Scientific Computation Platform for Geophysical Geodesy

The geodetic concepts and ideas that we have studied and summarized for 20 years are highly recommended here. These concepts and ideas are very beneficial to improve the level of geodetic understanding, and are also the basic principles and important basis for capturing, analyzing and solving many geodetic science and application problems. It is recommended to read, analyze and think about the content of this column frequently.

✍ Principles of geodetic collaborative monitoring and deep fusion

(1) Using scientific consistent geophysical models, rigorous uniform numerical standards, and analytic compatible geodetic and geodynamic algorithms, unify the space-time monitoring datum and reference epoch to construct the theoretical basis and necessary conditions for geodetic collaborative monitoring.

(2) Based on the principle of geophysical geodesy, deeply fuse or constrainedly assimilate multi-source heterogeneous geodetic data, and then by reconstructing the geodetic or geodynamic space-time relationship between various variations, realize collaborative monitoring with various geometric and physical geodetic technologies.

(3) For the same type of multi-source geodetic monitoring variations, the basic geodetic constraints or joint adjustment methods with additional monitoring datum parameters as needed are used to deep fusion.

(4) For different types of geodetic monitoring variations, physical geodetic, solid geophysical, or environmental geodynamic constraints with additional dynamic parameters as needed are used to deep fusion.

(5) The purpose of geodetic collaborative monitoring is not only to improve the spatiotemporal monitoring capability, but also to further reveal the geodynamic structure and characteristics of the monitored objects and support the massive integration, intelligence and automation of Earth observation.

Earth Gravity field, Geoid and Height datum

✋ Concepts and quantitative criterions for terrain effect

The terrain effect has always been a very tricky problem in physical geodesy and geophysical gravity exploration. According to the basic requirements of physical geodesy, PAGravf4.5 puts forward the convenient and practical quantitative criterion for terrain effects, which can give a reliable basis for effectively improving the performance and role of terrain effect in physical geodesy and geophysical gravity exploration.

The terrain effect on gravity field element has three key elements: The adjustment mode of the terrain or crustal masses, type of gravity field element effected by the terrain, and location of gravity field element (positional relationship with the terrain masses).

According to the different adjustment modes of terrain masses, the terrain effects usually include the local terrain effect, terrain Bouguer effect, ocean Bouguer effect, crustal equilibrium effect, terrain Helmert condensation and residual terrain effect, etc.

In physical geodesy, computation of the terrain effects on gravity field elements serves two basic purposes. One is to improve the griding performance of discrete field elements, and the other is to separate terrain ultrashort wave components for the gravity field approach. Accordingly, PAGravf4.5 defines the quantitative criteria for the selection of the terrain masses adjustment modes and algorithm parameters (such as integral radius, minimum degree of residual terrain model and spatial resolution, etc.).

(1) In order to improve the griding performance of discrete field elements, it is expected to improve the smoothness of discrete field elements after the terrain effect removed. In this case, the optimal criterion for terrain effect is that the standard deviation of discrete field elements would decrease after the terrain effect removed. This quantitative criterion is also applicable for geophysical gravity exploration purposes.

(2) The terrain effect is expected to consist of only ultrashort wave components for gravity field approach purpose, so the optimal criterion is that the standard deviation of field elements would decrease, and the statistical mean of terrain effects in the range of tens of kilometers is small after the terrain effect removed.

(3) The ratio D/ε of difference D between the maximum and minimum of terrain effects on a certain field element and its standard deviation ε, reflects the outlier of ultrashort wave signal in this mode of terrain effect. D/ε is large, which means that the proportion of ultrashort wave signals is small, but the signal is large. It is beneficial to improve the data processing performance to process this type of field element using this mode of terrain effect.

(4) When the sizes of several modes of terrain effects are roughly same, the greater the ratio of the standard deviation of terrain effect on gravity disturbance to the standard deviation of terrain effect on height anomaly, the richer the ultrashort wave components of terrain effect, and the more favorable it is for geoid refinement.

Among the above four guideline criterions defined by PAGravf4.5, the first two are the binding regulations, which are globally applicable and need to be followed. The latter two can be the technical references and should be employed appropriately based on further analysis. These criterions will no longer be appropriate when gravity field observations are so scarce that their space statistical representation is severely underrepresented.

The statistical properties of terrain effects vary significantly with the terrain and gravity field nature in the target area. It is recommended to calculate, compare, and analyze different modes of terrain effects on various field elements and their differences each other in advance, and then summarize the spectral domain character of the terrain effects and properties of the effect on different types of target field elements to design a calculation scheme with high adaptability based on these analysis results.

The performance of terrain effect is closely related to the complexity of the local terrain, the short-wave figure of the gravity field and the space distribution of the observations. When the terrain complexity is low, the local gravity field structure is not complex and the space distribution of the observations is dense, there is a possibility that the performance of the griding or gravity field approach cannot be further improved using any mode of terrain effect.

✋ The uniqueness of the geoid and PAGravf4.5ʹs realization

The Stokes boundary value problem requires that there are no masses outside the geoid, and the terrain masses should be compressed into the geoid under the condition of keeping the disturbing geopotentials unchanging outside the Earth surface. PAGravf4.5 believes that there is some a way to compress the terrain masses that the disturbing geopotential between ground and geoid are equal to the analytical continuation value of the disturbing geopotential outside the Earth surface, thereby the corresponding geoidal height is the analytical continuation solution of the geoid.

Various terrain effect on gravity disturbances and that on height anomalies computed by PAGravf4.5 satisfy the Hotine integral formula. For example, the terrain Helmert condensation effect on gravity disturbances (direct effects) and that on height anomalies (indirect effects) satisfy the Hotine integral formula. Therefore, no matter whether you choose the local terrain effect, terrain Helmert condensation, or residual terrain effect, the regional geoid refined by PAGravf4.5 programs with the terrain effect remove-restore scheme is the analytical continuation solution of the geoid.

Obviously, the geoidal height determined from satellite gravity field or directly calculated from a global geopotential coefficients model are all the analytical continuation solution of the geoid. Maintaining the uniqueness of the geoid solution, PAGravf4.5 can deeply integrate satellite gravity, geopotential coefficient model and regional gravity field data, and then theoretically strictly approach the gravity field and geoid.

✋ Classification and solution of gravity field boundary value problem

Most of the direct geodetic observations are based on the plumb line and level surface namely the natural coordinate system. For the sake of convenience, PAGravf4.5 divides the external boundary value problem into Stokes problem and Molodensky problem according to whether the inner normal of the boundary surface coincides with the plumb line, that is, whether the boundary surface is an equipotential surface.

(1) Stokes boundary value problem. The boundary value problem whose boundary surface is an equipotential surface, whose boundary value is a linear combination with the disturbing potential and its partial derivatives with respect to coordinates is called the Stokes boundary value problem. The inner normal lines of the boundary surface in the Stokes problem coincide with the plumb lines.

(2) Molodensky boundary value problem. The boundary value problem whose boundary surface is not an equipotential surface, whose boundary value is a linear combination with the disturbing potential and its partial derivatives with respect to coordinates is called the Molodensky boundary value problem. The inner normal lines of the boundary surface in the Molodensky problem do not coincide with the plumb lines.

When the boundary surface is not an equipotential surface, any one of the following three methods can be employed to solve the Molodensky boundary value problem.

(1) Analytically continuate the gravity field elements on the boundary surface to the equipotential surface close to the boundary surface. In this case, the new boundary surface becomes the equipotential surface, and the boundary value problem into the Stokes problem, and then solve the Stokes problem.

(2) Correct the gravity field elements on the boundary surface from the direction of the inner normal line of boundary surface to the direction of plumb line, so that the boundary value problem becomes a Stokes problem, and then solve the Stokes problem.

(3) Directly solve the Molodensky boundary value problem with the non-equipotential surface as the boundary surface. Which is not recommended by PAGravf4.5 due to the generally low accuracy of the solution.

PAGravf4.5 suggests that the geodetic boundary value problem is mainly solved according to the Stokes boundary value theory, and the Molodensky boundary value theory is mainly employed for the reduction of gravity field data into the equipotential surface, and also for error analysis for processing of gravity field data on the non-equipotential surface.

✋ Geopotential, geoid, and height datum

(1) The gravimetric geoid is the solution of the boundary value problem, whose geopotential is a constant and equal to the normal potential of the ellipsoidal surface, and the ellipsoidal surface is also the starting surface of the geoidal height.

(2) Global geopotential W₀ is an appoint geopotential for the global height datum in IERS numerical standards, which can be calculated from a latest geopotential model and sea surface height observations according to the Gaussian geoid appoint define.

(3) The appoint geopotential W₀ in IERS numerical standards has no direct geodetic relationship with the geopotential of the gravimetric geoid namely normal potential of the ellipsoidal surface.

(4) The zero normal height surface always coincides with the zero orthometric height surface everywhere, namely whether the orthometric or normal height system, the zero-height surface is the only one whose geopotential is constant.

(5) Essentially, the geoid determined according to the geodetic boundary value theory is the realization of the constant geopotential Wɢ in the Earth coordinate system, namely determining of the ellipsoidal height of the geoid.

(6) PAGravf4.5 recommends that the geoidal geopotential Wɢ or U₀ should replace the empirical appoint W₀ in the IERS numerical standard. The latter is calculated from the global geopotential model and satellite altimetry data according to the Gaussian geoid definition.

Whether for realizing of global height datum or for refining of regional height datum, if 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. It can effectively ensure the analytic rigor of gravity field approach in the realization and unification of height datum.

✋ Relationship between the height system and gravity field


✋ Analytical orthometric system is more suitable

(1) Let the gravity value of the move point between ground and geoid be equal to the gravity value analytically continued to the move point from the outer gravity field, and the resulting orthometric height is called the analytical orthometric height.

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.

(2) The geoidal height is the ellipsoidal height of the geoid, which is the solution of the Stokes boundary value problem in the Earth coordinate system. The measurement scale of the geoidal height is the geometric scale of the Earth coordinate system, while the measurement scale of the analytical orthometric height difference in the vertical direction is also strictly expressed by the geometric scale. Thus, the analytical orthometric height is consistent with the geometric scales of the Earth coordinate system and the geoidal height.

(3) The analytical orthometric height is not directly related to the terrain density, which can be continuously refined with the latest gravity field data. 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 compatible and 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.

height system 01.jpgheight system 02.jpg

✋ The problem of quasi-geoid as height datum surface

The geoid can be uniquely determined or continuously refined according to its geopotential value, and it can be employed as the orthometric height starting surface, which meets the requirements of the uniqueness of the geodetic datum. However, it is not theoretically rigorous to regard the quasi-geoid as the normal height starting surface.

(1) The zero normal height surface is the equipotential surface whose geopotential is equal to the geopotential at the height datum zero-point. It is the geoid, not the so-called quasi-geoid.

(2) Two points with the same latitude and longitude but different heights have different height anomalies. Therefore, if the normal height is considered to start from the quasi-geoid, there must be two different starting points in the vertical direction.

(3) In most cases, the measurement points will not be just on the specific ground elevation digital model surface employed in the quasi-geoid modelling. It is necessary to add a gradient (or gravity disturbance) correction for the height anomaly at the measurement point from the quasi-geoid model. see the section 5.1 for the calculation procedure.

PAGravf4.5 downplays the concept of quasi-geoid and does not regard so-called quasi-geoid as the starting surface of normal height. The height anomalies in PAGravf4.5 are strictly in one-to-one correspondence with their spatial locations.

Solid Earth Deformation and Geodetic Effects

✋ Tidal deformations of solid Earth and tidal effects on geodetic variations

(1) The external celestial bodies, ocean tide, and atmospheric tide excite the periodic deformation of the solid Earth and the periodic change of the gravity field, which are called as the tidal deformation of the solid Earth.

(2) The geodetic variations caused by the external celestial bodies, ocean tides, and atmospheric tides are usually called the tidal effects on the geodetic variations.

(3) The geodetic tidal effects include the solid Earth tidal effects and the load tidal effects. The geodetic solid Earth tidal effects are excited by the external celestial bodies, and the tidal load effects are excited by the ocean tide and atmospheric tide.

(4) The geodetic tidal effects can be modeled and accurately removed or restored anytime and anywhere. The geodetic tidal effect is equal to the negative value of the geodetic tidal correction.

The geodetic reference frame with only some tidal effects removed but non-tidal effects neglected is still stationary (unchanged with time). For example, a precision leveling network or a gravity control network, if its observations have been corrected only using some tidal effects, is still stationary.

✋ Non-tidal deformation of solid Earth and their effects on geodetic variations

(1) In the Earth surface system, surface non-tidal load variations such as soil and vegetation water, lake water, glacier and snow, groundwater, atmosphere, and sea level variations can induce the external geopotential variations, and then excite solid Earth deformation, which is manifested as ground displacement, gravity, and tilt variations. Which can be called as the load deformation of the solid Earth, and also takes the form of the variation of the Earth’s gravity field with time.

(2) Groundwater use, underground mining, underground construction, glacier or ice sheet melting, and other natural or artificial surface mass adjustments can break the mechanical balance state of the surface rock and soil layer, and then the surface rock and soil layer will tend slowly to another equilibrium state under the action of its own gravity or internal stress. The process causes plastic or viscous vertical deformation which is also called as the isostatic vertical deformation.

(3) The load deformation is excited by the surface environment load variations, and acts on the entire solid Earth. Which is an elastic deformation and can be quantitatively represented by the load Love numbers. The isostatic vertical deformation is induced by environmental geology change, whose dynamic action is in the underground rock and soil and transmitted by the rock and soil own as the mechanical medium. The isostatic deformation is a slow plastic or viscous vertical deformation.

(4) Non-tidal effects are difficult to be modeled and are generally measured using geodetic techniques. In most fast or real-time geodetic applications, short-time forecast estimations of the pole shift are adopted. The pole shift is the instantaneous loaction shift of the Earth pole at the current epoch relative to a certain reference epoch (such as epoch J2000.0) after removing all solid earth tides and load tidal effects. Neither the pole shift nor geocentric movement should not include various tidal effects.

The geodetic reference frame that needs to account for non-tidal effects can be only dynamic, and the reference value of the dynamic reference frame corresponds to a specific and unique reference epoch time. The reference value at the current epoch time is equal to the sum of the reference value at the reference epoch time and the non-tidal effect correction. The correction here is equal to the difference between the non-tidal effects at the current epoch and that at the reference epoch. The correction process is also called as the (non-tidal effects) epoch reduction.

✋ 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 Earth geopotential variation called as the direct effect, and then by Earth elastic dynamic action, causes the solid Earth deformation simultaneously to generate an additional geopotential variation called as the indirect effect. The load vertical deformation synchronizes with the time of the load redistribution, whose time-varying characteristics are similar to that of the surface load variations, showing 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 gravity or internal stress, the rock and soil layer slowly approach another equilibrium state. For example, the compaction effect of the rock and soil layers with voids in the ground after the loss of water and the expansion effect after water infiltration, the 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 located 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 the equilibrium adjustment is about 45˚, that is, 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 the dynamic action location. The isostatic vertical deformation is the opposite of its acceleration rate sign in a relatively long period of time (several years), and linear time variation in a short period of time (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 decays rapidly to zero with the distance of the calculated point away from the fault zone. On a centennial timescale, the tectonic vertical deformation rate remained basically unchanged.

✋ Representation and approach principles of load deformation field

(1) The load deformation field is a form of geodetic non-tidal load effects, which can be represented by the variations of the Earth’s gravity field with time. The relationship between the load effects on the parameters of the Earth’s gravity field is completely consistent with the relationship between the parameters. 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 quantities on the surface or outside 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 approached by the remove-restore process based on a GCM. Similarly, the regional load deformation field or temporal gravity field can also be approached by the remove-restore process based on an LCM.

(3) The approach theory of Earth’s gravity field is linear, so that Earth’s gravity field can be approached 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 approoach method. The total deformation effects of various environmental loads (atmospheric pressure, land water, sea level change, etc.) are equal to the deformation effects of the sum of these loads.

(4) The approach methods of Earth's gravity field can be summarized into two categories, namely, the Stokes / Hotine integral method (geodetic boundary value solution) in the spatial domain and the spherical basis function (surface spherical function, radial basis function, spline function, etc.) 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 of load Green 's function integral constraint in spatial domain and spherical basis function approach in spectral domain, which can also effectively integrate global or regional multi-source heterogeneous 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 objects. Which is the smallest requirement for the collaborative monitoring of multi-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 of one kind of geodetic quantity.

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 is 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 multi-source InSAR vertical variation time series, separate the ground vertical deformation, 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 the ground subsidence, earthquakes, geological disasters, ground stability variations, solid Earth deformation, groundwater variations, and 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 ground stability weakening mainly include that the ellipsoidal height increases, gravity decreases, horizontal gradient of the height or gravity variation is large, and the inner product of the tilt variations and terrain slope vector is greater than zero.

(3) According to the geological disasters that occurred, optimize and synthesize various geodetic ground stability variation grid time series to adapt to the local environmental geology, and then consolidate regional stability variations monitoring capabilities.