. As the order of the magnetic field derivative increases, the source effect attenuates faster at deeper layers. Therefore, the full tensor magnetic gradient better reflects changes in the magnetic field of materials near the Earth's surface. Thus, in geophysical exploration, magnetic anomalies are used to study the distribution of magnetic materials, and vertical or horizontal gradients of magnetic anomalies are used to study the distribution of local magnetic materials.

[0057] This invention provides a method for calculating a singularity-free full tensor magnetic gradient model based on ellipsoidal harmonics, the method comprising:

[0058] Calculate the first derivative of the magnetic field potential function based on the ellipsoidal harmonic function expansion at any point outside the Earth in the ellipsoidal coordinate system to obtain the ellipsoidal harmonic function expression of the magnetic vector. Then calculate the derivative of each component of the ellipsoidal harmonic function expression to obtain the expression of each component of the full tensor magnetic gradient.

[0059] Based on the expressions for each component of the full tensor magnetic gradient, a singular-free full tensor magnetic gradient map constructed using ellipsoidal harmonics is plotted by inputting latitude, longitude, and altitude on a global or regional scale.

[0060] In the ellipsoidal coordinate system, the magnetic potential field V(μ,δ,λ) at any point outside the Earth is calculated using formula (1):

[0061]

[0062] In the ellipsoidal coordinate system, μ is the semi-minor axis of the confocal ellipsoid, λ is the longitude, and δ is the colatitude; R = 6371.2 km is the conventional geomagnetic reference radius; i 2 =-1; N is the maximum truncation order of the model expansion; Let represent the Gaussian ellipsoidal harmonic coefficients, and n and m represent the order and degree of the ellipsoidal harmonic coefficients, respectively; where and These are the first-class semi-normalized Legendre functions and the second-class semi-normalized Legendre functions, respectively.

[0063] The components V of the full tensor magnetic gradient are calculated using formulas (2)-(7). xx V yy V zz V xy V xz V yz :

[0064]

[0065]

[0066]

[0067]

[0068]

[0069]

[0070] in E is the linear eccentricity; in the expression... These are the renormalized second-type semi-normalized Legendre function and its first and second derivatives, respectively.

[0071] In the above calculations, to obtain the component calculations, we need 1. the renormalized second-type semi-normalized Legendre function, and its first and second derivatives, to meet the calculation requirements of each component of the full tensor magnetic gradient.

[0072] 2. Calculate the first and second derivatives of the semi-normalized Legendre function of the first kind. To ensure that the equation does not contain singular values ​​when calculating the poles, the full tensor magnetic gradient expressions Vyy, Vxy, and Vxz containing sine functions in the denominator are replaced with expressions that do not contain sine functions in the denominator, thus satisfying the requirement of singularity-free calculation of the full tensor magnetic gradient;

[0073] 3. Based on spherical harmonic coefficients With ellipsoidal harmonic coefficient The transformation relationship is obtained, and the transformed ellipsoidal harmonic coefficients are calculated. The obtained ellipsoidal harmonic coefficients are then substituted into the expressions of each component of the full tensor magnetic gradient.

[0074] Among them: the renormalized second-kind semi-normalized Legendre function, and its first and second derivatives, including:

[0075] Since the second-kind semi-normalized Legendre function contains complex numbers, to simplify the calculation, the second-kind semi-normalized Legendre function is renormalized, and its first and second derivatives are calculated to meet the calculation requirements of each component of the full tensor magnetic gradient. Let... This represents the renormalized second-type semi-normalized Legendre function and its first and second derivatives.

[0076] The renormalized second-type semi-normalized Legendre function can be represented by the following form:

[0077]

[0078] After calculating and rearranging the above formula, the final renormalized semi-normalized Legendre function of the second kind is expressed as follows:

[0079] In the formula, k is the integer exponent of the hypergeometric series.

[0080] The first derivative of the seminormalized Legendre function of the second kind Represented as:

[0081]

[0082] Second derivative of the seminormalized Legendre function of the second kind Represented as:

[0083]

[0084] Calculate the first and second derivatives of the semi-normalized Legendre function of the first kind, including:

[0085] The first type of semi-normalized Legendre function is recursively expanded using P. n The differential equation identity of (x) is simplified and analyzed based on the special values ​​of m, leading to the derivation of the non-singularity calculation formula for the geomagnetic tensor ellipsoidal harmonic function. The denominator of the expression no longer contains the sine function of the geocentric colatitude, thus solving the problem of V. yy V xy V yz There exists The problem of singular value computation, the first kind of seminormalized Legendre function and its first and second derivatives The results were obtained through the following calculations:

[0086] The first type of seminormalized Legendre function

[0087]

[0088] in The Kroneker symbol represents...

[0089]

[0090] The first-order derivative of the seminormalized Legendre of the first kind is expressed as:

[0091]

[0092] The second derivative of the first-kind seminormalized Legendre is expressed as:

[0093]

[0094] (1) Expression of the ellipsoidal harmonics of the full tensor magnetic gradient V yy V xy Vyz In the equation, the singular term that exists in the denominator is:

[0095]

[0096]

[0097]

[0098] According to the spherical harmonic coefficient With ellipsoidal harmonic coefficient The transformation relationship is determined, and the harmonic coefficients of the transformed ellipsoid are calculated.

[0099] (1) Spherical harmonic coefficient With ellipsoidal harmonic coefficient The conversion relationship is as follows:

[0100]

[0101] in Represents the ellipsoidal harmonic coefficients; Let represent the spherical harmonic coefficients; S is the integer part of nm / 2; 0 ≤ p ≤ S; where g and h have the same conversion relationship, which can be simplified as:

[0102]

[0103]

[0104]

[0105] Based on the Laplace equation, this invention holds that Vxx + Vyy + Vzz = 0. Therefore, the closer Vxx + Vyy + Vzz is to 0 numerically, the higher the calculation accuracy of the singularity-free formula. Using the method of this invention, a singularity-free full tensor magnetic gradient calculation based on ellipsoidal harmonics is performed. An altitude of 0.2 km is set, and a full tensor magnetic gradient model at an altitude of 0.2 km is established (e.g., ...). Figure 1 Vxx; such as Figure 2 Vyy; as Figure 3 Vzz; as Figure 4 Vxy; as Figure 5 Vxz; as Figure 6 (Vyz). The sum of the three components of the full tensor magnetic gradient, Vxx, Vyy, and Vzz, varies from -9.63E-15 to 9.91E-15 nT / km. Therefore, the full tensor geomagnetic gradient obtained using the method of this invention can be considered to satisfy the Laplace equation. Furthermore, each component of the full tensor magnetic gradient obtained by the above method exhibits good orthogonality in the pole region.

[0106] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A singular-free full-tensor magnetic gradient model calculation method based on ellipsoidal harmonics, characterized in that, The method includes: Calculate the first derivative of the magnetic field potential function based on the ellipsoidal harmonic function expansion at any point outside the Earth in the ellipsoidal coordinate system to obtain the ellipsoidal harmonic function expression of the magnetic vector. Then calculate the derivative of each component of the ellipsoidal harmonic function expression to obtain the expression of each component of the full tensor magnetic gradient. Based on the expressions of each component of the full tensor magnetic gradient, by inputting the latitude, longitude and altitude of the global or regional range, a singular-free full tensor magnetic gradient map constructed based on ellipsoidal harmonic functions is drawn. Wherein: the magnetic potential field of any point outside the earth is calculated by formula (1) in the ellipsoidal coordinate system : （1）， In the ellipsoidal coordinate system, μ is the semi-minor axis of the confocal ellipsoid, λ is the longitude, and δ is the colatitude; R = 6371.2 km is the conventional geomagnetic reference radius. =-1; N is the maximum truncation order of the model expansion; Let represent the Gaussian ellipsoidal harmonic coefficients, and n and m represent the order and degree of the ellipsoidal harmonic coefficients, respectively; where and These are the first-class semi-normalized Legendre functions and the second-class semi-normalized Legendre functions, respectively. The full tensor magnetic gradient components are calculated by equations (2)-(7) , , , , , : （2）， （3）， （4）， （5）， （6）， （7）， wherein , ; E is the linear eccentricity; the expression , , are the second kind of semi-normalized Legendre functions and their first and second derivatives, respectively, after regularization. The renormalized second-order semi-normalized Legendre function is calculated as follows: （8）， The formula (8) is calculated and arranged, and the second type of semi-normalized Legendre function is , （9）， In the formula, k is the integer exponent of the hypergeometric series; Computing the first derivative of the second kind of semi-normalized Legendre functions is: （10）， Second derivative of the second kind of semi-normalized legendre functions is: （11）， The first type of seminormalized Legendre function （12）， wherein is the Kroneker symbol, denoted as （13）， The first-order derivative of the seminormalized Legendre of the first kind is: （14）， The second derivative of the first-kind seminormalized Legendre is: （15）， Replace the full tensor magnetic gradient components according to formulas (16)-(18). Full tensor magnetic gradient components Full tensor magnetic gradient components In, there exist singular terms ; ; ;in: （16）， （17）， （18）， Gauss ellipsoidal harmonic coefficients are derived from the spherical harmonic coefficients , are obtained after conversion, specifically including: converting the relationship （19）， wherein denotes an ellipsoidal harmonic coefficient; denotes a spherical harmonic coefficient; S is the integer part of n-m / 2; 0≤p≤S; wherein the conversion relationship of g, h is the same, and is simply written as: （20）， （21）， （22）。

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