# Heft 41/1990

# Schriftenreihe

des Instituts für Geodäsie

###

Heft 41/1990

VI, 148 S. Auflage: 300 ISSN: 0173-1009 |

Abstract |
II |

Foreword |
III |

Acknowledgements | IV |

Table of Contents | V |

1. Introduction |
1 |

2. Definition of the problem |
5 |

3. The influence of seismic structures on the gravity field:The geophysicist's view |
8 |

3.1 Computation of the gravity effect of given model bodies | 9 |

3.2 Computation of geoid undulations caused by model mass anomalies |
11 |

3.3 Estimation of the normal values for crustal parameters | 12 |

3.4 Two-dimensional model computations | 15 |

3.5 Three-dimensional model computations | 18 |

3.6 Velocity-density systematics | 25 |

4. A brief survey of proposed (geodetic) approaches inthe past |
34 |

5. The integrated geodesy adjustment model |
37 |

6. Density in an integrated geodetic approach |
42 |

6.1 Minimum norm solutions | 42 |

6.2 The attenuated white noise statistical gravity model | 46 |

6.3 A new approach to links gravity and density | 50 |

6.4 Reference (normal) density models | 58 |

7. Seismic and seismological data in an integrated approach |
61 |

7.1 Simple seismic velocity-density relationship in a homogeneous medium |
61 |

7.2 Seismic wave motion in an inhomogeneous medium | 63 |

7.3 A possibility of evaluating the autocavariance function of density anmoalies from seismic observations |
69 |

8. Test data |
72 |

8.1 Test network "Rossdorf" | 72 |

8.2 Test area "European Alps" | 76 |

9. Numerical investigations |
106 |

9.1 Prediction of gravity anomalies in test area "Rossdorf" | 106 |

9.2 Prediction of gravity anomalies in the European Alps | 114 |

10. Conclusions |
119 |

Appendix A | 121 |

References | 139 |

**Abstract**

*(nur in englischer Sprache)*

In the approximation of the earth's gravity field, in particular, in gravity prediction, geophysical information in the form of density and seismic data is mainly used indirectly in geodesy for gravity smoothing procedures. Moreover, mostly globally averaged model assumptions are used for density and Mohorovičić discontinuity.

Following the need for an improved knowledge of the gravity field the study tries to outline possible models and algorithms how to use nowadays available digital (surface) density models as well as seismic velocities and displacements in unified prediction approaches. Central role hereby plays the establishment of a (numerically reasonable) mathematical/physical relationship between the geophysical observations and data, respectively, and the gravity (disturbing) potential. Since these considerations are closely connected with the so-called gravimetric inverse problem, extended computations were carried out in the European Alps in order to verify already established (model) realationships.

After discussing various possiblities for the consideration of density in an integrated geodetic adjustment, a new approach is presented using the physical relationship, namely Newton's attraction integral, for the construction of the necessary auto- and crosscovariances when treating anomalous density as a stationary random process. Isostatic response theory as developed by DORMAN / LEWIS as a generalization of Vening Meinesz' isostasy model is introduced in the derivation and also proposed as deterministic predictor. The empirical relationship between seismic velocities and density as well as gravity is thoroughly investigated. Starting from the wave equation for an inhomogeneous medium seismological displacements are used forming a stochastic process.

For the various numerical investigations geophysical and geodetic/gravimetric data were collected in a local area as well as over the European Alps. Geophysical models of the Moho, the seismic basement, and the depth of the lithosphere as well as a threedimensional P wave velocity model are developed.

Numerical tests in gravity prediction are carried out mainly using the attenuated white noise gravity covariance model and quasi-harmonic inversion.

**Foreword**

*(nur in englischer Sprache)*

This report was prepared by Günter W. Hein in cooperation with the research associates Dr. Bernd Eissfeller, Dipl.-Ing. Michael Ertel and Dipl.-Ing. Klaus Hehl. The geophysical parts were contributed by Prof. Dr. Wolfgang Jacoby and Dipl.-Geoph. Dirk Czerwek, Universität Mainz, Institut für Geowissenschaften. In particular, chap. 3 is based on their work. The final report was edited by Dipl.-Geogr. Monika Jennert.

The study was partially sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under grant AFOSR-87-0271. It is administrated by the Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, with Dr. Christopher Jekeli, Scientific Program Officer.

**Conclusions**

*(nur in englischer Sprache)*

The study presents some first trials to use density and seismic data in a direct way in gravity field approximation, in particular gravity prediction. Although the geodesists are primarily not interested in the inverse problem, it is necessary to deal with it to some extent. The geophysicist's view on the structure of the earth's crust and its knowledge through seismic can help to improve the geodetic gravity prediction algorithms. In particular, empirical relationship between seismic velocity and density might be a valuable replacement of complex wave propagation formulas (see chapter 3).

In the theoretical part a few new approaches are presented which seem to be very promising. Since methods developed up till now use rather mathematical than (realistic) physical constraints for the covariance propagation needed in prediction algorithms of collocation type, special emphasis was put to take advantage of Newton's attraction integral for that purpose. The generalized isostatic response theory of __DORMAN / LEWIS __(__1970__) seems to be a promising candidate in the gravity prediction applications.

Although the quasi-harmonic inversion is based on physically not realistic constraints, it may offer in some cases reasonable prediction results. __HELLER__'s 1976 developed attenuated white noise statistical gravity model seems to be more realistic when introducing multiple layers of the earth'Äs crust. In particular, the proposal of __JORDAN __(__1978__) to extend this covariance theory to regional isostatic compensation might result in an even more realistic model for the terrain and crust-mantle contrasts.

In the consideration of seismic data for gravity prediction (chapter 7) the work of the geophysicist __CHERNOV __(__1960__) is appreciated. His statements about density and the elastic Lamé parameters at being randomly distributed in space forming a stochastic process allowed the incorporation of density and seismic data in the integrated geodesy adjustment model and the formulation of covariance functions according to seismic wave motion in an inhomogeneous medium.

The detailed numerical investigations in chapter 8 in a local as well in the European alps brought more insight in the statistical behaviour of density and present various characteristics of covariance functions. Some first gravity predictions using density and the quasi-harmonic inversion method as well as the attenuated white noise statistical gravity covariance model show reasonable results.

Since this report can represent only some initial work in this field, it is recommended to continue this study, in particular, by realizing numerically (i) the proposed covariance model in chapter 6.3 in connection with the generalized isostatic response theory, (ii) the extension of HELLER's covariance model to regional compensation as proposed by __JORDAN __(__1978__), and (iii) the inclusion of seismic data based on the statistical assumptions found by __CHERNOV __(__1960__).

There is no doubt that density and seismic information will strengthen and improve gravity prediction, in particular in areas where gravity gaps are. However, the best method still has to be found in the future. It is hoped that this report provides a plattform for further work.

__References:__

CHERNOV, Lev A. (1960): Wave Propagation in a Random Medium. McGraw-Hill, New York, VIII, 168 p.

DORMAN, Leroy M. / LEWIS, Brian T. R. (1970): Experimental Isostasy. 1. Theory of the Determination of the Earth's Isostatic Response to a Concentrated Load. Journal of Geophysical Research, Vol. 75, Issue 17, pp. 3357-3365.

HELLER, Warren G. / JORDAN, Stanley K. (1979): Attenuated White Noise Statistical Gravity Model. Journal of Geophysical Research: Solid Earth, Vol. 84, Issue B9, pp. 4680-4688.

JORDAN, Stanley K. (1978): Statistical Model for Gravity, Topography, and Density Contrasts in the Earth. Journal of Geophysical Research: Solid Earth, Vol. 83, Issue B4, pp. 1816-1824.