Scientific Computation Software for Geophysical Geodesy

Solid Earth Deformation and Geodetic Effects

Tidal deformations of solid Earth and tidal effects on geodetic quantities

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

(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 tides and atmospheric tides.

(4) The geodetic tidal effects can be modeled and can be 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 quantities

(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. This is called the load-deformation of the solid Earth, which 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 slowly tend 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 isostatic vertical deformation.

(3) The load-deformation is excited by the surface environment load variations, and act 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 is transmitted by the rock and soil own as the mechanical medium. The isostatic deformation is a slow plastic or viscous vertical deformation.

(4) The pole shift is the instantaneous loaction shift of the Earth pole relative to a certain reference epoch (such as epoch J2000.0) after removing all solid earth tides and loading tidal effects. Neither the pole shift nor geocentric movement include various tidal effects. 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 geodetic reference frame that needs to account for non-tidal effects can only be 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 a correction. The correction is equal to the difference of the non-tidal effects between at the current epoch time and at the reference epoch time. The correction process is also called the (non-tidal effects) epoch reduction.

✋Representation and approaching of load deformation and temporal gravity field

(1) The non-tidal load effects can be uniquely represented by the variations of the Earth’s gravity field with time. The relationship between the non-tidal load effects is completely consistent with the relationship between the parameters of the Earth’s gravity field.

(2) Global Earth gravity field can be represented by a geopotential coefficients model (GCM). Similarly, the global load-deformation field (namely temporal global gravity field) can be represented by a global surface load spherical harmonic coefficients model (LCM).

(3) Using a geopotential coefficients model, you can calculate various gravity field quantities on the surface or outside Earth. Similarly using a global load spherical harmonic coefficients model, you can calculate load effects on various geodetic quantities outside the solid Earth.

(4) 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.

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

There are three forms of ground vertical deformation (or ground subsidence), namely, the elastic loading 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 the non-loading vertical deformations, both of which are plastic vertical deformations.

(1) The loading vertical deformation is excited by the surface mass redistribution which 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 loading vertical deformation synchronizes with the time of the loading redistribution, whose time-varying characteristics are similar to 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 by underground engineering, and plastic isostatic rebound of the rock and soil layer 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.

✋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 variations time series as the constraints on the multi-source InSAR vertical variations 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 is the useful information needed for monitoring of 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 reduction from the regional grids 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 reduction mainly include that the ellipsoidal height increases, the gravity decreases, the 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 a variety of geodetic ground stability variation grids time series to adapt to the local environmental geology, and then consolidate regional stability variations monitoring capabilities.

✋Analytical compatibility between various geodetic algorithms

The consistency and analytical compatibility between various geodetic algorithms are the concrete manifestation for the requirement of geodetic theory and 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 quantities. (2) Compatibility between different types of geodynamic influences of one kind of geodetic quantitity.

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).

Earth Gravity field, Geoid and Height datum
Principles of Selecting Topographic Effects

In the local gravity field approach, the topographic effects computation has two basic purposes: one is to grid the discrete anomalous field elements for numerical integral operations, and the other is to separate the ultrashort wave components for the integral of anomalous gravity field.

In order to improve the gridding accuracy of the discrete field elements, it is required that the discrete field elements becomes smoother after removing the topographic effects. In this case, the criteria for selecting some a type of topographic effects is that after the topographic effects are removed, the standard deviation of the discrete field elements becomes smaller.

When the field elements are integrated, it is required that the topographic effects only have the ultrashort wave components. In this case, the criteria for selecting some a type of topographic effect is that after the topographic effects are removed, the standard deviation of the field elements are smaller, and the average value of the topographic effects is also small in the range of more than tens of kilometers.

Topographic Masses Effects for Physical Geodesy

The concept of topographic masses effects of anomalous gravity field elements have 3 essential key factors, which are the adjustment manner of the topographic or crustal masses, the type of the anomalous field elements affected, and location of the computed point relative to the masses.

According to different masses adjustment manners, the topographic effects commonly include: local terrain effect, crustal isostasy effect, topographic Bouguer masses effects, sea water Bouguer condensation effect, topographic Helmert condensation, residual topographic masses effect and so on.

Similar to the remove-restore scheme using an earth geopotential model, a remove-restore scheme using the topographic effects can be adopted to improve the local gravity field approach performance and algorithm stability when the topographic effects outside the earth are harmonic.

The uniqueness of the geoid and its realization

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

The topographic effects of gravity disturbances and the topographic effects of height anomalies computed by PALGrav4.0 satisfy Hotine integral formula. For example, the topographic Helmert condensation of gravity disturbances (direct effects) and the Helmert condensation of height anomalies (indirect effects) satisfy the Hotine integral formula. Therefore, no matter whether you choose local terrain effect, topographic Helmert condensation, or residual topographic effect, the regional geoid refined by the PALGrav4.0 programs with the topographic effects remove-restore scheme is the analytical continuation solution of geoid.

Obviously, the geoid height determined by the satellite gravity field or directly calculated by a geopotential coefficients model are all an analytical continuation solution of the geoid. By maintaining the uniqueness of the geoid solution, PALGrav4.0 can deeply integrate satellite gravity, geopotential coefficients model and regional gravity field data, and then theoretically strictly approach the local gravity field and geoid.

Uniqueness of integral solutions and its roles

The integral solutions of local gravity field are unique, which can guarantee the harmonic invariance of the local gravity field and the analytical invariance of the field spatial signature. The numerical integral solution of the local gravity field have the same properties and functions on the regional scale as the earth geopotential coefficients model on the global scale.

Since the numerical integral solutions are unique and harmonic, the remove-restore schemes with the previous numerical integral solutions as the reference field can be constructed and then play a scale controlling role to approach the current local gravity field and to refine a very small area geoid. So that the seamless splicing problem between the adjacent geoid models will become disappeared, and the current geoid model can be rigorously compatible with the previous geoid model.

Geopotential, Geoid and Height Datum

The gravimetric geoid, which is the solution of the boundary value problem, whose geopotential is a constant which is equal to the normal geopotential of the ellipsoidal surface which is the starting surface of the geoidal height.

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

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.

The appoint geopotential W₀ has no direct geodetic relationship with the geopotential of the gravimetric geoid namely normal geopotential of the ellipsoidal surface.

Analytical orthometric system is more suitable

Let the gravity value of the moving point between the ground and the geoid be equal to the gravity value analytically continued to the moving point by the outer gravity field, and the resulting orthometric height is called the analytical orthometric height. The difference between the analytical orthometric height and the Helmert orthometric height on global ground is no more than 10cm.

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. So its measurement scale is the geometric scale of the earth coordinate system. The measurement scale of the analytical orthometric height difference in the vertical direction is also strictly expressed by the geometric scale. It can be found that the analytical orthometric height is consistent with the geometric scales of the earth coordinate system and the geoid height.

The analytical orthometric height is not directly related to the terrain density, and 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.