minor project ppt (2)

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

Earth structureMODEL

PARAMETER ‘M’

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)

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

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

GAUSS NEWTON METHOD ON SYNTHETIC DATA:

The noise free synthetic data have been generated using the following expressions: gs=4πρzR3/3(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

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.

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

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

FIELD DATA

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

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

FIELD DATA

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

Radius of spherical ore body:

R= (3Z2Gmax/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.

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.

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.

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