CN106646648A - Two-step integration direct method for determining ground level plane based on band-limited airborne vector gravity - Google Patents

Abstract

The invention relates to a two-step integration direct method for determining ground level plane based on band-limited airborne vector gravity, and the main technical characteristics of the method are that the method comprises the steps: calculating a band-limited airborne disturbing potential based on the band-limited airborne vector gravity according to a generalized horizontal boundary value theory; directly calculating a band-limited marine disturbing potential based on the band-limited airborne disturbing potential according to the Dirichlet theory, and converting the band-limited marine disturbing potential into the ground level plate through a Bruns formula. The method is reasonable in design, provides a new solution for the calculation of the ground level plane based on the airborne vector gravity, and achieves a function of determining the ground level plane based on the airborne vector gravity. The precision of the obtained ground level plane meets the demands of engineering application.

Description

The two steps integration direct method that aviation vector gravitational determines geoid is limited based on band

Technical field

The invention belongs to airborne vector gravimetry technical field, especially a kind of to be determined based on band limit aviation vector gravitational The two steps integration direct method of geoid.

Background technology

Airborne vector gravimetry technology is with aircraft as carrier, with inertial navigation system and global navigation satellite positioning system The inertial acceleration information than force information and carrier is united as sensor and obtains respectively, by the joint such as Coordinate Conversion and filtering Calculate the vector gravitational on aviation height.Geoid is an occluding surface for representing the figure of the earth, be defined as with entirely Ball without the optimal closely sealed terrestrial gravitation equipotential surface of the static mean sea level of tide, while be also can reflect earth's internal structure with it is close The physical surface of degree distribution characteristicss.Determine that geoid is one of main purpose of airborne vector gravimetry.

At present, determine that the research of geoid is adopted similar to astronmical leveling principle based on aviation vector gravitational, by The horizontal component of vector gravitational is converted into survey line section integration the disturbing potential on course line, by disturbing potential downward continuation to big ground water Behind quasi- face, then undulation of the geoid is calculated using Bruns formula, but, what said method was obtained is level surface relative to the earth, Need to add and be only geoid after benchmark, it is difficult to meet the needs of through engineering approaches application.

The content of the invention

It is an object of the invention to overcome the deficiencies in the prior art, there is provided one kind is determined big based on band limit aviation vector gravitational The two steps integration direct method of ground-level, realizes that aviation vector gravitational determines the computing function of geoid and the earth for obtaining Level surface precision meets through engineering approaches application demand.

The present invention solves existing technical problem and takes technical scheme below to realize：

It is a kind of that the two steps integration direct method that aviation vector gravitational determines geoid is limited based on band, comprise the following steps：

It is step 1, theoretical based on the horizontal boundary values of broad sense, calculate band limit boat using band limit aviation vector gravitational on enroute altitude Empty disturbing potential；

It is step 2, theoretical using Dirichlet, band limit sea disturbance position is directly calculated based on band limit aviation disturbing potential, pass through Bruns formula will be converted into geoid with limit sea disturbance position.

Further, the formula of step 1 calculating with limit aviation disturbing potential is：

In formula：T^b(r, θ, λ) is band limit aviation disturbing potential, and r is the earth's core at aviation calculating point to footpath, and θ and λ is aviation meter The colatitude and longitude at point is calculated, b is band limit order, and GM is Gravitational coefficient of the Earth, and R is earth radius, and L is far field truncation funcation Maximum order, l is the reference to gravitational field model order removed, C_n(H,ψ₀) it is band limit aviation Vector operation band limit aviation disturbing potential Far field truncation funcation, H be aviation height, ψ₀It is to integrate radius, T_n(θ, λ) is the Laplace harmonic functions of disturbing gravity position, π It is pi, N is measuring point number in integration radius,WithRespectively band limit aviation vector gravitational measuring point j North-south component and thing component,It is the north-south component with limit aviation vector gravitational measuring point j and thing point Amount calculates the kernel function with limit aviation disturbing potential, ψ_jBe observation station and calculate point between spherical angle away from Δ σ_jIt is integration unit plane Product.

Further, the north-south component with limit aviation vector gravitational measuring point j and thing component calculate band limit aviation disturbance The kernel function of positionComputing formula be：

θ in formula_jAnd λ_jIt is the colatitude and longitude at aviation measuring point j, P_n(cosψ_j) it is Legendre function.

Further, the far field truncation funcation C that aviation disturbing potential is limited with limit aviation Vector operation band_n(H,ψ₀) calculating Formula is：

In formula, R_nm(ψ₀) be Legendre function integral function, be expressed as：

Further, the step 2 directly calculates the formula with sea disturbance position is limited and is based on band limit aviation disturbing potential：

In formula：T^b(R, θ, λ) is that band limits sea disturbance position, Q_n(H,ψ₀) it is that band limit aviation disturbing potential calculating band limit sea is disturbed The far field truncation funcation of dynamic position, T^b(r,θ_j,λ_j) limit aviation disturbing potential, Y for band at measuring point j^b(R,ψ_j, r) it is band limit boat at measuring point j Empty disturbing potential calculates the kernel function with limit sea disturbance position；

Wherein, kernel function Y with limit sea disturbance position is calculated with limit aviation disturbing potential at measuring point j^b(R,ψ_j, calculating r) is public Formula is：

Wherein, the far field truncation funcation Q with limit sea disturbance position is calculated with limit aviation disturbing potential_n(H,ψ₀) computing formula For：

Further, the step 2 is based on the formula that geoid is calculated with limit sea disturbance position：

In formula：N^b(R, θ, λ) is geoid, and γ is normal gravity.

Advantages of the present invention and good effect are：

The present invention is first depending on the horizontal boundary values theory of broad sense, calculates band limit aviation based on band limit aviation vector gravitational and disturbs Position；Then it is theoretical according to Dirichlet, band limit sea disturbance position is directly calculated based on band limit aviation disturbing potential, it is public by Bruns Formula will be converted into geoid with limit sea disturbance position.The method is provided to calculate geoid based on aviation vector gravitational New solution；By experimental verification, the precision of the geoid obtained based on the method can meet engineer applied Demand.

Description of the drawings

Fig. 1 a are 4km band limit aviation disturbing gravities north and south horizontal component schematic diagram；

Fig. 1 b are 4km band limit aviation disturbing gravity thing horizontal component schematic diagrams；

Fig. 2 is geoid schematic diagram；

Fig. 3 is the difference value schematic diagram of the geoid that the present invention is obtained and standard value.

Specific embodiment

The embodiment of the present invention is further described below in conjunction with accompanying drawing.

The present invention is a kind of new method that geoid is calculated based on aviation vector gravitational data, main to include in following Hold：It is theoretical according to the horizontal boundary values of broad sense, band limit aviation disturbing potential is calculated based on band limit aviation vector gravitational；According to Dirichlet Theory, based on band limit aviation disturbing potential band limit sea disturbance position is directly calculated, and will be turned with limit sea disturbance position by Bruns formula Turn to geoid.

To make the objects, technical solutions and advantages of the present invention become more apparent, develop simultaneously embodiment pair below in conjunction with accompanying drawing Present invention elaborates.

Using the air strips limit aviation vector gravitational north and south of the global order of high-order gravity field model EGM2008 simulation calculations 2160 Horizontal componentWith thing horizontal componentBased on data, the flying height of the aviation vector gravitational is 4000 meters, area The latitude in domain is 36 ° to 41 °, and longitude is 247 ° to 252 °, and resolution is 2.0 '.In order to simulate airborne vector gravimetry data The error of result, band limit aviation vector gravitational horizontal component is introduced respectively observation error white noise error (σ=± 3mGal)。

Table 1 is with limit aviation horizontal component statistical table/mGal

Fig. 1 a and Fig. 1 b sets forth 4km bands limit aviation disturbing gravity north and south horizontal component and 4km band limit aviation disturbances Gravity thing horizontal component.

The effectiveness of direct method, base are integrated in order to check based on two steps that geoid is determined with limit aviation vector gravitational Corresponding geoid N is calculated in EGM2008 gravity field models^b(R, θ, λ), zoning is 38 ° to 39 °, and longitude is 249 ° to 250 °, resolution is 2.0 '.

The geoid of table 2 statistics/centimetre

Type	Minima	Maximum	Meansigma methodss	Standard deviation	Middle error	Number
							Geoid	-46.516	50.115	2.878	19.533	19.744	900

Fig. 2 gives geoid schematic diagram.

This aviation vector gravitational data are calculated using the present invention calculate concretely comprising the following steps for geoid：

It is step 1, theoretical based on the horizontal boundary values of broad sense, calculate band limit boat using band limit aviation vector gravitational on enroute altitude Empty disturbing potential, specific formula for calculation is：

In formula：T^b(r, θ, λ) is band limit aviation disturbing potential, and r is the earth's core at aviation calculating point to footpath, and θ and λ is aviation meter The colatitude and longitude at point is calculated, b is band limit order, and GM is Gravitational coefficient of the Earth, and R is earth radius, and L is far field truncation funcation Maximum order, l is the reference to gravitational field model order removed, C_n(H,ψ₀) it is band limit aviation Vector operation band limit aviation disturbing potential Far field truncation funcation, H be aviation height, ψ₀It is to integrate radius, T_n(θ, λ) is the Laplace harmonic functions of disturbing gravity position, π It is pi, N is measuring point number in integration radius,WithRespectively band limit aviation vector gravitational measuring point j North-south component and thing component,It is the north-south component with limit aviation vector gravitational measuring point j and thing point Amount calculates the integration kernel function with limit aviation disturbing potential, ψ_jIt is the spherical angle that calculates between point of aviation measuring point and aviation away from Δ σ_jIt is Integration unit area.

Wherein, the north-south component and thing component with limit aviation vector gravitational measuring point j calculates the product with limit aviation disturbing potential Pyrene functionComputing formula be：

θ in formula_jAnd λ_jIt is the colatitude and longitude at aviation measuring point j, P_n(cosψ_j) it is Legendre function.

Wherein, the far field truncation funcation C of aviation disturbing potential is limited with limit aviation Vector operation band_n(H,ψ₀) computing formula be：

R in formula_nm(ψ₀) be Legendre function integral function, be expressed as：

Dirichlet is theoretical for step 2, foundation, and the band limit aviation disturbing potential obtained based on step 1 directly calculates band limit sea Disturbing potential, geoid will be converted into by Bruns formula with limit sea disturbance position, and specific formula for calculation is：

In formula：T^b(R, θ, λ) is that band limits sea disturbance position, Q_n(H,ψ₀) it is that band limit aviation disturbing potential calculating band limit sea is disturbed The far field truncation funcation of dynamic position, T^b(r,θ_j,λ_j) limit aviation disturbing potential, Y for band at measuring point j^b(R,ψ_j, r) it is band limit boat at measuring point j Empty disturbing potential calculates the kernel function with limit sea disturbance position.

Wherein, kernel function Y with limit sea disturbance position is calculated with limit aviation disturbing potential at measuring point j^b(R,ψ_j, calculating r) is public Formula is：

Wherein, the far field truncation funcation Q with limit sea disturbance position is calculated with limit aviation disturbing potential_n(H,ψ₀) computing formula For：

According to Bruns formula, geoid N is calculated based on band limit sea disturbance position^bThe formula of (R, θ, λ) is：

In formula：γ is normal gravity.

Step 2 result of calculation is compared with the geoid of standard, statistics such as following table：

Table 3 based on two steps integrate direct method geoid compare with standard value statistical table/centimetre

	Minima	Maximum	Meansigma methodss	Standard deviation	Middle error	Number
							Fiducial value	-9.028	4.478	-0.075	2.212	2.213	900

Fig. 3 gives the difference value that the geoid with standard value of direct method are integrated based on two steps, by table 3 and Fig. 3 As can be seen that on 4000 meters of aviation height, using two steps integration direct method (σ=± 3mGal) containing white noise error is based on Band limit aviation vector gravitational to calculate the precision of geoid be 2.213 centimetres, fully meet the requirement of through engineering approaches application.

It is emphasized that embodiment of the present invention is illustrative, rather than it is determinate, therefore present invention bag The embodiment for being not limited to described in specific embodiment is included, it is every by those skilled in the art's technology according to the present invention scheme The other embodiment for drawing, also belongs to the scope of protection of the invention.

Claims (6)

1. it is a kind of based on band limit aviation vector gravitational determine geoid two steps integration direct method, it is characterised in that include with Lower step：

It is step 1, theoretical based on the horizontal boundary values of broad sense, calculate band limit aviation using band limit aviation vector gravitational on enroute altitude and disturb Dynamic position；

It is step 2, theoretical using Dirichlet, band limit sea disturbance position is directly calculated based on band limit aviation disturbing potential, pass through Bruns formula will be converted into geoid with limit sea disturbance position.

It is 2. according to claim 1 that the two steps integration direct method that aviation vector gravitational determines geoid is limited based on band, It is characterized in that：The step 1 calculates the formula with limit aviation disturbing potential：

$\begin{array}{c}{T}^b(r,\mathrm{\θ},\mathrm{\λ})-\frac{GM}{2R}\underset{n=l}{\overset{L}{\Σ}}n(n+1)C_n(H,{\mathrm{\ψ}}_0)T_n(\mathrm{\θ},\mathrm{\λ})\\ =\frac{GM}{4\mathrm{\π}r}\underset{j=1}{\overset{N}{\Σ}}\[{\mathrm{\δg}}_N^b\left(r,{\mathrm{\θ}}_j,{\mathrm{\λ}}_j\right),{\mathrm{\δg}}_E^b\left(r,{\mathrm{\θ}}_j,{\mathrm{\λ}}_j\right)\]{\[G_{NT}^b\left({\mathrm{\ψ}}_j\right),G_{ET}^b\left({\mathrm{\ψ}}_j\right)\]}^T{\mathrm{\Δ\σ}}_j\end{array},L\≤l+b$

In formula：T^b(r, θ, λ) is band limit aviation disturbing potential, and r is the earth's core at aviation calculating point to footpath, and θ and λ is that aviation calculates point The colatitude and longitude at place, b is band limit order, and GM is Gravitational coefficient of the Earth, and R is earth radius, and L is the maximum of far field truncation funcation Order, l is the reference to gravitational field model order removed, C_n(H,ψ₀) it is with the remote of limit aviation Vector operation band limit aviation disturbing potential Area's truncation funcation, H be aviation height, ψ₀It is to integrate radius, T_n(θ, λ) is the Laplace harmonic functions of disturbing gravity position, and π is round Frequency, N is measuring point number in integration radius,WithRespectively with the south of limit aviation vector gravitational measuring point j Northern component and thing component,It is the north-south component with limit aviation vector gravitational measuring point j and thing component meter Calculate the kernel function with limit aviation disturbing potential, ψ_jBe observation station and calculate point between spherical angle away from Δ σ_jIt is integration unit area.

It is 3. according to claim 2 that the two steps integration direct method that aviation vector gravitational determines geoid is limited based on band, It is characterized in that：The north-south component with limit aviation vector gravitational measuring point j and thing component are calculated with limit aviation disturbing potential Kernel functionComputing formula be：

$\left\{\begin{array}{c}{G}_{NT}^b\left({\mathrm{\ψ}}_j\right)=-\underset{n=l}{\overset{l+b}{\Σ}}\frac{2n+1}{n(n+1)}\frac{\∂P_n\left({\mathrm{cos\ψ}}_j\right)}{\∂{\mathrm{\θ}}_j}\\ G_{ET}^b\left({\mathrm{\ψ}}_j\right)=\frac{1}{{\mathrm{sin\θ}}_j}\underset{n=l}{\overset{l+b}{\Σ}}\frac{2n+1}{n(n+1)}\frac{\∂P_n\left({\mathrm{cos\ψ}}_j\right)}{\∂{\mathrm{\λ}}_j}\end{array}\right.$

θ in formula_jAnd λ_jIt is the colatitude and longitude at aviation measuring point j, P_n(cosψ_j) it is Legendre function.

It is 4. according to claim 2 that the two steps integration direct method that aviation vector gravitational determines geoid is limited based on band, It is characterized in that：The far field truncation funcation C that aviation disturbing potential is limited with limit aviation Vector operation band_n(H,ψ₀) computing formula For：

$C_n(r,{\mathrm{\ψ}}_0)=\underset{m=l}{\overset{l+b}{\Σ}}\frac{2m+1}{m(m+1)}{R}_{nm}\left({\mathrm{\ψ}}_0\right),l\≤n\≤L$

In formula, R_nm(ψ₀) be Legendre function integral function, be expressed as：

$R_{nm}\left({\mathrm{\ψ}}_0\right)={\∫}_{x={\mathrm{\ψ}}_0}^{\mathrm{\π}}{P}_n\left(\cos x\right)P_m\left(\cos x\right)\sin xdx.$

It is 5. according to claim 1 that the anti-solution of two steps integration that aviation vector gravitational determines geoid is limited based on band, It is characterized in that：The step 2 directly calculates the formula with sea disturbance position is limited based on band limit aviation disturbing potential：

$T^b\left(R,\mathrm{\θ},\mathrm{\λ}\right)-\frac{GM}{2R}\underset{n=l}{\overset{L}{\mathrm{\Σ}}}{\left(\frac{R}{r}\right)}^{n+1}{Q}_n\left(H,{\mathrm{\ψ}}_0\right)T_n\left(\mathrm{\θ},\mathrm{\λ}\right)=\frac{1}{4\mathrm{\π}}\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{T}^b\left(r,{\mathrm{\θ}}_j,{\mathrm{\λ}}_j\right)Y^b\left(R,{\mathrm{\ψ}}_j,r\right){\mathrm{\Δ\σ}}_j,L\≤l+b$

In formula：T^b(R, θ, λ) is that band limits sea disturbance position, Q_n(H,ψ₀) it is that band limit aviation disturbing potential calculates band limit sea disturbance position Far field truncation funcation, T^b(r,θ_j,λ_j) limit aviation disturbing potential, Y for band at measuring point j^b(R,ψ_j, it is r) that the aviation of band limit is disturbed at measuring point j Dynamic position calculates the kernel function with limit sea disturbance position；

Wherein, kernel function Y with limit sea disturbance position is calculated with limit aviation disturbing potential at measuring point j^b(R,ψ_j, computing formula r) For：

$Y^b(R,{\mathrm{\ψ}}_j,r)=\underset{n=l}{\overset{l+b}{\Σ}}(2n+1){\left(\frac{r}{R}\right)}^{n+1}{P}_n\left({\mathrm{cos\ψ}}_j\right)$

Wherein, the far field truncation funcation Q with limit sea disturbance position is calculated with limit aviation disturbing potential_n(H,ψ₀) computing formula be：

$Q_n(H,{\mathrm{\ψ}}_0)=\underset{m=l}{\overset{l+b}{\Σ}}(2m+1){\left(\frac{r}{R}\right)}^{m+1}{R}_{nm}\left({\mathrm{\ψ}}_0\right),l\≤n\≤L.$

It is 6. according to claim 1 that the anti-solution of two steps integration that aviation vector gravitational determines geoid is limited based on band, It is characterized in that：The computing formula of the step 2 is：

$N^b(R,\mathrm{\θ},\mathrm{\λ})=\frac{T^b(R,\mathrm{\θ},\mathrm{\λ})}{\mathrm{\γ}}$

In formula：N^b(R, θ, λ) is geoid, and γ is normal gravity.