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Gradient Modeling with Gravity and DEM Gradient Modeling with Gravity and DEM by Lizhi Zhu Report No. 483 Geodetic Science and Surveying The Ohio State University Columbus, Ohio 43210 June 2007 Gradient Modeling with Gravity and DEM by Lizhi Zhu Report No. 483 Geodetic Science and Surveying The Ohio State University Columbus, Ohio 43210 June 2007 ABSTRACT This study deals with the methods of forward gravity gradient modeling based on gravity data and densely sampled digital elevation data and possibly other data, such as crust density data. In this study, we develop an improved modeling of the gravity gradient tensors and study the comprehensive process to determine gravity gradients and their errors from real data and various models (Stokes’ integral, radial-basis spline and LSC). Usually, the gravity gradients are modeled using digital elevation model data under simple density assumptions. Finite element method, FFT and polyhedral methods are analyzed in the determination of DEM-derived gravitational gradients. Here, we develop a method to model gradients from a combination of gravity anomaly and DEM data. Through a solution on the boundary value problem of the potential field, the gravity anomaly data are combined consistently with the forward model of DEM to yield nine components of the gravity gradient tensor. As a result, forward gravity gradients can be synthesized using both geodetic and geophysical data. We use two different methods to process gravity data. One is the regular griding method using kriging and least squares collocation, and the other one is based on fitting splines or wavelet functions. For DEM data, we use finite elements, polyhedra and wavelets or splines to compute the gradients. The second Helmert condensation principle and the remove-restore technique are used to connect DEM and gravity data in the determination of gravity gradients. Modeling of the gradients thus, particularly at some altitude above ground, from surface gravity anomalies is based on numerical implementations of solutions to boundary-value problems in potential theory, such as Stokes’ integral, least-squares collocation, and some Fourier transform methods, or even with radial-basis splines. Modeling of this type would offer a complementary if not alternative type of support in the validation of airborne gradiometry systems. We compare these various modeling techniques using FTG (full tensor gradient) data by Bell Geospace and modeled gradients, thus demonstrating techniques and principles, as well limitations and advantages in each. The Stokes’ integral and the least-squares collocation methods are more accurate (about 3 E at altitude of 1200 m) than radial-basis splines in the determination of gravity gradient using synthetic data. Furthermore, the comparison between the modeled data and real data verifies that the high resolution (higher than 1 arcmin) gravity data is necessary to validate the gradiometry survey data. Ground and airborne gradiometer systems can be validated by analyzing the spectral properties of modeled gradients. Also, such modeling allows the development of survey parameters for such instrumentation and can lead to refined high frequency power spectral density models in various applications by applying the appropriate filter. ii PREFACE This report was prepared for and submitted to the Graduate School of the Ohio State University as a dissertation in partial fulfillment of the requirements of the Ph.D. degree. The work was supported with sponsorship from the National Geospatial-Intelligence Agency under contract no. NMA401-02-1-2005. iii ACKNOWLEDGMENTS I wish to express my sincere gratitude to my adviser Dr. Christopher Jekeli for proposing this topic of my dissertation and his patient support and guidance to complete this research. I also want to thank Dr. C.K. Shum and Dr. Ralph R.B. von Frese for reviewing my dissertation. iv TABLE OF CONTENTS ABSTRACT........................................................................................................................ ii PREFACE .......................................................................................................................... iii ACKNOWLEDGMENTS ................................................................................................. iv CHAPTERS 1. INTRODUCTION ...........................................................................................................1 1.1 Gravity and Gradiometry .........................................................................................................1 1.2 Background ..............................................................................................................................2 1.3 Chapter descriptions.................................................................................................................4 2. FORWARD GRADIENT MODEL BASED ON DEM...................................................6 2.1 Gravitational Gradient..............................................................................................................6 2.2 Gradients from Terrain Elevation.............................................................................................7 2.3 Finite elements .........................................................................................................................9 2.3.1 DEM data.....................................................................................................................9 2.3.2 The right rectangular prisms ........................................................................................9 2.3.2 Direct Numerical Integral ..........................................................................................14 2.3.3 Polyhedron Method....................................................................................................18 2.3.4 The extent of DEM and the reference surface............................................................20 2.4 Fourier Transform ..................................................................................................................24 2.4.1 Parker’s method .........................................................................................................24 2.4.2 Forsberg’s method......................................................................................................26 2.4.3 Discrete Fourier transform .........................................................................................27 2.5 Numerical analysis.................................................................................................................29 2.6 Conclusion .............................................................................................................................42 3. GRADIENT MODEL BASED ON GRAVITY.............................................................44 3.1 Introduction............................................................................................................................44 3.2 Stokes’ integral in spherical approximation ...........................................................................44 3.3 Stokes’ integral in planar approximation................................................................................49 3.4 Spherical spline model ...........................................................................................................52 3.4.1 Introduction to spherical spline..................................................................................52 3.4.2 Interpolation of gravity and gradients ........................................................................54 3.4.3 Derivatives of Kernel Function..................................................................................55 3.5 Least–squares collocation ......................................................................................................57 3.6 Analyzing gradient model using simulated fields ..................................................................58 3.6.1 Simulated field ...........................................................................................................58 3.6.2 Modeled Gradient disturbances..................................................................................60 3.7 Conclusion .............................................................................................................................75 v 4. GRADIENT MODEL BASED ON GRAVITY AND DEM .........................................77 4.1 Introduction............................................................................................................................77 4.2 Principle of the gradient model ..............................................................................................78 4.2.1 Gradients from residual gravity anomalies ................................................................78 4.2.2 Gravity gradient from DEM and Gravity anomalies..................................................79 4.3 Airborne gradiometry survey in Parkfield, CA ......................................................................81 4.3.1 Gradiometer data:.......................................................................................................81 4.3.2 Post processing of gravity gradient data ....................................................................84 4.3.3 Crossover point analysis ............................................................................................85 4.4 Gravity gradient model ..........................................................................................................87
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