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1.
Summary The general problem of determining the figure of the earth leads to the solution of the geodetic boundary value problem. By its discrete approximation we obtain the discrete disturbing potential that maintains all properties of the original problem. Thus, the discrete approximation of the disturbing potential can be used in studying the behaviour of the earth's gravity field outside the disturbing masses. The deflections of the vertical are one of the quantities describing the behaviour of the earth's gravity field. A method for their computation from the discrete solution of the geodetic boundary value problem is put forth and estimates for its accuracy are given.  相似文献   

2.
Summary Following Molodensky's suggestions anomalies of the vertical gradient of gravity were used to achieve a greater accuracy in the determination of the figure of the Earth by gravimetrical methods. The existing methods of computing this quantity do not take into account inclinations of the physical surface of the Earth. Using the Laplace equation, the second derivative ∂2 T/∂v 2 (1) of the disturbing potentialT is expressed by the second derivatives ofT along the tangentsτ 1 andτ 2 to the physical surface of the Earth in mutually perpendicular planes and by the derivatives of gravity anomalies (2). The derivatives ∂2 T/∂τ 1 2 and ∂2 T/∂τ 2 2 have been determined using the Molodensky method [4] of solving his integral equation for the single layer density. In the zero approximation, the Noumerov formula [2] was obtained; however, the results obtained using this formula should be referred to the physical surface of the Earth, not to the Listing geoid. The correction of the first approximation is given by formula (16). The second vertical derivative of gravity anomalies can be determined using the expression (20).   相似文献   

3.
Solution of the gradiometric boundary value problems leads to three integral formulas. If we are satisfied with obtaining a smooth solution for the Earth’s gravity field, we can use the formulas in regional gravity field modelling. In such a case, satellite gradiometric data are integrated on a sphere at satellite level and continued downward to the disturbing potential (geoid) at sea level simultaneously. This paper investigates the gravity field modelling from a full tensor of gravity at satellite level. It studies the truncation bias of the integrals as well as the filtering of noise of data. Numerical studies show that by integrating T zz with 1 mE noise and in a cap size of 7°, the geoid can be recovered with an error of 12 cm after the filtering process. Similarly, the errors of the recovered geoids from T xz,yz and T xx-yy, 2xy are 13 and 21 cm, respectively.  相似文献   

4.
The following Poisson’s equation with the Stokes’ boundary condition is dealt with $$\left\{ \begin{gathered} \nabla ^2 T = - 4\pi Gp outside S, \hfill \\ \left. {\frac{{\partial T}}{{\partial h}} = \frac{1}{\gamma }\frac{{\partial y}}{{\partial h}}T} \right|_s = - \Delta g, \hfill \\ T = O\left( {r^{ - 3} } \right) at infinity, \hfill \\ \end{gathered} \right.$$ whereS is reference ellipsord. Under spherical approximation transformation, the ellipsoidal correction terms about the boundary condition, the equation and the density in the above BVP are respectively given. Therefore, the disturbing potentialT can he obtained if the magnitudes aboveO(ε4) are neglected.  相似文献   

5.
The Boundary Element Method (BEM), a numerical technique for solving boundary integral equations, is introduced to determine the earth's gravity field. After a short survey on its main principles, we apply this method to the fixed gravimetric boundary value problem (BVP), i.e. the determination of the earth's gravitational potential from measurements of the intensity of the gravity field in points on the earth's surface. We show how to linearize this nonlinear BVP using an implicit function theorem and how to transform the linearized BVP into a boundary integral equation using the single layer representation. A Galerkin method is used to transform the boundary integral equation using the single layer representation. A Galerkin method is used to transform the boundary integral equation into a linear system of equations. We discuss the major problems of this approach for setting up and solving the linear system. The BVP is numerically solved for a bounded part of the earth's surface using a high resolution reference gravity model, measured gravity values of high density, and a 50 50 m2 digital terrain model to describe the earth's surface. We obtain a gravity field resolution of 1 1 km2 with an accuracy of the order 10–3 to 10–4 in about 1 CPU-hour on a Siemens/Fujitsu SIMD vector pipeline machine using highly sophisticated numerical integration techniques and fast equation solvers. We conclude that BEM is a powerful numerical tool for solving boundary value problems and may be an alternative to classical geodetic techniques.  相似文献   

6.
A new gravimetric, satellite altimetry, astronomical ellipsoidal boundary value problem for geoid computations has been developed and successfully tested. This boundary value problem has been constructed for gravity observables of the type (i) gravity potential, (ii) gravity intensity (i.e. modulus of gravity acceleration), (iii) astronomical longitude, (iv) astronomical latitude and (v) satellite altimetry observations. The ellipsoidal coordinates of the observation points have been considered as known quantities in the set-up of the problem in the light of availability of GPS coordinates. The developed boundary value problem is ellipsoidal by nature and as such takes advantage of high precision GPS observations in the set-up. The algorithmic steps of the solution of the boundary value problem are as follows:
- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of the ellipsoidal centrifugal field for the removal of the effect of global gravity and the isostasy field from the gravity intensity and the astronomical observations at the surface of the Earth.
- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the gravity intensity and the astronomical observations at the surface of the Earth the effect of the residual masses at the radius of up to 55 km from the computational point.
- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and ellipsoidal centrifugal field for the removal from the geoidal undulations derived from satellite altimetry the effect of the global gravity and isostasy on the geoidal undulations.
- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the geoidal undulations derived from satellite altimetry the effect of the water masses outside the reference ellipsoid within a radius of 55 km around the computational point.
- Least squares solution of the observation equations of the incremental quantities derived from aforementioned steps in order to obtain the incremental gravity potential at the surface of the reference ellipsoid.
- The removed effects at the application points are restored on the surface of reference ellipsoid.
- Application of the ellipsoidal Bruns’ formula for converting the potential values on the surface of the reference ellipsoid into the geoidal heights with respect to the reference ellipsoid.
- Computation of the geoid of Iran has successfully tested this new methodology.
Keywords: Geoid computations; Ellipsoidal approximation; Ellipsoidal boundary value problem; Ellipsoidal Bruns’ formula; Satellite altimetry; Astronomical observations  相似文献   

7.
General inverse of Stokes, Vening-Meinesz and Molodensky formulae   总被引：1，自引：0，他引：1  
The undulation of the geoid, the gravity anomaly and the deflection of the vertical are the three basic observations describing the shape and the gravity field of the earth. The Stokes’ formula that computes the undulation of the geoid using the gravity anomaly on the geoid under spherical approximate conditions was first put forward by Stokes[1]. According to Stokes’ theory, The Vening-Meinesz formula that computes the meridian and the prime vertical components of the deflection of the ve…  相似文献   

8.
A spherical approximation makes the basis for a majority of formulas in physical geodesy. However, the present-day accuracy in determining the disturbing potential requires an ellipsoidal approximation. The paper deals with constructing Green’s function for an ellipsoidal Earth by an ellipsoidal harmonic expansion and using it for determining the disturbing potential. From the result obtained the part that corresponds to the spherical approximation has been extracted. Green’s function is known to depend just on the geometry of the surface where boundary values are given. Thus, it can be calculated irrespective of the gravity data completeness. No changes of gravity data have an effect on Green’s function and they can be easily taken into account if the function has already been constructed. Such a method, therefore, can be useful in determining the disturbing potential of an ellipsoidal Earth.  相似文献   

9.
In this article, we first reviewed the method of boundary integral equation (BIEM) for modelling rupture dynamics of a planar fault embedded in a 3-D elastic half space developed recently (ZHANG and CHEN, 2005a,b). By incorporating the half-space Green's function, we successfully extended the BIEM, which is a powerful tool to study earthquake rupture dynamics on complicated fault systems but limited to full-space model to date, to half-space model. In order to effectively compute the singular integrals in the kernels of the fundamental boundary integral equation, we proposed a regularization procedure consisting of the generalized Apsel-Luco correction and the Karami-Derakhshan algorithm to remove all the singularities, and developed an adaptive integration scheme to efficiently deal with those nonsingular while slowly convergent integrals. The new BIEM provides a powerful tool for investigating the physics of earthquake dynamics. We then applied the new BIEM to investigate the influences of geometrical and physical parameters, such as the dip angle (δ) and depth (h) of the fault, radius of the nucleation region (Rasp), slip-weakening distance (Dc), and stress inside (Ti) and outside (Te) the nucleation region, on the dynamic rupture processes on the fault embedded in a 3-D half space, and found that (1) overall pattern of the rupture depends on whether the fault runs up to the free surface or not, especially for strike-slip, (2) although final slip distribution is influenced by the dip angle of the fault, the dip angle plays a less important role in the major feature of the rupture progress, (3) different value of h, δ, Rasp, Te, Ti and Dc may influence the balance of energy and thus the acceleration time of the rupture, but the final rupture speed is not controlled by these parameters.  相似文献   

10.
Using approximate boundary conditions, expressions for electromagnetic fields have been derived for a thin, highly resistive layer lying between two homogeneous layers excited by an electric dipole grounded on the surface of the earth. The variations of the fields with the parameter T/T1 (ratio of the transverse resistance of the thin layer to the transverse resistance of the first layer) were studied in relation to frequency, time, the normalized separation source—receiver, and the angle between the source and the radius to the observation point. For a value of h2/h1 (ratio of thickness of second layer to the thickness of the first layer) approximately equal to 0.2, the general three-layer medium case gives the same results as this approach. It was found that the electric fields have a very strong dependence on the parameter T (transverse resistance) which characterizes the thin, highly resistive layer. However, the magnetic fields depend only very weakly on this parameter.  相似文献   

11.
An ellipsoidal Neumann type geodetic boundary-value problem (GBVP) for the computation of disturbing potential on the surface of the Earth based on the surface gravity disturbance as the boundary data is formulated. The solution methodology of the GBVP can be algorithmically summarized as follows: (i) using global navigation satellite systems (GNSS) coordinates of the gravity stations, the surface gravity disturbances are generated as the boundary data. (ii) Applying the deflection correction to the gravity disturbances to arrive at the derivative of the surface disturbing potential along the ellipsoidal normal. (iii) Removing the low frequencies part of the gravity field using harmonic expansion to degree and order 110. (iv) Using the short wavelength part of the corrected gravity disturbances derived in the previous section as the boundary data within the constructed GBVP to derive the short wavelength disturbing potential over the Earth surface. (v) The computed shortwave length signals of disturbing potentials are converted to disturbing potential values by restoring the removed effects.  相似文献   

12.
Tsunami created by spreading submarine slides and slumps with spatially variable final uplift are investigated in the near-field using a kinematic model. It is shown that for velocities of spreading comparable to and smaller than the long period tsunami velocity (g is the acceleration due to gravity and h is the ocean depth), the models with spatially uniform final uplift of the accumulation and depletion zones provide good approximation for the tsunami amplitudes in the near-field. For spreading velocities 2–5 times greater than cT, and for applications that use wavelengths of the order of the source dimensions, the spatial variability of the final uplift has to be considered in estimation of the high-frequency tsunami amplitudes in the near-field.  相似文献   

13.
Boundary value problem (BVP) plays a funda-mental role in physical geodesy that aims at determin-ing the earth’s shape and its external gravity field. TheMolodensky BVP and the Stokes BVP are typical inphysical geodesy, and the gravity anomaly is a kind ofbasic data. With the wide use of GPS, measurementaccuracy of the earth’s surface can reach one centime-ter, while that of the gravity measurement can reachμgals. Hence, it is necessary to establish a new kind ofBVP which can satisfy…  相似文献   

14.
The simultaneous solution of the Planck equation (involving the widely used “dual-band” technique) using two shortwave infrared (SWIR) bands allows for an estimate of the fractional area of the hottest part of an active lava flow (f h) and the background temperature of the cooler crust (T c). The use of a high spectral and spatial resolution imaging spectrometer with a wide dynamic range of 15 bits (DAIS 7915) in the wavelength range from 0.501 to 12.67 μm resulted in the identification of crustal temperature and fractional areas for an intra-crater hot spot at Mount Etna, Italy. This study indicates the existence of a relationship between these T c and f h extracted from DAIS and Landsat TM data. When the dual band equation system is performed on a lava flow, a logarithmic distribution is obtained from a plot of the fractional area of the hottest temperature vs. the temperature of the cooler crust. An entirely different distribution is obtained over active degassing vents, where increases in T c occur without any increase in f h. This result indicates that we can use scatter plots of T c vs. fh to discriminate between different types of volcanic activity, in this case between degassing vents and lava flows, using satellite thermal data.  相似文献   

15.
The effects of variable speeds of spreading of submarine slides and slumps on near-field tsunami amplitudes are illustrated. It is shown that kinematic models of submarine slides and slumps must consider time variations in the spreading velocities, when these velocities are less than about 2cT, where is the long period tsunami velocity in ocean of constant depth h. For average spreading velocities greater than 2cT, kinematic models with assumed constant spreading velocities provide good approximation for the tsunami amplitudes above the source.  相似文献   

16.
Proposed is a new definition of earthquake response spectra, which takes account of the number of response cycles N. The Nth largest amplitude of absolute acceleration response of a linear oscilator with natural period T and damping ratio h, which is subjected to ground motion at its base, is defined as SA(T, h, N). By defining a reduction factor η(T, h, N) as SA(T, h, N)/SA(T, h, 1), characteristics of η(T, h, N) were investigated based on 394 components of strong motion records obtained in Japan. Two practical empirical formulae to assess the reduction factor η(T, h, N) are proposed.  相似文献   

17.
Gravity gradients can be used to determine the local gravity field of the Earth. This paper investigates downward continuation of all elements of the disturbing gravitational tensor at satellite level using the second-order partial derivatives of the extended Stokes formula in the local-north oriented frame to determine the gravity anomaly at sea level. It considers the inversion of each gradient separately as well as their joint inversion. Numerical studies show that the gradients Tzz, Txx, Tyy and Txz have similar capability of being continued downward to sea level in the presence of white noise, while the gradient Tyz is considerably worse than the others. The bias-corrected joint inversion process shows the possibility of recovering the gravity anomaly with 1 mGal accuracy. Variance component estimation is also tested to update the observation weights in the joint inversion.  相似文献   

18.
The redistribution of air masses induces gravity variations (atmospheric pressure effect) up to about 20 μgal. These variations are disturbing signals in gravity records and they must be removed very carefully for detecting weak gravity signals. In the past, different methods have been developed for modelling of the atmospheric pressure effect. These methods use local or two-dimensional (2D) surface atmospheric pressure data and a standard height-dependent air density distribution. The atmospheric pressure effect is consisting of the elastic deformation and attraction term. The deformation term can be well modelled with 2D surface atmospheric pressure data, for instance with the Green's function method. For modelling of the attraction term, three-dimensional (3D) data are required. Results with 2D data are insufficient.From European Centre for Medium-Range Weather Forecasts (ECMWF) 3D atmospheric pressure data are now available. The ECMWF data used here are characterised by a spacing of Δ and Δλ = 0.5°, 60 pressure levels up to a height of 60 km and an interval of 6 h. These data are used for modelling of the atmospheric attraction term. Two attraction models have been developed based on the point mass attraction of air segments and the gravity potential of the air masses. The modelling shows a surface pressure-independent part of gravity variations induced by mass redistribution of the atmosphere in the order of some μgal. This part can only be determined by using 3D atmospheric pressure data. It has been calculated for the Vienna Superconducting Gravimeter site.From this follows that the gravity reduction can be improved by applying the 3D atmospheric attraction model for analysing long-periodic tidal waves including the polar tide. The same improvement is expected for reduction of long-term absolute gravity measurements or comparison of gravity measurements at different seasonal times. By using 3D atmospheric pressure data, the gravity correction can be improved up to some μgal.  相似文献   

19.
The vertical hydraulic conductivity (Kv) of a stream or lake sediment is often determined in the field using standpipe tests. Calculation of Kv is based on the assumption that the hydraulic head in the pipe is equal to that of the stream or lake stage. In this work, a modified equation for Kv is developed for the standpipe test which is applicable when this assumption is not valid. The equation involves not only the hydraulic head at different times but also the difference in the hydraulic head (a) between the groundwater level and river stage. The effects of certain factors on Kv, such as the ratio of the hydraulic head at different times (h1/h2), the difference a, and the initial water table height (h0), are also discussed. The results show that when h1/h2 is constant, the relative error (Er) in Kv increases with the ratio a/h2. Furthermore, if a/h2 < 0.05, then for any value of h1/h2, Er is less than 5% using the modified equation. Also, if a/h2 is large, hydraulic head readings with larger h1/h2 ratios must be used to avoid large Er values. The results of a field test also indicate that the error in Kv decreases as the value of h0 increases. Copyright © 2013 John Wiley & Sons, Ltd.  相似文献   

20.
The Platanares geothermal area in western Honduras consists of more than 100 hot springs that issue from numerous hot-spring groups along the banks or within the streambed of the Quebrada de Agua Caliente (brook of hot water). Evaluation of this geothermal area included drilling a 650-m deep PLTG-1 drill hole which penetrated a surface mantling of stream terrace deposits, about 550 m of Tertiary andesitic lava flows, and Cretaceous to lower Tertiary sedimentary rocks in the lower 90 m of the drill core.Fractures and cavities in the drill core are partly to completely filled by hydrothermal minerals that include quartz, kaolinite, mixed-layer illite-smectite, barite, fluorite, chlorite, calcite, laumontite, biotite, hematite, marcasite, pyrite, arsenopyrite, stibnite, and sphalerite; the most common open-space fillings are calcite and quartz. Biotite from 138.9-m depth, dated at 37.41 Ma by replicate 40Ar/39 Ar analyses using a continuous laser system, is the earliest hydrothermal mineral deposited in the PLTG-1 drill core. This mid-Tertiary age indicates that at least some of the hydrothermal alteration encountered in the PLTG-1 drill core occured in the distant past and is unrelated to the present geothermal system. Furthermore, homogenization temperatures (Th) and melting-point temperatures (Tm) for fluid inclusions in two of the later-formed hydrothermal minerals, calcite and barite, suggest that the temperatures and concentration of dissolved solids of the fluids present at the time these fluid inclusions formed were very different from the present temperatures and fluid chemistry measured in the drill hole.Liquid-rich secondary fluid inclusions in barite and caicite from drill hole PLTG-1 have Th values that range from about 20°C less than the present measured temperature curve at 590.1-m depth to as much as 90°C higher than the temperature curve at 46.75-m depth. Many of the barite Th measurements (ranging between 114° and 265°C) plot above the reference surface boiling-point curve for pure water assuming hydrostatic conditions; however, the absence of evidence for boiling in the fluid inclusions indicates that at the time the minerals formed, the ground surface must have been at least 80 m higher than at present and underwent stream erosion to the current elevation. Near-surface mixed-layer illite-smectite is closely associated with barite and appears to have formed at about the same temperature range (about 120° to 200°C) as the fluid-inclusion Thvalues for barite. Fluid-inclusion Th values for calcite range between about 136° and 213°C. Several of the calcite Th values are significantly lower than the present measured temperature curve. The melting-point temperatures (Tm) of fluid-inclusion ice yield calculated salinities, ranging from near zero to as much as 5.4 wt. % NaCl equivalent, which suggest that much of the barite and calcite precipitated from fluids of significantly greater salinity than the present low salinity Platanares hot-spring water or water produced from the drill hole.  相似文献   

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