The basic principles, main methods and all the formulas in physical geodesy and Earth gravity field have been realized completely in PAGravf4.5 to popularize higher education. Many long-term puzzles such as various terrain effects on various observations, full-element analytical modelling on gravity field, fine gravity prospecting from heterogeneous observations, external accuracy index measurement and computational performance control have been effectively solved to strengthen application of gravity field.
Precise Approach of Earth Gravity Field and Geoid (PAGravf4.5) has five subsystems, includes data analysis and preprocessing calculation of Earth gravity field, computation of various terrain effects on various field elements outside geoid, precision approach and full element modelling on Earth gravity field, optimization, unification, and application for regional height datum as well as editing, calculation, and visualization tools for geodetic data files.
(1) Solves the analytical compatibility and rigorous unified computation problems of various modes of terrain effects on various types of field elements, to fulfill the requirements of gravity field data processing in various cases and comprehensively improve the geophysical gravity exploration.
(2) Sets up the scientific and complete gravity field approach system with the positive-inverse integral in spatial domain and SRBF approach in spectral domain, to realize the full element analytical modelling in full space outside geoid from heterogeneous observations.
(3) Develops some ingenious physical geodetic algorithms based on the analytical relationship between Earth gravity field and height datum, to improve and unify the regional height datum, and consolidate and expand the applications of Earth gravity field.
✍ Please download PAGravf4.5_win64en.exe and PAGravf4.5_win64en.2 together into a folder before installing (Window 10). If the Windows system security software isolates the file PAGravf4.5_win64en.exe, please find the file and restore the trust.
(1) Develops the unified analytical algorithm system for various modes of terrain effects on various gravity field elements outside the geoid, effectively to synthesize various heterogeneous observations for the geophysical gravity exploration and gravity field data processing.
(2) Sets up the gravity field approach system with the spatial domain integration algorithms based on boundary value theory and spectral domain radial basis function approach algorithms to realize the full element analytical modelling in full space from various heterogeneous observations in the different altitudes, cross-distribution, and land-sea coexisting cases.
(3) Presents the quantitative selection criteria for terrain effects according to physical geodesy, solves the fine gravity prospecting problem from heterogeneous observations, and develops some ingenious physical geodetic algorithms to improve and unify the regional height datum.
(4) Realizes the detection of observation gross errors, measurement of external accuracy indexes, control of computational performance and assessment of model result quality to strengthen application capacity of Earth gravity field.
PAGravf4.5 independently developed a complete set of terrain effect algorithm system to realize various modes of terrain effects on different types of gravity field elements on the geoid or in its outer space.
(1) The set of algorithms are rigorous in theory, the numerical integral has no calculation error, and the accuracy of the fast FFT algorithms is controllable.
(2) There are various modes of terrain masses adjustment, the type of gravity field element affected by terrain can be arbitrary and the field element can be located on the geoid or in its outer space.
(3) Strictly follows the analytical relationship of gravity field between the terrain effects on different types of field elements.
(4) Makes full use of the analytical compatibility between different modes of terrain effects, so that the algorithm codes can be short and concise.
From the terrain effect formulas in sections 7.5 to 7.8, it is easy to see that many algorithm formulas are very similar. Some terrain effect algorithms only adjust some parameters and call the same codes.
(1) The fixed integral radius of gravity field
Limiting the definition domain of kernel function, PAGravf4.5 executes the gravity field integral operation with the given radius, including numerical integral and FFT integral algorithm (kernel function windowing), to coordinate and unify various gravity field approach algorithms. Two-dimensional FFT adopts the modified planar two-dimensional kernel function, and its calculation accuracy is not significantly different from that of one-dimensional FFT in the range of latitude 10°.
(2) The calculation point and the move point (integral running area element)
The coordinates of geodetic points are expressed by latitude and longitude and ellipsoidal height. For example, the position of boundary surface, measurement point, calculation point, and integral move point (area element or volume element) are expressed by geodetic coordinates. The integral cell grid position is the geodetic coordinates of the center of the cell grid, and the integral radius is calculated by geodetic coordinates.
(3) The equipotential boundary surface
Most gravity field integral formulas are derived from Stokes boundary value theory, such as the Hotine integral, Vening Meinesz integral, radial gradient integral formula, etc. The solution of Stokes boundary value problem requires that the boundary surface is an equipotential surface, that is, the anomalous gravity field elements should be located on some an equipotential surface.
In PAGravf4.5, the accuracy of the ellipsoidal height employed as the boundary surface is not less than 10 m can meet most requirements. The boundary surface can be constructed from a 360-degree global geopotential coefficient model, which can also be replaced by the ellipsoidal height grid of normal or orthometric equiheight surface in near-Earth space.
PAGravf4.5 proposes three key technical countermeasures to make the spherical radial basis functions (SRBF) approach algorithm independent of the observation error, avoid the spectral leakage of the undetermined target field element, and improve the analytical performance of the SRBF algorithm, to ensure the analytical approach of gravity field using SRBF.
(1) Using the edge effect suppression method to instead of normal equation regularization.
PAGravf4.5 proposes the algorithm to improve the performance of parameter estimation of spherical radial basis functions (SRBF) coefficients by suppressing edge effects. When the SRBF center is located at the margin of the calculation area, let that the SRBF coefficient is equal to zero as the observation equation to improve the stability and reliability of unknown SRBF coefficient estimation.
After the edge effect suppression method employed, the normal equation need not be regularized. Which can keep the analytical nature of the SRBF approach algorithm and the analytical nature of gravity field from being affected by the observation errors.
(2) Employing the cumulative SRBF approach method to achieve the best approach of the gravity field.
The target field elements are equal to the convolution of the observations and the filter SRBF. When the target field elements and the observations are of different types, it is difficult for one SRBF to effectively match the spectral center and bandwidth of the observations and the target field element at the same time, which would make the spectral leakage of the target field element. In addition, the SRBF type, the minimum and maximum degree of Legendre expansion and the SRBF center distribution also all affect the approach performance of the gravity field. Therefore, only the optimal estimation of the SRBF coefficient with the burial depth as the parameter is not enough to ensure the best approach of the gravity field.
PAGravf4.5 proposes a cumulative SRBF approach scheme according to the linear additivity of the gravity field to replace the optimal estimation scheme of SRBF coefficients with the burial depth as the parameter to solve the key problem above. Using the multiple cumulative SRBF approach scheme, it is not necessary to determine the optimal burial depth.
When each SRBF approach of gravity field employs a SRBF with different spectral figure, the cumulative SRBF approach can fully resolve the spectral domain signal of the target field element by combining multiple SRBF spectral centers and bandwidths, and then optimally restore the target field element in space domain.
The character of cumulative SRBF approach scheme of gravity field: the essence of each SRBF approach is to employ the previous approach results as the reference gravity field, and then refine the residual target field element by remove -restore scheme.
(3) Proposing the cofactor matrix diagonal standard deviation method.
PAGravf4.5 proposes a cofactor matrix diagonal standard deviation method to combinate different types of heterogeneous observations for estimation of the SRBF coefficients, instead of the common variance component estimation method. This method replaces the residual observations variance in the iterative process of the variance component estimation with the diagonal standard deviation of the cofactor matrix, so that the properties of the parameter estimation solution are only related to the space distribution of the observations from being affected by the observation errors. Which is conducive to combination of various types of observations with extreme differences in space distribution, such as a very small number of astronomical vertical deflections or GNSS-levelling data.
In this case, the normal equation does not also need to be iteratively calculated, which conducive to improve the analytical nature of the SRBF approach algorithm.
The typical technical features of SRBF approach program in PAGrav4.5: (1) The analytical function relationships between gravity field elements are strict, and the SRBF approach performance has nothing to do with the observation errors. (2) Various heterogeneous observations in the different altitudes, cross-distribution, and land-sea coexisting cases can be directly employed to estimate the full element models of gravity field without reduction, continuation, and griding. (3) Can integrate very little astronomical vertical deflection or GNSS-levelling data, and effectively absorb the edge effect. (4) Has the strong capacity in detection of observation gross errors, measurement of external accuracy indexes, and control of computational performance.
PAGravf4.5 has the high-precision analytical computation capacity of various terrain effects on any type of gravity field element. At the same time, it has the full-element analytic modelling function on gravity field from various heterogeneous observations. The combination of the two can effectively solve the fine computation problem of gravity exploration that can deeply fuse all the gravity field information in multi-source heterogeneous observations.
In any region of the world, you can accurately calculate the land-sea unified complete Bouguer gravity anomaly (disturbance), complete Bouguer vertical deflection, complete Bouguer gravity gradient, and classical Bouguer / isostatic gravity anomaly (disturbance) from heterogeneous observations such as gravity, gravity gradient, satellite altimetry, (astronomical) vertical deflection, GNSS-leveling etc. in the different altitudes, cross-distribution, and land-sea coexisting cases.
The general computation process for fine gravity prospecting from heterogeneous observations is as follows.
(1) Select the calculation area, target calculation surface (terrain equiheight surface is recommended here) for gravity exploration and obtain (or collect) all the gravity field and geodetic observations as much as possible.
(2) Call the related programs in the subsystem [Precision approach and full element modelling on Earth gravity field] to determine the high-resolution grid of gravity field element corresponding to the target prospecting model on the calculation surface.
(3) Call the related programs in the subsystem [Computation of various terrain effects on various field elements outside geoid] to determine the terrain effect grid on gravity field element corresponding to the target prospecting model on the calculation surface.
(4) By subtracting the terrain effect grid from the gravity field element grid directly, you can obtain the target gravity prospecting model which have deeply fuse all the gravity field information in heterogeneous observations.
If the SRBF approach method is employed in step (2), the whole gravity prospecting modelling process above are all strictly analytical, which can effectively avoid signal attenuation and distortion of gravity field and the problem of terrain effect difficult to control in traditional reduction, continuation, and gridding.
The computation of fine gravity prospecting with the analytic relations of gravity field as strong constraints by PAGravf4.5 can deeply fuse all the gravity field information from heterogeneous observations in the different altitudes, cross-distribution and land-sea coexisting cases, whose observation mode can be on terrestrial, marine, aviation and satellite.
The algorithm system in PAGravf4.5 is scientific and rigorous, and there are several schemes to calculate the same terrain effect. Various gravity field elements can be solved from any type of field element outside. The field element type can be traditional or uncommon such as satellite tracking satellite and satellite orbit perturbation, and the applicable space can be on the geoid or in its outer space.
The approach of Earth gravity field is linear operation. The terrain effect and gravity field approach algorithms in PALGravf4.5 are all linear. Therefore, Any program in PAGravf4.5 can output the error distribution characters of the target field elements with the simulated spatial noises as observations. PAGravf4.5 has strong error analysis capacity, which can be the important means to optimize the gravity prospecting computation and gravity field approach scheme.
One mode of terrain effect, there are also multiple computation schemes to be chose. For example, to compute the land-sea complete Bouguer effect, PAGravf4.5 has three schemes and programs to be chose.
For the specific data processing and modelling purpose of gravity field, there are multiple PAGravf4.5 programs, different parameter settings or multiple schemes to be chose. In the actual application, you should investigate the gravity data situations and the nature of gravity field in the target region, carefully select, test and analyze the related PAGravf4.5 algorithms, parameters and schemes to design and optimize the computation technology route.
(1) Performance test for terrain effect algorithm
The terrain effect optimization criterion proposed by PAGravf4.5 according to the basic principles of physical geodesy, can greatly reduce the complexity of terrain effect analysis, and provide a concrete and feasible technical route for effectively playing the key role of terrain effect in geophysical gravity exploration and gravity field approach.
The statistical properties of terrain effects vary significantly with the local terrain, gravity field nature and observations distribution in the computation area. PAGravf4.5 terrain effect computation subsystem present some cases in a difficult mountain area, and the ratio of the maximum and minimum values to the standard deviation of various terrain effects on various field elements was statistically analyzed. The statistical analysis results of these cases show that, when ignoring the observation distribution and gravity field nature, the local terrain effect is favorable for the gravity data processing, the terrain Helmert condensation is favorable for the processing of gravity gradient data, and the residual terrain effect is more favorable for the refinement of geoid.
Before the computation, it is necessary to comprehensively and carefully test and analyze the technical route of terrain effects according to the local terrain, gravity field nature and available gravity resources in the target area according to the quantitative criteria of terrain effect, so as to ensure that the terrain effect algorithm and parameter settings can be based on some evidences. Only then can the applicability and technical level of the terrain effect processing scheme be significantly improved.
(2) Performance test for gravity field approach algorithm
The performance of most gravity field approach algorithms and their parameter settings can be tested and verified with an ultra-high degree global geopotential coefficient model. Many program samples of PAGravf4.5 take the 540th degree EGM2008 geopotential coefficient model as the reference gravity field, and then employ the anomalous gravity field calculated by the 541 to 1800th degree model as the reference true values for testing and verification.
The test outline of the PAGravf4.5 program algorithm: Take some residual anomalous field elements calculated by the 541 to 1800th degree EGM2008 geopotential coefficients as the observations, call the PAGravf4.5 programs or functions to be verified, and obtain the calculated values of the target residual field elements. Then compare the difference between the calculated values and the target reference values calculated by the EGM2008 model, to evaluate the technical performance of the algorithm programs and their parameter settings in PAGravf4.5.
PAGravf4.5 can compute various types of field elements on the geoid or in its outer space from some a type of field elements located on a certain boundary surface and can also cyclically calculate the same type of field elements on the boundary surface. Comparing the difference and similarity between the observed field elements and the calculated field elements obtained by the cyclical computation, the algorithm character and performance of the relevant programs and functions called in the computation process can be analyzed.
(3) Performance test for gravity field approach using SRBF
The best approach scheme of gravity field using SRBF is related to the observation situations, nature of gravity field and algorithm parameters. The program of [Gravity Field approach by spherical radial basis functions and its performance analysis] can be employed to comprehensively analyze the spectral center and bandwidth of the observation, target field element and SRBF in different parameters combination case. According to the principle of fully resolving the spectrum of the target field element, design and optimize the scheme and relevant parameters for SRBF approach of gravity field in advance.
(4) Notes on the PAGravf4.5 program examples
In the examples of terrain effect programs of PAGravf4.5, a typical difficult mountainous area with an average altitude of 4000m and terrain relief of more than 3000m is selected to facilitate the display of the details of terrain effect and its algorithm characters. Similarly, in the gravity field integral examples of PAGravf4.5, the region with typical complex features where the short-wave signal of the gravity field is rich (residual gravity disturbance after the 540-degree reference model value removed, and the space variation exceeds 300mGal) is selected to facilitate the display of the detailed features of the local gravity field and its approach algorithm.
The statistical results of these samples roughly show the basic performance of the corresponding algorithms. Since the main purpose of these samples is to introduce the computation process, there is no statistical analysis and optimization of the algorithm itself and parameter settings. Therefore, there is still greater potential, which need be further explored by the user in combination with the specific situations.
(1) Ignoring the terrain correction and direct effect concepts
In the classic terrain correction, the correction object is only the terrestrial gravity. PAGravf4.5 need deal with various modes of terrain effects on various types of gravity field elements on the geoid or in its outer space. The classic direct effect is the effect of terrain mass on gravity (gravity disturbance or gravity anomaly), and the indirect effect is the effect of terrain mass on geopotential (disturbing potential, height anomaly or geoid). PAGravf4.5 need deal with various terrain effect on full elements gravity field elements. The concept of terrain correction, direct and indirect effect can no longer meet the needs of PAGravf4.5.
PAGravf4.5 adopts the concept of terrain effect uniformly, and strictly distinguishes the adjustment mode of terrain masses, the type of field elements effected and the location of field elements. For example, the local terrain, terrain Helmert condensation and residual terrain effects on the ground gravity disturbance, external gravity disturbance and geoidal gravity disturbance include 3 × 3 = 9 different terrain effect quantities.
The terrain effect on various types of gravity field elements is equal to the negative value of its terrain correction. For example, the local terrain effect is equal to the negative value of classic local terrain correction.
(2) Recommending W₀=Wɢ as the global geopotential
PAGravf4.5 proposes that the scale parameters (GM, a) of global geopotential coefficient model, the second-degree zonal harmonic coefficient C̅₂₀ and the rotation mean angular velocity ω should be employed as the four basic parameters of the normal ellipsoid. In this case, the second-degree zonal harmonic term of anomalous gravity field is always zero, which is beneficial to improve the performance of the gravity field approach.
The geoid determined according to the geodetic boundary value theory is essentially the realization of the constant geopotential Wɢ in the Earth coordinate system, namely determining of the ellipsoidal height of the geoid. The geoidal geopotential Wɢ is always equal to the normal potential U₀ of the normal ellipsoid. PAGravf4.5 suggests that the geoidal geopotential Wɢ replaces the appoint empirical W₀ in the IERS numerical standard. The latter is calculated from the global geopotential model and satellite altimetry data according to the Gaussian appoint geoid definition.
Whether for realizing of global height datum or for refining of regional height datum, when the geoidal geopotential Wɢ is the global geopotential W₀, it can not only effectively reflect the unique invariance of geodetic datum, but also make full use of physical geodesy (space geodesy) technology and method to approach the global geopotential with infinite precision. Which can ensure the analytic rigor of gravity field approach method in the realization and unification of height datum.
(3) Recommending the analytical orthometric system
The analytic orthometric height of the global ground points is closer to the normal height, and the difference is about 60 cm from the Helmert orthometric height at 3000 m altitude. The analytical orthometric height is not directly related to the terrain density, which can be continuously refined with the latest gravity field data. The analytical orthometric height is consistent with the geometric scales of the Earth coordinate system and the geoid height.
On the view of uniqueness, repeatability, and measurability of geodetic datum, analytical orthometric height is more suitable for height datum purpose than other types of orthometric height. Different from the Helmert orthometric system, the analytic orthometric system and normal height system are analytically consistent with each other and supported by the rigorous gravity field theory, they can also be directly employed for the moon and Earth-like planets.