Christopher Jekeli

THE DETERMINATION OF GRAVITATIONAL POTENTIAL DIFFERENCES FROM SATELLITE-TO-SATELLITE TRACKING

A new, rigorous model is developed for the difference of gravitational potential between two close earth-orbiting satellites in terms of measured range-rates, velocities and velocity differences, and specific forces. It is particularly suited to regional geopotential determination from a satellite-to-satellite tracking mission. Based on energy considerations, the model specifically accounts for the time variability of the potential in inertial space, principally due to earth’s rotation. Analysis shows the latter to be a significant (±1 m2/s2) effect that overshadows by many orders of magnitude other time dependencies caused by solar and lunar tidal potentials. Also, variations in earth rotation with respect to terrestrial and celestial coordinate frames are inconsequential. Results of simulations contrast the new model to the simplified linear model (relating potential difference to range-rate) and delineate accuracy requirements in velocity vector measurements needed to supplement the range-rate measurements. The numerical analysis is oriented toward the scheduled Gravity Recovery and Climate Experiment (GRACE) mission and shows that an accuracy in the velocity difference vector of 2×10−5 m/s would be commensurate within the model to the anticipated accuracy of 10−6 m/s in range-rate.

Spherical harmonic analysis, aliasing, and filtering

The currently practiced methods of harmonic analysis on the sphere are studied with respect to aliasing and filtering. It is assumed that a function is sampled on a regular grid of latitudes and longitudes. Then, transformations to and from the Cartesian plane yield formulations of the aliasing error in terms of spherical harmonic coefficients. The following results are obtained: 1) The simple quadratures method and related methods are biased even with band-limited functions. 2) A new method that eliminates this bias is superior to Colombos method of least squares in terms of reducing aliasing. 3) But, a simple modification of the least-squares model makes it identical to the new method as one is the dual of the other. 4) The essential elimination of aliasing can only be effected with spherical cap averages, not with the often used constant angular block averages.

Airborne Vector Gravimetry Using Precise, Position-Aided Inertial Measurement Units

Vector gravimetry using a precise inertial navigation system continually updated with external position data, for example using GPS, is studied with respect to two problems. The first concerns the attitude accuracy requirement for horizontal gravity component estimation. With covariance analyses in the space and frequency domains it is argued that with relatively stable uncompensated gyro drift, the short-wavelength gravity vector can be estimated without the aid of external attitude updates. The second problem concerns the state-space estimation of the gravity signal where considerable approximations must be assumed in the gravity model in order to take advantage of the ensemble error estimation afforded by the Kalman filter technique. Gauss-Markov models for the gravity field are specially designed to reflect the attenuation of the signal at a specific altitude and the omission of the long-wavelength components from the estimation. With medium accuracy INS/GPS systems, the horizontal components of gravity with wavelengths shorter than 250 km should be estimable to an accuracy of 4–6 mgal (µg); while high accuracy systems should yield an improvement to 1–2 mgal.

Results of airborne vector (3‐d) gravimetry

Gravity field modeling using airborne vertical component gravimetry has made significant strides over the last decade. We demonstrate the feasibility of extending this to three-dimensions using data from inertial navigation systems (INS) and the Global Positioning System (GPS). A significant advantage of measuring the horizontal gravity components is that the geoid can be determined in profiles by direct along-track integration, thus not only adding strength to conventional methods, but reducing the required area of survey support, especially along model boundaries. As such, the ultimate limitation of the method is in the quality of the INS and GPS sensors. In our test case, all three components of the gravity vector were determined over a profile in the Canadian Rocky Mountains. Differences between available truth data and the computed gravity components have standard deviations of 7–8 mGal (horizontal) and 3 mGal (vertical). These standard deviations include uncertainties in the truth data (<5 mGal, for horizontal; 1.3 mGal, for vertical). The resolution in the computed values is about 10 km. These analyses have demonstrated for the first time that the total gravity vector can be determined from airborne INS and GPS to reasonable accuracy and resolution, without any external orientation information, nor prior statistical hypothesis on the gravity signature, using medium-accuracy INS and geodetic quality GPS receivers.

Gravity Requirements for Compensation of Ultra-Precise Inertial Navigation

Precision inertial navigation depends not only on the quality of the inertial sensors (accelerometers and gyros), but also on the accuracy of the gravity compensation. With a view toward the next-generation inertial navigation systems, based on sensors whose errors contribute as little as a few metres per hour to the navigation error budget, we have analyzed the required quality of gravity compensation to the navigation solution. The investigation considered a standard compensation method using ground data to predict the gravity vector at altitude for aircraft free-inertial navigation. The navigation effects of the compensation errors were examined using gravity data in two gravimetrically distinct areas and a navigation simulator with parameters such as data noise and resolution, supplemental global gravity model noise, and on-track interpolation method. For a typical flight trajectory at 5 km altitude and 300 km/hr aircraft speed, the error in gravity compensation contributes less than 5 m to the position error after one hour of free-inertial navigation if the ground data are gridded with 2 arcmin resolution and are accurate to better than 5 mGal.

Ground-vehicle INS/GPS vector gravimetry

For geophysical purposes, gravity is measured in many ways, from static-point observations, using a gravimeter, to mean-value determinations from gravimeter and gravity gradiometer data collected by airplanes, ships, and satellites. We tested estimates of vertical and horizontal components of the gravity vector by combining Global Positioning System (GPS) data with a Honeywell H764G inertial navigation system (INS) on a land vehicle traversing highways in southwestern Montana. The estimation methods were based on techniques applied successfully to airborne INS/GPS data. In addition, we used wavelet denoising and wavenumber correlation procedures to enhance the estimates. Analyses of multiple traverses along the roads verified levels of repeatability as good as 0.64 mGal (all numerical accuracy values refer to standard deviations) in the vertical gravity-disturbance component. Control data, interpolated onto each road segment from an available database of gravity values, had an accuracy better than 2–4 mGa...

Analytical solutions of the Dirichlet and Neumann boundary-value problems with an ellipsoidal boundary

Abstract. Analytical solutions of exterior boundary-value problems with an ellipsoidal boundary are derived with an accuracy of O(ɛ4) for both the Dirichlet and the Neumann boundary-value problems, where ɛ2 is the square of the second eccentricity of the ellipsoid. In addition, an arithmetic example is implemented to verify that the analytical solutions improve the accuracy from O(ɛ2) to O(ɛ4) compared to the spherical approximation. The solutions are given as integrals of closed-form, analytic Greens functions and are particularly suited to local applications.

A numerical study of the divergence of spherical harmonic series of the gravity and height anomalies at the earth's surface

The problem of the divergence of the geopotential spherical harmonic series at the earths surface is investigated from a numerical, rather than a theoretical, approach. A representative model of the earths potential is devised on the basis of a density layer, which, in the spherical approximation, generates a gravity field whose harmonic constituents decay according to an accepted degree variance model. This field, expanded to degree 300, and a topographic surface specified to a corresponding resolution of 67 km are used to compute the differences between truncated inner and outer series of the gravity and height anomalies at the surface of the earth model. Up to degree 300, these differences attain RMS values from 0.33 μgal to 86 μgal for the gravity anomaly and from 0.32 μm to 410 μm for the height anomaly, in areas ranging respectively from near the equator to the vicinity of the pole. In addition to these values, there is an expected truncation effect, caused by the neglect of higher degree components of the inner series, of about 30 mgal and 36 cm, respectively. The field is then subjected to a Gaussian filter which effectively cuts off information at degree 300 (at the 5% level). The RMS error to degree 300 is thereby reduced by factors of 10 to 20, with a concomitant reduction in the truncation effect to about 0.3 mgal and 0.7 cm.

Potential Theory and the Static Gravity Field of the Earth

Classical potential theory and Newtons law of gravitation form the foundation of determining and modeling the Earths gravitational field. This chapter offers a mathematical review and derivation of models used for terrestrial and satellite-based measurements. The approach is based on developing solutions to a boundary-value problem. A section is devoted to the measurement of gravitational field quantities, especially using satellite techniques, which currently are receiving the most attention in view of their application to global studies related to climate, sea level, and polar ice sheet mass balance. A discussion of the geoid, its accessibility and determination, rounds out the chapter.

Omission Error, Data Requirements, and the Fractal Dimension of the Geoid

The newest global geopotential model, EGM08, yields significantly improved height anomaly (and geoid undulation) estimates, but not yet at the level of 1cm accuracy. Achieving this goal requires higher resolution gravimetric data (among other advancements, both theoretical and numerical). To determine the necessary data resolution, a statistical approach using the power spectral density (psd) of the height anomaly may be used to relate resolution to standard deviation in omission error. Kaula’s rule was the first such relationship based on a power-law approximation to the psd. It is shown that the Earth’s topography, whose fractal nature implies a power-law attenuation of its psd, and which in many cases is linearly correlated with the gravity anomaly on the basis of Airy’s isostatic assumption, can be used to design approximations to the psd of the local height anomaly, thus leading to estimates of the data resolution required to support the 1cm accuracy level.