1. Interpretation of Spherical Gravity Anomaly
using a Non-linear
(Gauss-Newton) Technique
Under the guidance of: Presented by:
Dr. Dinesh Kumar Charu Kamra
(Professor)
Department of Geophysics

2. Earth structure
MODEL PARAMETER
‘M’
RESULT
(geophysical data)
DATA VECTOR ‘D’
GEOPHYSICAL INVERSE PROBLEM
The geophysical problems are inverse problems where one
starts with the end result (geophysical data) and work
backwards to know the earth’s structure.
Linear Problem: The problem in which equation
d=Gm ; can be represented in
explicit form.
For e.g. Ti=a+bZi ,can be solved linearly
Non-linear problem: The problem in which equation
d=Gm; can not be represented in
explicit form .
For e.g. Ti=a+bz+cz2 ; cannot be solved linearly(can be
solved by iterative techniques)

3. study
Objective of the study
To interpret the available field gravity data by applying the iterative technique “Gauss
Newton Method “ to the synthetic data of gravitational field due to a spherical ore body.
GAUSS NEWTON METHOD: We have investigated method namely Gauss Newton
(G-N), for non-linear inverse problem.
Derivation of the formula:
We know that d=f(m)
where,
d is the observed data and f(m) is function of model parameters .
In Gauss Newton technique, f(m) is expanded around the initial model (m0 ) using
Taylor’s series.
If y represents the difference between the observed data and the synthetic data
estimated from the initial model.
METHODOLOGY

6. After every iteration in each technique, the root mean square (rms) error has been
estimated using the following expression
rms= √ ∑(dobs-dsyn)2 /n
Where,
n is the number of data points
dobs is observed data and dsyn is synthetic data

7. GAUSS NEWTON METHOD ON SYNTHETIC
DATA:
The noise free synthetic data have been generated using
the following expressions:
gs=4πρzR3/3(x2+z2)3/2 ……………………..(1)
where = 6.67×10-8 cm3/gsec2 is a gravitational constant,
R is the radius of the ore body,
z is the depth of center of ore body (sphere),
x is the horizontal distance,
ρ is the density contrast.
Figure 1 shows the gravity profile generated using
above equation with parameters
z = 500 m, R = 200 m and ρ = 0.4 g/cm3
Figure1 : Gravity profile due to spherical
ore body

8. The non-linear Gauss-Newton method have been applied to this profile to extract the
three parameters (z), (R) and (ρ).Take density contrast as ‘a priori’ information and
resolve the other two parameters (z and R).
We generated synthetic data (dsyn) using equation (1)with initial guess R=800 and Z=400
to check the applicability of Gauss Newton Technique.
X= (AT A) -1 AT y ……………(2)
Where A: is the matrix of partial derivatives with respect to the R and Z i.e. dg/dR and
dg/dZ .
AT: is the transpose of matrix A
y: is the matrix of difference between synthetic (dsyn) and observed data (dobs);
X: is the matrix of unknown correction applied to model parameters.
After every iteration , the root mean square (rms) error has been estimated using the
following expression
rms= √ ∑(dobs-dsyn)2 /n
Where,
n is the number of data points.
dobs is observed data and dsyn is synthetic data.

9. Next we examine the performance of techniques by adding the 10% error to the
synthetic data
Figure 2: Gravity profile for spherical ore body with 10% noise

10. Table1. Performance of the techniques on synthetic data in case of spherical ore body.
Figure 3 : The graph between number of iterations and RMS error

11. FIELD DATA
Figure 4: Bouguer gravity map, Humble salt dome, Harris county, Texas
(Nettleton,1962,p.1839). Courtesy AAPG

12. Figure 5: Gravity profile along AA’ for the contour map shown in above figure
FIELD DATA

13. DIRECT INTERPRETATION
Limiting depth: Limiting depth refers to
the maximum depth at which the top of a
body could lie and still produce an
observed gravity anomaly.
Z=1.3X1/2, Where,
X1/2: half-width from observed gravity
profile
Half-width method.
The half-width of an anomaly (x1/2) is the
horizontal distance from the anomaly
maximum to the point at which the
anomaly has reduced to half of its
maximum value (Fig. 6). Figure 6: limiting depth using half
width method

14. Radius of spherical ore body:
R= (3Z
2
Gmax/4Πγρ)
1/3
Now, after examine the profile across line AA’ ,we get z=5632 m and
R=3140 m
RESULT OF FIELD DATA :
The initial value for R and z are found to be 3140 m. and 5632 m respectively.
The G-N method gives the final value as R = 4581 m. and Z = 8414 m. with root mean
square error of 0.53535.

15. CONCLUSIONS:
• The performance of (Gauss -Newton,method) to solve non-linear
problems have been examined by taking synthetic gravity data with
and without noise.
• The error increase by adding10% noise to the synthetic data
• Method does not resolve all the three parameters simultaneously.
Therefore, one parameter has been taken as ‘a priori’ information.
• A field gravity data has been interpreted using Gauss-Newton
method.

16. REFERENCES:
• Meju , M.A. (1994) Geophysical Data Analysis: Understanding Inverse Problem
Theory and Practice, SEG publications, 296p.
• Nettleton, L.L. (1962) Gravity and magnetics for geologists and seismologists: Bull.
AAPG, 46(10), 1815-1838.
• Nettleton, L.L. (1971) Gravity and magnetics for geologists and seismologists:
geophysical monograph series, SEG publcations, 121p.
• Philip Kearey, Michael Brooks An Introduction to Geophysical
Exploration,Department of Earth Sciences,page 140-141.