1. JOINT MODELLING OF GRAVITY & GRAVITY
GRADIENTS AND IT’S APPLICATION IN
GEOLOGICAL INTERPRETATION
A Thesis Submitted to the Andhra University
for the degree of
DOCTOR OF PHILOSOPHY
in
GEOPHYSICS
By
CHANDRA PRAKASH DUBEY
CSIR – National Geophysical Research Institute
Hyderabad – 500 007, India
Under the guidance of
DEPARTMENT OF GEOPHYSICS
COLLEGE OF SCIENCE & TECHNOLOGY
ANDHRA UNIVERSITY
April, 2015
Dr. Virendra Mani Tiwari, FNASc
Internal Supervisor
Gravity and Magnetic Studies Group
CSIR-National Geophysical
Research Institute
Hyderabad, Telangana, India
Prof. Paluri Rama Rao
External Supervisor
Department of Geophysics
Andhra University,
Vishakhapatnam
Andhra Pradesh, India

2. Dedicated to my grandparent Lt. Shri Kamala Prasad Dubey,
Smt. Kailashi Devi.................

4. DECLARATION
This is to certify that the thesis entitled “JOINT MODELING OF GRAVITY &
GRAVITY GRADIENTS AND IT’S APPLICATION IN GEOLOGICAL
INTERPRETATION”, submitted by me to the Department of Geophysics, Andhra
University, Vishakhapatnam for the award of the degree of PhD is a bonafide record of
research work carried out by me. The contents of this thesis, in full or in parts, have not been
submitted to any other Institute or University for the award of any degree.
The literature related to the problem investigated has been cited. Due
acknowledgements have been made wherever necessary.
Date: Chandra Prakash Dubey, CSIR-SRF
Place: Hyderabad Ph.D (Geophysics)
CSIR-NGRI, Hyderabad &
Department of Geophysics,
Andhra University, Vishakhapatnam.

5. v
ACKNOWLEDGEMENT
I would like to express my heartfelt indebtedness to my supervisor Dr. Virendra
Mani Tiwari, without whom the present research is not possible. Dr. Tiwari provided the
best opportunity to work with him in NGRI and was supportive of my excursions into
many different areas apart from my PhD work. His teachings and scientific temper greatly
influenced me in my research work.
My deepest thanks to Prof. P. Rama Rao who agreed to guide me as my co-
supervisor from Andhra University whose continuous support made possible the
fulfilment of my aspirations. Prof. Rao always encouraged me to pursue my ideas, and
willingly offered suggestions for improvement.
I acknowledge Prof. Dr. Hans Jürgen Gotze, IFG, CAU, Kiel, Germany for giving
me an excellent exposure to overseas research problems during my visit as a DAAD
Fellow to their Institute. I am also grateful to Dr. Sabine Schmidt for her continuous
guidance and support for IGMAS+ and allowing me to access to the historical torsion
balance measurements. I thank Mr. Ben, Mr. Peter, Mr. Nils and Ms. Christiana at IFG,
CAU, Kiel for their contribution and support.
My sincere thanks to Dr. D. C. Mishra, Dr. Bijendra Singh, Dr. M. R. K.
Prabhakar Rao, Dr. A. P. Singh, Dr. Ch. V. Raju, and Dr. Niraj Kumar, scientists of
Gravity working group at NGRI, Hyderabad who always encouraged me to do quality
research and provided me with invaluable skills and tools by way of regular group
interactions. Particularly, I would like to thank Dr. D. C. Mishra and Dr. Prabhakar Rao
for the time and effort they made in reading my thesis and for constructive suggestions
which greatly improved my thesis. Help rendered by my colleagues of gravity and
magnetic studies group at different stages of this work is unforgettable. I am particularly
thankful to my colleagues Mr. Narakula Srinivas Rao, Tehnical Assistant, Gravity Group
for his restless help whenever needed, and Mrs. Jyothi, PhD Student, Gas Hydrate Group
for the motivation and encouragement all the time.
I thank my friend Ms. Alka, Assistant Manager, Syndicate Bank, Hyderabad for
her cheerful support during last stages of the thesis. Last but not the least; I thank my
grandparents and my family for their selfless cooperation, warmth and affection shown
towards me during the course of my research. They have supported and encouraged me
throughout my studies and I am extremely grateful for their patience and love. I also wish
to thank my parents for their time to time support and encouragement. I also wish to
thank my sweet nephew Shivansh and my brothers and sisters for their keen and constant
encouragement.
(Chandra Prakash Dubey)

6. vi
TABLE OF CONTENTS
Acknowledgements v
Table of Contents vi
Abstract ix
Research Publications xii
List of Figures xiv
List of Tables xx
1. Introduction.............................................................................................................. 1
1.1. Problem definition and research aims.................................................................................... 2
1.2. Motivation.............................................................................................................................. 4
1.3. The conventional modeling of single geophysical field and its limitations.......................... 4
1.4. Joint Modeling of Different Potential Field Data.................................................................. 6
1.4.1. The Parameter/Structural Approach for Modeling..................................................... 7
1.5. Thesis outline…………………............................................................................................. 8
2. Fundamentals of Gravity and Gravity Gradiometry........................................... 10
2.1. Introduction............................................................................................................................. 10
2.2. Newtonian Gravity.................................................................................................................. 11
2.2.1. Gravitational Force........................................................................................................ 11
2.2.2. Gravitational Field......................................................................................................... 12
2.2.3. Gravitational Potential.................................................................................................. 13
2.3. Gravity Gradient...................................................................................................................... 16
2.4. Historical Background of Gravity Measurements................................................................... 18
2.4.1. Types of Gravimeter......................................................................................................... 18
2.4.2. The Simple Pendulum....................................................................................................... 19
2.4.3. Comparison of Absolute and Relative Methods............................................................... 20
2.4.4. Gravity Measurements..................................................................................................... 20
2.5. Historical Background of Gravity Gradient Measurements................................................... 22
2.5.1. Construction and Design of an Eötvös Torsion Balance................................................. 25
2.5.2. Early application of Torsion Balance Measurements...................................................... 26
2.5.3. Drawbacks of Torsion Balance Measurements................................................................ 28
2.6. Advantage Measurements of Gravity....................................................................................... 28
2.6.1. Gravity Gradiometry vs. Conventional Gravimetry......................................................... 30
2.7. Gravity and Gravity Gradiometry Modeling............................................................................ 31

7. vii
2.8. Application and Opportunities................................................................................................. 31
2.8.1. Geophysical Application.................................................................................................. 32
2.8.2. Geodetic Application........................................................................................................ 33
2.8.2. Opportunities.................................................................................................................... 33
3. Computational Algorithm of Gravity & Gravity gradient Tensor and their
Applications..............................................................................................................
34
3.1. Introduction.............................................................................................................................. 34
3.2. Forward Modeling.................................................................................................................... 35
3.3. Model Calculation of Regular Geometries............................................................................... 36
3.3.1. 2D Geometries: Semi Infinite Horizontal Slab and Vertical Sheet/Dike......................... 36
3.3.2. 3D Geometries.................................................................................................................. 38
3.3.2.1. Solid Sphere and Vertical Cylinder........................................................................ 38
3.3.2.2. Rectangular Lamina or prism................................................................................ 42
3.3.2.3. Invariants of prism................................................................................................. 47
3.3.3. Normal Fault Model......................................................................................................... 48
3.4. Application of Algorithm over Two Different Geographical Regions.................................... 49
3.4.1. Case Study 1. Structural Analysis of Wichita Uplift using Rectangular Prism: Horst
Model................................................................................................................................
49
3.4.2. Case Study 2. Analysis of Irregular 3D Geometries using Vertical Rectangular
Prisms or Infinite Cell: Vredeforte Dome, South Arica...................................................
52
3.5. Strategy behind the Generation and Computation of Irregular Shaped Geometries................ 56
3.6. Summary................................................................................................................................ 58
4. Joint Modelling of Gravity and Gravity Gradient for Shallow Structures........ 59
4.1. Introduction.............................................................................................................................. 59
4.2. Torsion Balance Data over Saxony Basin, Germany…........................................................... 61
4.2.1. Operational Details.......................................................................................................... 63
4.3. Geology of Study Area and Database Information.................................................................. 65
4.3.1. Salt Tectonics in the Lower Saxony Basin....................................................................... 65
4.3.2. Database Information of Wathlingen- Hänigsen Area................................................... 67
4.4. 3D Forward Modelling of Gravity, Gravity Gradient and Curvature...................................... 70
4.4.1. Analytical Forward Modeling.......................................................................................... 72
4.4.2. 3D Density Modelling of the Wathlingen - Hänigsen Salt Dome.................................... 74

8. viii
4.5. Summary.................................................................................................................................. 81
5. Modelling of Gravity Data for Deeper Lithosphere Structures: Bay of
Bengal.....................................................................................................................
83
5.1. Introduction.............................................................................................................................. 83
5.2. Tectonics and Sedimentary Evolution of the Bay of Bengal................................................... 86
5.3. Utilized Data Sets..................................................................................................................... 89
5.3.1. Bathymetry and Gravity data........................................................................................... 89
5.3.2. Sediment Thickness Data................................................................................................. 89
5.3.2.1. Density-Depth Relationship for BOB Sediments................................................... 91
5.4. 3D Density Model of Bay of Bengal........................................................................................ 93
5.4.1. Gravity Effect of Bathymetry............................................................................................ 94
5.4.2. Gravity Effect of Sediments.............................................................................................. 96
5.4.3. Crustal Gravity Anomaly................................................................................................. 100
5.5. 3D Crustal and Lithospheric Structure.................................................................................... 102
5.5.1. Crust mantle interface (CMI)........................................................................................... 102
5.5.2. Lateral Density Variation (LDV) in Upper Mantle.......................................................... 104
5.5.2.1. Analysis of LDV in Terms of Mass Distribution.................................................. 105
5.5.3. 3D Lithospheric Thickness............................................................................................... 112
5.6. Structural Analysis of BoB from Computed Gravity Gradient................................................ 118
5.6.1. Integrated Analysis of Seismic and Gravity Gradient over the 850
East Ridge............... 126
5.7. Summary.................................................................................................................................. 128
6. Results and Discussion............................................................................................ 132
6.1. Local Structural settings........................................................................................................... 133
6.2. Regional Structural Settings..................................................................................................... 135
6.3. The Implications of Jointly Models for Subsurface Characterisation...................................... 136
6.3.1. The Gravity Gradient and Curvatures vs Gravity Field.................................................. 137
6.3.2. The Joint Modeling Algorithm and Their Application..................................................... 137
6.4. Suggestions for Further Research............................................................................................ 138
7. Appendix................................................................................................................... 140
8. References................................................................................................................. 159

9. ix
ABSTRACT
New measuring instruments of Earth’s gravity gradient tensors (GGT) have offered a
fresh impetus to gravimetry and its application in the shallow and deep subsurface exploration.
Several efforts are made to provide a thorough understanding of the complex properties of the
gravity gradient tensor and mathematical formulations to compute GGT. However, there are not
many open source softwares in this regard. The understanding of the tensor properties is
imperative and provides fundamental guidelines in the development of a three dimensional
geological model. From the last few decades, numerous attempts have been made on modelling
and deciphering the three dimensional complex geometries like Salt domes and other complex
tectonic environments using gravity and gravity gradiometry. Recent advancement of gravity
gradiometry on different platforms like airborne and space using satellite opened up a path way
for their use both in exploration and geodynamics so as to understand the complexity below the
earth’s surface, even though the interpretation techniques of gravity gradiometer are still in
developing mode.
In the above said context, the author in the present thesis developed a MATLAB
computational algorithm to calculate the gravity field and full gravity gradient tensor for
undulated surface followed by regular geometries like, an infinite horizontal slab, vertical sheet,
solid sphere, vertical cylinder, normal fault model, and rectangular lamina or conglomerations of
such bodies and the results are compared with responses using professional software based on
different computational schemes. Real subsurface geometries are approximated through infinite
cell of vertical rectangular lamina to characterize the response due to complex geological
structure of interest. Thus, these computational algorithms help to understand the complex
behavior of gravity gradient responses even for simple geometries that produces complex pattern
of anomalies like monopole, dipole, tripole, and quadrupole responses and further to correlate
these responses with real geological data to infer simple as well as complex subsurface
structures. The geological application of this algorithm is demonstrated over a horst-type
structure of Oklahoma Aulacogen, USA and Vredefort Dome, South Africa, where measured
GGT data are available.
The thorough understanding of the behavior of gravity gradients through MATLAB
computation of different geometries mentioned above was used in the second part of the thesis to

10. x
derive a 3D density model of salt domes. For this, the real gravity and reprocessed gradient data
of the Wathlingen salt dome situated in the southern part of the Northwest German basin was
made use for joint modelling. Numerous attempts are made on modelling of salt tectonics and
several models are developed to explain the ascent of salt diapirs using different geophysical
parameters like gravity, magnetic, seismic etc. Since, salt diapirs are salt structures that pierce
their overburden. Such diapirism occurs in variety of different tectonic settings including passive
continental margins, intra – cratonic basin or fold and thrust belts. The structures produced by
salt tectonics can have a strong impact on the formation of hydrocarbon reservoirs. A 3D density
model of the Wathlingen salt dome, situated in the southern part of the Northwest German basin
is derived from the joint modelling of reprocessed torsion balance measurements. Gravity,
gravity gradients (Wzx, Wzy), curvature (Wxy) derived from both horizontal gravity gradients,
and Horizontal Directive Tendency (HDT) are jointly modeled to decipher the geometrical
structure of the salt dome. The model is constrained by geological and borehole information. It is
found that Wathlingen salt dome is a mushroom structured salt body, which is 14 km long, 4 km
to 8 km wide extending up to ~ 4 km depth. The top mushroom structure of the salt is
horizontally spread up to ~ 8 km. It would not have been possible to derive the complex 3D
structure from modelling of gravity data alone.
After the above successful effort in mapping salt dome features, which is of residual scale
(few tens of km) the present study is extended to a regional scale (over hundreds of km) to
address regional geodynamics. The regional scale study utilizes satellite gravity data derived
from satellite altimetry over the region of Bay of Bengal, which is known for large sediment
thickens to reveal hidden structures beneath sea floor. The basement depth and crustal thickness
are important parameters when interpreting the tectonic or thermal evolution of a margin,
however depth to basement, basement architecture and crustal structure remain poorly
determined in the deep water parts of the Bay of Bengal. While regional-scale interpretation of
the new and existing 2D seismic reflection data has provided important new insights into the
structural framework of the basin, a number of outstanding issues remain which cannot be
addressed using seismic data alone as they do not have information up to Moho or below it. The
main goal of the study is an attempt to determine this sedimentary cover in oceanic crust and
structural trends in a part of Bengal fan between the latitudes 0o
00’ to 22o
00’ N and the
longitude of 80o
00’ to 94o
00’ E, through the integrated analysis and interpretation of gravity

11. xi
observations and computed gravity gradients. Integrated interpretation leads to an improved
representation of lithospheric structures in Bay of Bengal. For quantitative interpretation of the
sources of the anomaly, a 3D density model of the oceanic lithosphere is designed by means of
forward modelling and inversion constrained by prior information to investigate the crust mantle
interface and detailed lithosphere properties. As a result of the current investigation, a regional
structural map of the basement is constructed. In addition, two major ridges, 85o
and 90o
east
ridges get easily identified using gradient components. This study also, reveals that, the depth to
the basement surface ranges from 7 to more than 29 km, delineating promising deep sedimentary
layer of around 22 km thick. Modelling results indicated that, the lateral density variation is so
steep throughout the study area and also the thick sediment load changes the lateral property of
deeper layer below crust. The last part of the thesis therefore demonstrates that satellite derived
gravity data can be used to model regional gravity anomaly produced by lithospheric properties
and structures evolve over the geological time period.

12. xii
RESEARCH PUBLICATIONS
Thesis Papers:
Dubey, C. P., Gotze, H. J., Schmidt, S., Tiwari, V. M., 2014, A 3D model of the Wathlingen
salt dome in the Northwest German Basin from joint modelling of gravity, gravity gradient,
and curvature. Interpretation, Vol. 2, No. 4, p. 1-13, doi: 10.1190/INT-2014-0012.1.
Dubey, C. P., and Tiwari, V. M., 2015, MATLAB Algorithm for Computation of gravity
field and its gradient: some applications; Computer and Geosciences. (Submitted after minor
revision).
Dubey, C. P., and Tiwari, V. M., and Rao, P. R., Detailed lithosphere structure of Bay of
Bengal and their tectonic implications. Lithosphere (To be Submitted)
Dubey, C. P., Tiwari, V. M., and Rao, P. R., Mapping subsurface structures below Bay of
Bengal Basin using full gravity gradient tensor. SEG-Interpretation (To be Submitted).
Supplementary Paper:
Gupta, H.……..Tiwari, V.M. ……Dubey, C. P.…….. Nayak, S., 2014, Investigations related
to scientific deep drilling to study reservoir-triggered earthquakes at Koyna,
India, International Journal of Earth Sciences, DOI 10.1007/s00531-014-1128-0.
Tiwari, V. M., Mishra, S., Dubey, C. P., and Raju, D. Ch. V., 2015, 3D Structural setting
beneath Koyna – Warna region from airborne gravity gradients and magnetic data; (In
pipeline).
International Conference Proceedings:
Dubey, C. P., Gotze, H. J., Schmidt, S., and Tiwari, V. M., 2014, A 3D Model of the
Wathlingen Salt Dome in the Northwest German Basin from joint modelling of Gravity,
Gravity Gradient, and Curvature; Geophysical Research Abstracts, Vol. 16, EGU 2014-
13916, 2014, EGU General Assembly 2014.
Dubey, C. P., 2014, Glimpses of SEG NGRI Student Chapter at CSIR-NGRI, Hyderabad,
India and my personal research; 84th SEG Annual Meeting, Denver, Colorado, USA, 2014.

13. xiii
Mishra, S., Dubey, C. P., and Tiwari, V. M., 2014, 3D Structural setting beneath Koyna –
Warna region from airborne gravity gradients and magnetic data, NGRI – RC Meeting 2014,
Hyderabad India.
Dubey, C. P., and Tiwari, V. M., 2013 Imaging 3D complex geological structures from
Gravity Gradiometry, NGRI – RC Meeting 2013, Hyderabad, India.
Dubey, C. P., and V. M. Tiwari, 2011, Computation of gravity potential and full gravity
gradient tensor of arbitrary geometry and application, IGU 2011, Vishakhapatnam, India.
Dubey, C. P., and V. M. Tiwari, 2010, A computational method to calculate the gravity
potential and full gravity gradient tensor of arbitrary shape bodies and its applications, ICON-
GSCESS 2010, BHU, Varanasi, India.