Scientific Computation Platform for Geophysical Geodesy

Precise Approximation of Earth Gravity Field and Geoid (PAGravf4.5) system comprehensively encompasses the fundamental principles, core methodologies, and exhaustive formalism of physical geodesy and Earth's gravity field theory. Its development is driven by a commitment to enhancing the pedagogical landscape in higher education.

PAGravf4.5 effectively resolves a series of persistent technical challenges, including the rigorous computation of external terrain effects, full-element analytical modeling of the gravity field, geophysical exploration modeling using multi-source heterogeneous data, and the precise determination of external accuracy metrics alongside computational performance control. By addressing these complexities, the system aims to consolidate and expand the practical utility of Earth's gravity field theory and data resources, while cultivating the analytical mindset and conceptual framework necessary for exploring and resolving complex geodetic problems through the principles of physical geodesy.

Precise Approach of Earth Gravity Field and Geoid (PAGravf4.5) is engineered to rigorously adhere to the theoretical foundations of physical geodesy. Its overarching scientific objectives are:

(a) To comprehensively resolve the complex diverse terrain effects on various gravity field elements both on and outside the geoid;

(b) To realize full-space, all-element analytical modeling of the gravity field and centimeter-level geoid;

(c) To achieve closed-loop analytical operations for various external anomalous gravity field elements;

(d) To significantly expand the computational capabilities and practical application scope of Earth’s gravity field.

These objectives are realized through:

(a) Resolving Analytical Compatibility of Diverse TerrainEffects: Overcomingthe challenge of achieving analytical compatibility and rigorously unifiedcomputation for diverse terrain effects on all gravity field elements.

(b) Constructing an Integrated Dual-Domain AlgorithmicFramework: Establishing a robust framework that synergizes spatial-domain boundary-value integrals with spectral-domain SRBF approximations to enable comprehensive modeling using complex, multi-source data.

(c) Developing Ingenious Algorithms for Height DatumOptimization and Unification: Developing algorithms to facilitate analytical modeling of gravity exploration and the pivotal optimization and unification of regional height datums, strengthening the linkage between gravity field theory and height systems.

✍ 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.

(a) Unified Analytical Computation Framework for Diverse Terrain Effects: A rigorously developed framework capable of precisely computing diverse terrain effects on a broad spectrum of gravity field elements on and outside the geoid, supporting advanced gravity exploration modeling.

(b) Integrated Dual-Domain Gravity Field Approximation Framework: A powerful hybrid computational framework combining spatial and spectral methods, achieving all-element analytical modeling and capable of processing and fusing multi-source, heterogeneous data.

(c) Quantitative Selection Criteria and Ingenious Algorithm Development: Introduces quantitative selection criteria for terrain effects and deploys a comprehensive suite of algorithms, with a paramount application directed toward the optimization and unification of regional height datums.

(d) Comprehensive Quality Control, Accuracy Assessment, and Performance Enhancement: Implements built-in functionalities for gross error detection, measurement of external accuracy indices, and control over computational quality, addressing long-standing technical bottlenecks.

A Comprehensive Analytical Algorithm Framework for Terrain Effects

To facilitate the computation of diverse terrain effects on various gravity fieldelements on or outside the geoid, PAGravf4.5 has developed a comprehensiveanalytical algorithm framework characterized by:

(1) Theoretical Rigor and Numerical Precision:

The algorithmic formulations are theoretically analytic and rigorous. Numericalintegration is implemented without computational error, while the accuracy ofFFT-based fast algorithms is also controllable.

(2) Diversity and Flexibility:

The frameworksupports a wide spectrum of terrain effect types (e.g., Local Terrain Effect,Terrain Bouguer Effect, Helmert Condensation Effect, and Residual TerrainEffect). The affected gravity field elements can be arbitrarily specified,including geopoential, height anomalies, gravity disturbances, verticaldeflections, and gravity gradients.

(3) Analytical Consistency:

Terrain effects on different types of gravity field elements strictly adhere to theintrinsic analytical relationships of the fieldelements. This ensures refined and internallyconsistent algorithmic formulations, preserving the fundamental physical andmathematical properties of the gravity field throughout all computations.

(4) Code Efficiency via Analytical Compatibility:

The frameworkfully exploits the analytical compatibility among terrain effects of differentnatures, enabling compact and efficient code implementation. As demonstrated inSections 7.5 – 7.8, many formulas share structural similarities. In PAGravf4.5,various terrain effect algorithms can be realized by invoking the same corecodebase with parameter adjustments, significantly enhancing code reusabilityand maintainability.

Technical Characteristics of Local Gravity Field Integral Algorithms

(1) Fixed Integration Radius:

PAGravf4.5 implements gravity field integration with a fixed radius by controlling thedomain of the integration kernel function. This applies to both numericalintegration and Fast Fourier Transform (FFT)-based algorithms (which employwindowed kernels). This approach harmonizes and unifies various gravity fieldapproximation methodologies. The 2D FFT utilizes a modified planar 2D kernel;within a latitude span of 10°, its computational accuracy shows no significantdeviation from the 1D FFT method.

(2) UniformRepresentation of Point Locations:

All pointlocations are uniformly expressed using ellipsoidal geodetic coordinates (latitude,longitude, and ellipsoidal height). This convention applies to boundarysurfaces, observation points, computation points, and moving integration points(area or volume elements). The location of an integration grid cell is definedby the coordinates of its centroid. All integration distances are computeddirectly from these coordinates, ensuring a consistent geometric foundation.

(3) Equipotential Boundary Surface:

Mostgravity field integral formulas (e.g., Hotine, Vening-Meinesz, and radialgradient integrals) derive from the solution to the Stokes Boundary ValueProblem (BVP), which requires the boundary surface to be an equipotentialsurface (e.g., the geoid).

In PAGravf4.5, since anomalous gravity field elements and residual field elements(after reference geopotential model removal) exhibit low sensitivity to minorpositional variations, the accuracy requirement for the ellipsoidal heightrepresenting the boundary surface may be relaxed. A height accuracy of no lessthan 10 meters is generally sufficient. This boundary surface can beconstructed using a 360-degree geopotential coefficient model. For near-Earthapplications, the ellipsoidal height grid of an orthometric (or normal)equi-height surface serves as an effective surrogate for the true equipotentialsurface.

Analytical Approximation of Gravity Field Using Multi-source Heterogeneous Data with SRBFs

PAGravf4.5 recommends three pivotal technical measures for the Spherical Radial BasisFunction (SRBF) approximation algorithm. These measures render the algorithm robust against observational gross errors, prevent spectral leakage in targetfield elements, enhance the analytical rigor of approximation models, andenable all-element gravity field modeling from multi-source heterogeneous data.

(1) Edge Effect Suppression as an Alternative to Normal Equation Regularization

PAGravf4.5 proposes an algorithm that enhances parameter estimation performance by suppressing edge effects. When an SRBFcenter (node) lies at or outside the boundary of the ccomputation region, its corresponding SRBF coefficient is constrained to zero. This significantly improves the stability and reliability of SRBF coefficient estimation. The addition of boundary conditions can weaken the need for Tikhonov-type regularization, which might otherwise distort the analytical relationships between field elements.

(2) Normal Equation Normalization for Fusion of Heterogeneous observation Systems

PAGravf4.5 recommends a highly versatile method for the deep fusion of multi-source heterogeneous observation systems via the normalization of normal equations from different observation groups. This technique effectively manages the integration of diverse observation systems with vastly different covariance structures relative to the unknown coefficients, facilitating gravity field approximation through the fusion of heterogeneous gravity field observations. The method completely decouples the influence of the observation system model (covariancestructure) from observation quality (errors or outliers). This separation ensures the fusion process remains immune to observational errors, type differences, or spatial distribution variations. This feature is particularly advantageous for fusing sparsely distributed data (e.g., limited astronomical vertical deflections or GNSS-leveling sites) and enables precise outlier detection.

(3) Cumulative SRBF Approximation for Optimal Gravity Field Recovery

A target field element is essentially the convolution of the observational field element with a SRBF filter. When target and observational elements differ in type, a single SRBF often fails to simultaneously match the spectral centers and bandwidths of both, inevitably causing spectral leakage. Moreover, factorsbeyond the Bjerhammar sphere burial depth (bandwidth parameter) – such as SRBF type, degree range, and SRBF node distribution – significantly impact gravity field approximation performance. Thus, optimizing approach based solely on burial depth is insufficient for optimal gravity field ecovery.

To address this, PAGravf4.5 proposes a Cumulative SRBF Approximation scheme. This scheme combines multiple SRBFs with distinct spectral centers and bandwidths to fully resolve the spectral signals of the target field element, thereby eliminating spectral leakage and achieving optimal field recovery. Each stage of residual approximation essentially treats the previous result as a reference gravity field, refining the residual target element according to the remove-restore principle.

Gravity Exploration Analytical Modeling using Multi-source Heterogeneous Data

PAGravf4.5 empowers users with high-precision analytical computation of diverse terrain effects on various gravity field elements on and outside the geoid. Concurrently, it features full-space, all-element gravity field analytical modeling capabilities that integrate multi-source, heterogeneous, varying-altitude, cross-distributed, and multi-type data encompassing terrestrial, maritime, aerial, and space-based observations. The synergy ofthese capabilities effectively resolves the analytical modeling challenges of geophysical gravity exploration under complex observational scenarios.

(1) Application Scenario

In any region globally, PAGravf4.5 can integrate multi-source heterogeneous data – including gravity, gravity gradients, (astronomical) vertical deflections, satellite altimetry, GNSS leveling, and satellite gravity – from spaceborne, airborne, terrestrial, and marine platforms. This facilitates the precise computation of land-sea unified Complete Bouguer gravity anomalies/disturbances, vertical deflections, and gradients, as well as unified Classical Bouguer and Isostatic anomalies/disturbances.

Gravity exploration analytical modeling using multi-source heterogeneous data can be achieved viathe following four-step workflow:

(a) Define Scope and Model Type: Delineate the target region, computation surface, and exploration model type. Acquire all available gravity and other geodetic data within and surrounding the target region.

(b) Generate Target Field Grid Model: Utilize modules from the subsystem “High-Precision Gravity Field Approximation and Full-element Modeling” to compute a high-resolution grid model of the target field element on the specified computation surface.

(c) Compute Terrain Effect Grid Model: Employ modules from the subsystem “Computation of Diverse Terrain Effects on Various Gravity Field Elements” to derive a high-resolution grid model of the terrain effect consistent with the chosen exploration model.

(d) Synthesize Exploration Model: Directly subtract the terrain effect grid [Step (c)] from the gravity field element grid [Step (b)]. The result is a gravimetric exploration model that fully integrates all available gravity field data.

(2) Summary of Advantages

This scheme deeply fuses multi-source heterogeneous geodetic data within an unified mathematical framework, strictly adhering to model definitions to achieve high-precision analytical modeling. By circumventing traditional gravity reduction, continuation, and gridding operations, it effectively mitigates issues such as signal attenuation, non-analytical distortion, and error propagation inherent in conventional methods.

Algorithm and Computational Technical Route Optimization

The PAGravf4.5 algorithm system is designed with scientific rigor andcomprehen-siveness. In principle, it supports multiple schemes for computing diverse terrain effects on any gravity field element across all domains (space,air, land, and sea). Furthermore, it can derive various gravity field elementsin external space from any single input type, ranging from traditional quantities (gravity anomalies, vertical deflections) to diverse derivatives from Satellite-to-Satellite Tracking (SST) or satellite orbital perturbations.The applicable spatial domain encompasses the geoid and its external space.

Gravity field approximation theory in external space is founded on linear space. Accordingly, all terrain effect and gravity field approximation algorithms inPAGravf4.5 are linear. This linearity implies that injecting simulated spatialnoise as the observed quantities yields an output that accurately characterizes the error distribution properties of the target field elements or terrain effects. Thus, PAGravf4.5 possesses robust capabilities for error simulation and analysis and can serve as a critical tool for optimizing gravity exploration modeling, field approximation algorithms, and technical routes.

Even for a single terrain effect type, multiple computational schemes are available in PAGravf4.5. For example, for the computation of land-sea unified Complete Bouguer effect, PAGravf4.5 can offer three distinct schemes and corresponding programs for user selection.

For specific objectives, users can select various algorithms, parameter sets, or diverse technical routes from PAGravf4.5. In practice, users should select, test, and evaluate the most suitable algorithms and relevant parameters based on the data conditions and local gravity field properties of the target region to optimize the technical route for their specific application.

Algorithm Performance and Parameter Testing Analysis

(1) Performance Testing and Analysis of Terrain Effect Algorithms

The Terrain Effect Optimization Criteria in PAGravf4.5 are grounded in the fundamental principles of physical geodesy. These criteria significantly reduce the complexity of terrain effect analysis and provide concrete, feasible technicalpathways to leverage the critical role of terrain effects in gravity exploration modeling and gravity field approximation.

The nature of terrain effects varies significantly depending on topographic relief, local gravity field structure, and gravity field element typology within the target region. In the cases of the PAGravf4.5 terrain effect subsystem, arugged mountainous region was selected for statistical analysis of the ratio between the range (max-min difference) and the standard deviation for diverse terrain effects on different types of field elements. Results indicate that, absent specific considerations of data distribution and field structure:

(a) LocalTerrain Effects are beneficial for gravity data processing;

(b) Helmert Condensation is advantageous for gravity gradient data processing;

(c) Residual Terrain Effects are superior for geoid refinement.

Prior toactual computation, it is imperative to conduct comprehensive and meticuloustesting of the terrain effect technical route for the target region. This process should be guided by quantitative criteria, accounting for topography characteristics, local gravity field structure, and available geodetic dataresources. Only by ensuring a well-founded selection of algorithms and parameters can the technical applicability of the terrain effect processing scheme be effectively enhanced.

(2) Performance Testing of Gravity Field Approximation Algorithms

The performance of most gravity field approximation algorithms and parameter settings can be validated using an ultra-high-degree geopotential coefficient model. In PAGravf4.5 local gravity field approximation examples, degrees 1–540 of the EGM2008 model serve as the reference field, while degrees 541–1800 constitute the residual anomalous field elements, providing a basis for statistical performance analysis.

Technical Workflowfor Performance Testing:

(a) Input Selection: Select residual anomalous field elements as observational inputs.

(b) Computation: Invoke the local gravity field approximation algorithm under test to compute values for another set of residual field elements.

(c) Validation: Compare the computed values against the reference truth values (directly derived from degrees 541–1800 EGM2008 model). The discrepancies provide a quantitative evaluation of the algorithm's technical performance and parameter sensitivity.

PAGravf4.5 possesses the capability to compute full-space, all-element fields from a single observation type and supports cyclic calculations of the same element at identical points. By analyzing differences between cyclically computed resultsand original observ-ations (reference truth), users can diagnose algorithmic characteristics, optimize configur-ations, and improve modeling schemes.

The module [All-element Gravity Field Modeling Using Multi-source Heterogeneous Data with SRBFs] inherently offers robust testing and validation capabilities.

For any specific objective, users may select from multiple algorithms, parameters, or technical routes. Practical implementation necessitates a thorough characterization of local data conditions and field properties to guide the rigorous selection, testing, and optimization of the most suitable approach.

(3) Built-in Case Studies in PAGravf4.5

PAGravf4.5 includes built-in case studies designed to demonstrate algorithmic capabi-lities under demanding conditions:

Terrain Effect Case: Selects a rugged region with an average elevation of 4,000 m and relief range exceeding 3,000 m to showcase detailed terrain effect characteristics.

Gravity Field approximation Case: Chooses a complex region rich in short-wavelength signals (residual gravity disturbances > 300 mGal after removing degrees ≤ 540) to illustrate local field approximation details.

Statistical results from these cases reflect the baseline performance of the corres-ponding algorithms. As their primary purpose is to illustrate computational workflows, they do not encompass exhaustive parameter optimization. Users are encouraged to explore further potential tailored to their specific application scenarios.