Scientific Computation Platform for Geophysical Geodesy

Geophysical Geodesy Scientific Computing Platform is architected upon a conceptual framework distilled from over two decades of my research. Its primary objective is to refine the foundational logic and disciplinary philosophy of geodesy. By rigorously emphasizing the intrinsic principles, axioms, and requirements of the field, the platform seeks to eliminate superfluous theoretical complexity, thereby facilitating the effective identification of fundamental scientific problems and the resolution of applied technical challenges. This approach aims to comprehensively elevate the cognitive understanding of geodesy.

While these concepts and philosophies are frequently overlooked or deemed too nuanced for detailed exposition in standard textbooks and academic literature, they constitute the fundamental principles and critical basis for capturing, analyzing, and resolving complex geodetic problems in both scientific research and production environments.

User Guidance: It is strongly recommended that users periodically review, analyze, and reflect upon the content presented in this section during their interaction with the platform. Deep engagement with these materials is essential to fully grasp and further evolve the scientific philosophies embedded within the system.

✍ Geodesy is the discipline that takes the Earth's body and Earth-fixed space as its primary objects of of observation and study. It is dedicated to determining the Earth's figure and gravity field, and to measuring the precise positions and relationships among points in Earth space. The discipline further extends to monitoring Earth deformation, temporal variations of the Earth's gravity field, kinematic trajectories of points and their spatio-temporal coherence. Fundamentally, geodesy serves as a metrological science for the precise measurement of the Earth and the monitoring of global change.

✍ Physical Geodesy is the theoretical geodesy focused on determining the Earth's figure and its external gravity field, and on investigating the theoretical and analytical relationships between the terrestrial reference system, height datums, and the Earth's gravity field. It provides the theoretical framework that leverages gravity field principles to synergize geometric and physical geodesy, thereby advancing the discipline of geodesy. Fundamentally, it serves as a metrological science committed to monitoring mass transport within the Earth system and temporal variations of the gravity field.

✍ The Metrological Essence andNormative Constraints of Geodesy

Modern geodesy is fundamentally a metrological science dedicated to the precise global measurement of the Earth and the monitoring of global changes within the Earth system. The intrinsic attributes of its observables and measurement elements – objectiveuniqueness, precise measurability, and spatio-temporal unity – constitute the foundational and normative constraints that permeate all sub-disciplines ofgeodesy.

To articulate the core essence and metrological character of geodesy, we proposetwo pivotal conceptual pillars:

(1) General Geodetic Elements

All general geodetic elements, encompassing a broad spectrum of observables, parameters, and derived quantities, must rigorously adhere to the requirementsof objective uniqueness and precise measurability. This ensures that every geodetic element within the geodetic framework is uniquely defined, reproducible, and quantifiable.

(2) Geodetic Datum Constants

Defined as time-invariant geodetic elements, geodetic datum constants must satisfy astringent set of criteria: unique invariance, precise measurability, and conceptual independence from specific epochs and measurement errors. These criteria constitute the starting datum requirements of geodesy.

There exist exactly nine such Geodetic Datum Constants. They are mutually statistically independent and share no analytical functional relationships, yet collectively support the entire edifice of geodesy.

(3) The Nine Fundamental Geodetic DatumConstants

The Nine Constants form a minimal yet complete set required to define and express the Earth's position, orientation, rotation, and gravity field within an inertialor quasi-inertial reference system. This canonical set comprises:

(a) Origin of the Earth-Fixed Reference System: The mean center of mass (COM) of the Earth (contributing 3 constants: COM coordinates).

(b) Orientation of the Earth-Fixed Axes:

● The Reference Pole (the mean shape pole of the Earth, defining the z-axis orientation; 2 constants).

● The Zero Meridian (the reference meridian, defining the -axis orientation, equivalent to the Non-Rotating Origin [NRO] of the Geocentric Celestial Reference System [GCRS]; 1 constant).

(c) Fundamental Physical Constants:

● The global geopotential value (W₀), conventionally defining the geoid in gravity field space.

● The geocentric gravitational constant ().

● The mean angular velocity of Earth's rotation ().

Upon their definition via international convention and realization through precise measurement campaigns, these nine datum constants acquire a conventional nature. From the nine foundational constants, other geodetic constants (e.g.,reference ellipsoid parameters) may be derived according to geodetic principles. These derived constants likewise inherit this conventional nature.

All these geodetic constants are characterized by:

● Unique invariance over long timescales (e.g., over decades);

● Conceptual freedom from notions of "error" and specific "epochs";

● Immunity to perturbations such as terrain effects, tidal effects, or loading deformationeffects.

The Geoid, Global Geopotential and Height Datum

(1) The Geoid, Global Geopotential, and Normal Gravity Field

In accordance with the classical Gauss-Listing definition, the geoid is defined as the equipotential surface that best approximates the global mean sea level. This definition serves essentially as a specified convention in geodesy, establishing an invariant reference starting value for vertical positioning.The geopotential value associated with this Gaussian geoid is termed the Earth's geopotential constant, commonly known as the global geopotential and denoted as W₀.

The geoid is a specific equipotential surface defined in Earth-fixed reference system by a conventional constant geopotential value, W₀. Its geometric positioning within a terrestrial reference system is expressed in terms of geoid (ellipsoidal) height. Consequently, the concept of the geoid can only be derived from the terrestrial reference system first. Computing the geoid undulation (geoidal height) necessitates the prior convention of a normal ellipsoid (or reference ellipsoid). A foundational geodetic requirementis that the normal geopotential (₀) on the surface of this chosen ellipsoidmust be equated to ₀ (i.e., ₀ = ₀), thereby rigorously linking the geometric starting surface to the physical gravity field.

The normal ellipsoid, which mathematically represents the normal gravity field, is defined by four fundamental geodetic constants. Among these, the geocentric gravitational constant () and the mean Earth angular velocity () are directly measurable physical quantities. The remaining two constants are: the Earth's semi-major axis (); and one parameter selected from the set comprising the dynamic form factor (₂), the normal geopotential on the ellipsoid surface (₀), or the geometric flattening (). For this latter pair, one must be specified to uniquely determine the other, ensuring a closed and consistent normal gravity field.

Both the normal gravity field and the global geopotential W₀ inherently possess a conventional nature. For geodetic elements endowed with this conventional nature, once their constant values are established by international agreement, they must remain uniquely invariant over extended periods (e.g., decades). This temporal stability is paramount, as it guarantees theuniqueness of all associated gravity field elements (such as geoidal height and gravity anomalies) and ensures the analytical compatibility and consistency of the functional relationships interconnecting these geodetic elements across different measurement epochs and techniques.

Consequently, by virtue of their conventional definition, the normal gravity field and the geoidal geopotential (W₀) are immune to perturbations such as various terrain effects, tidal effects, or loading deformations. They are conceptually devoid of "measurement error" (Error-Free) and independent of any specific epoch (Time-Invariant), serving as stable, absolute references within the global geodetic framework against which all time-variable and relative measurements are compared.

(2) Implementation Principle for the Global Geopotential W₀ Consistent with the Gravimetric Geoid

The realization of W₀ may follow a rigorous, multi-step procedure using a precise global mean sea surface height grid and a latest global geopotential model:

(a) Estimation of Semi-major Axis (): Utilizing an optimal global geopotential coefficient model (e.g., a high-degree EGM) in conjunction with a precise global mean sea surface height model (derived from satellite altimetry), the Earth's semi-major axis is estimated. This estimation strictly adheres to the Gauss-Listing convention, ensuring the defined geoid approximates the global mean sea level.

(b) Determination of Normal Ellipsoid Constants: Employing the measured second-degree zonal harmonic coefficient (C̅₂₀, related to ₂ and consistent with the chosen ) from the global geopotential model, along with the known values of and , the four fundamental constants defining the Earth's normal ellipsoid is chosed.

(c) Calculation of ₀ and Conventional Assignment: The four constants enable the analytical calculation of the normal geopotential ₀ on the ellipsoid surface. The final, crucial step is the conventional assignment: defining ₀ as the global geopotential constant ₀ (i.e., ₀ = ₀). This process achieves the scientifically desired equivalence: W₀ = U₀= Wɢ (where ɢ is the geopotential of the gravimetric geoid), thereby unifying the geometric and physical geoid definitions.

(3) Concept of the Analytical Geoid andAnalytical Orthometric Height

The so-called "true" geoid (or terrain-corrected geoid), which attempts to account for topographic mass effects through reduction, suffers from decimeter-level uncertainties in continental mountainous regions. These uncertainties stem from necessary approximations in terrain density models and assumptions regarding methods of terrain mass reduction (e.g., Helmertcondensation or complete Bouguer reduction). At the centimeter-level precision demanded, such a geoid fails to meet the fundamental constraints of uniqueness and precise measurability required for valid geodetic element. Consequently, the geoidal height derived following such terrain adjustments is metrologically undefined; it is neither uniquely determinable nor does it possess an accepted concept of accuracy.

By rigorous constraints to the invariance of the external disturbing potential field, PAGravf4.5 employs analytical terrain mass adjustment strategies butdoes need not to know the specific adjustment process. Within this framework, the Analytical Geoid, defined by the geopotential ₀, is established as a uniquely defined andprecisely measurable geodetic element. The corresponding Analytical OrthometricHeight can be derived by determining mean gravity via analytical continuation from external gravity, and thus also becomes a uniquely defined and precisely measurable geodetic element. Numerically, the Analytical Orthometric Height exhibits closer agreement with Normal Height than with the classical Helmert orthometric height.

Unlike the Helmert orthometric height, both the Analytical Orthometric Height and theNormal Height strictly satisfy the datum conditions for GNSS Replacing Leveling. Furthermore, these two height systems share a rigorous analytical functional relationship within the framework of gravity field theory. This robust and consistent theoretical foundation is not Earth-specific; these two height systems can be directly extended and applied to other celestial bodies, such as the Moon and terrestrial planets.

(4) Definition of Geoid Geometric Deformation

The geodetic elevation (orthometric or normal height) of any terrain point is an objective quantity uniquely defined by its geopotential number. For a deforming Earth, the geopotential W changes objectively with time due to there distribution of internal mass, inducing temporal variations in the geopotential number (and thus geometric elevation) of any point. Earth’s deformation directly causes discrepancies of geopotential spatial distribution at two time epochs. This means that in the Earth-fixed reference system, the geometric positions of the geoid (where ɢ remains unchanged) at two epochs differ, and this difference represents the geoid's geometric deformation.

The Quasi-Geoid and Height System Conceptual Updates

(1) Applicability of the Geoid as the Zero-Height Surface

Geodetic elevation (orthometric or normal height) is rigorously defined in the Earth-fixed reference system grounded in gravity field theory. As geodetic elements, they are characterized by their uniqueness and measurability. However, the physical quantity they represent – the geopotential number – is inherently approximate, a limitation dictated by the geodetic definition of orthometric (or normal) height.

For any ground point where the normal height is zero, both its orthometric height and geopotential number are likewise zero. Its geopotential equals the geoid'sgeopotential constant, ₀ (or ɢ). Consequently, this point must lie on the geoid. Therefore, the three surfaces – the zero orthometric height surface, the zero normal height surface, and the zero geopotential number surface – all coincide with the geoid. That is, regardless of whether one employs theorthometric height, normal height, or geopotential number system, the height starting datum surface is invariably the (global or regional) geoid (ZHANG Chuanyin, 2017). Consequently, the reference starting surface for all three aforementioned height systems can only be the geoid.

(2) The Inconsistency of Utilizing the Quasi-Geoid as the Reference Datum

In classical physical geodesy, the quasi-geoid is defined as the closed surface where the ellipsoidal height coincides with the height anomaly. It is often erroneously adopted as the reference starting surface for normal heights.However, this practice contradicts the rigorous geodetic definition of normal height.

Firstly, the geopotential of zero normal equiheight surface is equal to zero, which is the geoid, not the quasi-geoid. Secondly, for two points sharing identical latitude and longitudes but differing in altitude, their height anomalies are defferent. If the normal height system were referenced to the quasi-geoid, it would necessitate two non-coincident reference staring points at the vertical direction, thereby violating the uniqueness requirement of the normal height system.

Furthermore, actual observation points rarely coincide precisely with the Digital Elevation Model (DEM) surface used to construct the height anomaly model. For centimeter-levelprecision, a height-dependent correction, δζ, accounting for the height anomaly gradient (or gravity disturbance), must be applied.

Consequently, while the definition of the normal height system is theoretically rigorous, the use of the quasi-geoid as its reference surface is fundamentally flawed. PAGravf4.5 deliberately de-emphasizes the concept of the quasi-geoid and doesnot treat it as the reference starting surface for normal heights. Within thePAGravf4.5 framework, the height anomaly is strictly and uniquely correlated with its three-dimensional spatial position.

(3) Geometric Properties and ConceptualUpdates of Height Systems

Both the orthometric and normal height systems are defined within an Earth-fixed reference system. Consider two surfaces of constant orthometric height, h₁ and h₂, such that their difference ∆h₁₂ = h₂ - h₁ = C ≠ 0 is constant. The ellipsoidal height difference between the two surfaces is: ∆H₁₂ = H₂ - H₁ = (h₂ + N) - (h₁ + N) = h₂ - h₁ = C. This demonstrates that the ellipsoidal height difference between two constant orthometric height surfaces equals their orthometric height difference, independent of the geoidal height . Therefore, all orthometric equi-height surfaces are parallel to the geoid, implying they share an identical geometricshape with the geoid.

It is particularly necessary to emphasize that the orthometric height is the straight-line distance from a ground point to the geoid, measured along the direction perpendicular to the geoidal surface. Due to the influence of vertical deflections and the curvature of normal gravity lines, the actual plumb line is an irregular curve. The length of this curved plumb line from the ground to the geoid is greater than the straight-line distance. The conventional view that orthometric height is the length of this irregular plumbline from the ground point to the geoid lacks a foundation in geodetic theory and is incorrect. Globally parallel closed surfaces do not exist in external Earth space. Since parallelism holds only in infinitesimally local spaces, orthometric height possesses typical local properties.

Normal height is also unique and measurable. However, since the height anomaly ζ varies with elevation, normal equi-height surfaces are not strictly parallel in the Earth-fixed reference system. The signal of the height anomaly attenuates with increasing elevation. Consequently, the geometric shape of a normal equi-height surface becomes progressively smoother relative to the geoid as elevation increases.

In summary, Orthometric Height offers intuitive geometric measurement properties due to the parallelism of its equi-height surfaces with the geoid, while Normal Height aligns more closely with gravity field properties, as its equi-height surfaces smooth out with elevation increase, reflecting the attenuation of gravitational signals. Both height systems possess distinct advantages, limitations, and scientific applicability. Their coexistence is both necessaryand scientifically justified.

Classification and Solution Methodologies for Earth's Gravity Field Boundary Value Problems

Classifying the Earth's gravity field Boundary Value Problems (BVPs) exclusively within theframework of the classical first, second, and third BVPs of potential theory isinsufficient for characterizing their integral solutions in a modern geodeticcontext. In classical potential theory, these categories prescribe thepotential, its normal derivative (gravity disturbances), or the obliquederivative (gravity anomalies) on the boundary, respectively. However,

A comparative analysis of the derivation processes for the integral solutions ofthe Stokes problem (classically a third BVP) and the Hotine problem(classically a second BVP) reveals that their mathematical derivations are fundamentally identical. Conversely, comparing the Stokes problem with theMolodensky problem – both classically categorized as third BVPs – exposes significant discrepancies in both their derivation methodologies and the formsof their solutions. The Molodensky problem is demonstrably more complex thanthe Stokes problem.

Consequently, restricting the classification of external gravity field BVPs to the classical potential theory framework fails to meet the theoretical demands of modern gravity field approximation and obscures the intrinsic nature of these geodetic problems.

In practice, the fundamental nature of a gravity field BVP is dictated primarily by whether the boundary surface is an equipotential surface. Since most observational quantities (e.g., gravity, vertical deflections) are obtained in the local level (plumb line and horizontal plane) system at the observationpoint, PAGravf4.5 classifies all external BVPs – where the boundary value is any linear combination of partial derivatives of the disturbing potential in the – into two primary categories:

(a) Stokes-type Problem: Encompasses all BVPs where the boundary surface is an equipotential surface (e.g., the geoid or a level surface), and the boundary value is any linear combination of partial derivatives of the disturbing potential in the local horizontal system.

(b) Molodensky-type Problem: Encompasses all BVPs where the boundary surface is not an equipotential surface (e.g., the Earth's topographical surface), and the boundary value is any linear combination of partial derivatives of the disturbing potential.

When the boundary surface is non-equipotential (a Molodensky-type problem), it can beaddressed using one of the following four approaches:

(a) Analytical Continuation to an Equipotential Surface: Gravity field elements on the boundary (e.g.,gravity anomalies, vertical deflections) are analytically continued onto a nearby equipotential surface. This transformation converts the boundary surface into an equipotential surface, thereby reducing the problem to a solvable Stokes-type formulation.

(b) Correction from Surface Normal to Plumb LineDirection: A correction is applied to the boundary values to account for the deviation between the surface's inner normal and the plumb line (vertical) direction.This adjustment effectively transforms the problem into a Stokes-type BVP.

(c) Direct Integral Solution on a Non-Equipotential Surface: The Molodensky BVP is solved directly on the actual, irregular boundary surface(e.g., the Earth's surface).

(d) Spectral-Domain Least Squares Approximation: The Molodensky BVP is solved using spectral-domain least squares collocation or spherical radial basis function approximation methods, which circumvent the complexities associated with the physical boundary surface.

Obtaining a rigorous closed-form integral solution for the Molodensky-type problem is mathematically intractable in many practical scenarios. Given that the integral solution of the Stokes-type problem also circumvents the complexities of terrain condensation when computing ground height anomalies, it is recommended that gravity field BVPs be addressed primarily using Stokes-type solution methods. Nevertheless, the theory of the Molodensky-type problem retains value for applications such as the reduction of gravity data to an equipotential surface and the quantitative analysis of approximation errors introduced by processing data on non-equipotential surfaces.

Error Analysis and Accuracy Assessment in Gravity Field Approximation

Unlike discrete geometric observables, the Earth's gravity field is a continuous physical entity. Consequently, gravity field elements are mathematically defined as spatial averages over finite resolutions rather than point values. This distinction necessitates a shift from standard geometric error analysis to a framework based on Spatial Representativeness.

(1) Concepts of Observational Error and Target Element Accuracy

In gravity field approximation, both observational and target elements function as spatial averages determined by grid resolution or spectral truncation.

● Observational Element Error: Defined as the Spatial Representativeness Error. It quantifies the discrepancy between the true spatial average and the gridded observation. This error is governed not merely by instrument precision, but by observation density and the complexity of the local gravity field structure (historically linked to terrain representativeness).

● Target Element Accuracy: Defined as the Spatial Representativeness Accuracy. It reflects the fidelity of the approximated element at its effective resolution.

Crucially, representativeness is coupled with field complexity: identical resolutions yield higher representativeness in smooth fields than in complex ones. However, analytical correlations between grid cells ensure that the information content exceeds simple geometric resolution limits.

(2) Theoretical Basis for Error Analysis

Gravity field approximation is a linear transformation within a linear space where elements are analytically correlated via spatial integrals or spectral expansions. This high analyticity allows for the extraction of weak signals from noisy environments, distinguishing it from geometric geodesy.

(a) Spatial-Domain Integral Methods:

Accuracy assessment follows two schemes:

● Scheme I (Direct Derivation): Estimates the spatial representativeness error of boundary data (accounting for discretization) and propagates this error through the integral formula to the target element.

● Scheme II (Indirect Assessment): Constructs an integral algorithm using observed elements as reference truth. By statistically analyzing discrepancies between these references and computed integral values, one derives the representativeness error to estimate target accuracy.

(b) Spectral-Domain Least Squares Methods:

Used in Global Geopotential Models (GGMs) or Spherical Radial Basis Function (SRBF) models, this rigorous framework involves:

● Residual Analysis: Computing residuals between spectral estimates (spatially representative) and observations (reference truth) to extract spatial representativeness errors.

● Error Propagation: Deducing propagation algorithms from the least squares model to compute the accuracy of spectral coefficients and target elements.

These methods effectively quantify the Spatial Representativeness Error for any observation type (spaceborne, airborne, terrestrial, or marine). While they provide only an upper bound for intrinsic instrument error, they determine the decisive factor governing the fidelity and resolution of the final gravity field model.

✋ Types of ground vertical deformation and space-time quantitative natures

There are three forms of ground vertical deformation (or ground subsidence), namely, the elastic load vertical deformation, viscous or plastic isostatic vertical deformation, and plastic tectonic vertical deformation near the compressive geological fracture zone. The latter two are also called as the non-load vertical deformations, both of which are plastic vertical deformations.

(1) The load vertical deformation excited by the surface mass redistribution, firstly causes the geopotential variation called as the direct effect, and then excites the solid Earth deformation simultaneously by elastic dynamic action to induce an associated geopotential variation called as the indirect effect. The load vertical deformation synchronizes with the time of load redistribution, whose time-varying characteristics are similar to that of the surface load variations with the complex nonlinearity and quasi-periodicity.

(2) The isostatic vertical deformation usually manifests as a dynamic process. In the process, the original equilibrium state of the underground rock and soil layer is firstly destroyed by the geology dynamic action, and then under the action of the own gravity or internal stress, the rock and soil layer slowly approaches another equilibrium state. For example, the compaction effects of the rock and soil layers with voids in the underground after the loss of water and the expansion effects after water infiltration, deformation of the upper rock layer (wall rock deformation) caused after underground construction and ground plastic isostatic rebound after surface mass migration.

• Spatial quantitative characteristics of the isostatic vertical deformation

The dynamic action is inside the underground rock and soil layer, and the equilibrium adjustment object is the rock and soil layer above the dynamic action point. The space influence angle of equilibrium adjustment is about 45˚, so the spatial range of ground vertical deformation is approximately equal to the buried depth of the action point.

• Temporal quantitative characteristics of the isostatic vertical deformation

The duration of the equilibrium adjustment is approximately proportional to the burial depth of dynamic action point. The isostatic vertical deformation is the opposite of its acceleration sign in a relatively long-period of time (e,g, several years), and linear time variation in a short period of time (e.g. several months).

(3) The tectonic vertical deformation driven by the horizontal movement of the lithospheric plate only appears near the compressive fault zone. Whose spatial influence radius is equivalent to the depth of the fault, and the deformation declines rapidly to zero with the distance of the calculation point away from the fault zone. On a centennial timescale, the tectonic vertical deformation rate would remain basically unchanged.

✋ Representation and approach principles of load deformation field

(1) The load deformation field is a form of non-tidal geodetic load effects, which can be uniquely represented by the variations of Earth’s gravity field with time. The relationship between the non-tidal load effects on the elements of Earth’s gravity field is completely consistent with the relationship between the elements. Global Earth gravity field can be represented by a geopotential coefficient model (GCM). Similarly, the global load deformation field (namely temporal global gravity field) can be represented by a global surface load spherical harmonic coefficient model (LCM).

(2) From a geopotential coefficient model, you can calculate various gravity field elements on the ground or outside the solid Earth. Similarly, from a global load spherical harmonic coefficient model, you can calculate load effects on various geodetic variations outside the solid Earth. Regional gravity field (geoid) can be refined by the remove-restore scheme based on a GCM. Similarly, the regional load deformation field or temporal gravity field can also be refined by the remove-restore scheme based on an LCM.

(3) The approach theory of Earth’s gravity field is linear, so that Earth’s gravity field can be refined by the remove-restore scheme and cumulative iteration method. Similarly, the approach theory of load deformation field is also linear, so load deformation field can be also refined by the remove-restore scheme and cumulative approach method. The total effects of various environmental loads (atmospheric pressure, land water, and sea level variation, etc.) are equal to the deformation effects of the sum of these loads.

(4) The approach methods of the Earth's gravity field can be summarized into two categories, namely, the Stokes / Hotine integral method (geodetic boundary value problem solution) in the spatial domain and the spherical basis function (e.g. surface spherical function, radial basis function and spline basis function) approach method in the spectral domain, which can integrate various global or regional gravity field data. Similarly, for load deformation field (time-varying gravity field) approach or monitoring, there are two methods namely the load Green's function integral constraint in spatial domain and spherical basis function approach in spectral domain, which can also effectively fuse global or regional multi-source heterogeneous geodetic variations.

✋ Analytical compatibility between various geodetic algorithms

The consistency and analytical compatibility between various geodetic algorithms are the requirement of geodetic theory and concrete manifestation for the uniqueness of monitoring object. Which is the smallest requirement for the collaborative monitoring of various geodetic technologies and deep fusion of multi-source heterogeneous geodetic data.

Analytical compatibility between geodetic algorithms involves two issues: (1) Compatibility between various geodynamic influences for different types of geodetic variations. (2) Compatibility between different types of geodynamic influences for one kind of geodetic element.

The first type of compatibility is the basic requirement of geodetic theory. For example, the load effect on the normal height on a site is equal to the Hotine integral of the load effect on gravity disturbances. For another example, the solid tidal effect on the normal height on a site is equal to the sum of the effects on the ellipsoidal height and geoid.

The second type of compatibility is constrained by the solid deformation geodynamic equations (including constitutive equations).

✋ CORS and InSAR collaborative monitoring principle for vertical deformation

(1) Through the gross error detection, spatial filtering and time series analysis, the InSAR vertical variation can be separated into two parts, one part is the vertical deformation of the rock and soil layer several meters deep, and the other part is the expansion and contraction of the soil own. Only the former is compatible with most geodetic variations, while the latter is mainly affected by the temperature and rainfall and should not be regarded as a solid Earth deformation.

(2) Using the CORS network ellipsoidal height variation time series as the constraints on the multi-source InSAR vertical variation time series, separate the ground vertical deformation signal, and then realize the collaborative monitoring of the CORS network and multi-source InSAR.

(3) Only the vertical deformation of the rock and soil layer several meters deep are the useful information needed for monitoring of the ground subsidence, earthquakes, geological disasters, ground stability variations, solid Earth deformation, groundwater variations and other geodynamics.

✋ Continuous quantitative monitoring scheme of ground stability variations

(1) Construct the quantitative criteria for the ground stability weakening from the regional grid time series of the geodetic vertical deformation, ground gravity and tilt variations, and then continuous quantitatively monitor the ground stability variations.

(2) Quantitative criteria of the ground stability weakening mainly include that the ellipsoidal height increases, ground gravity decreases, horizontal gradient of the height or gravity variation is larger and the inner product of the tilt variations and terrain slope vector is greater than zero.

(3) According to the objective nature of ground stability reduction at the place and time of geological disaster, optimize and synthesize the ground stability variation grid time series to adapt to the local environmental geology, and then consolidate regional stability variations monitoring capabilities.