# 2D Talwani Inversion based on Hexagon Model

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1. INTRODUCTION
This study aims to determine the subsurface structure by performing an inversion of the measured gravity data on the surface. This is deemed necessary because to find out the anomaly pattern is not easy, there are many combinations of patterns that give the same anomalous response on the surface (ambiguity). So it is necessary to do research that is able to provide a unique solution in solving the ambiguity problem. Vatankhah et.al (2012) have conducted a similar study using the box model Talwani approach. As a result, the program created is able to map the subsurface structure by varying the smoothness constant and applying linear weighting techniques.

Figure 1. Research results Vatankhah et. al (2012)

This study uses a different approach, namely the hexagon model. This modified research is expected to provide better results.

2. Basic Theory
Talwani (1959) formulated a forward modeling equation using an n-sided polygon which is given as follows:
Where :
∆g = anomaly response (mGal)
G = 6,67×10-3 cms2
x = x direction (m)
z = depth (m)
∆ρ = the difference in density of the object to the environment (gr/cm3)

Backus and Gilbert (1967) introduced a technique capable of solving linear inversion equations, namely the Lagrangian multiplier. Boulanger and Chouteau (2001) used this method to invert 3D gravity data. The given Lagrangian multiplier equation is:

with A :
Matrix G is a Jacobian matrix and H is a matrix containing the smoothness constant (cH) multiplied by the Hessian matrix obtained through a finite difference approach in the horizontal direction (x) and depth (z), so that H is formulated as:
with and in the form of a square matrix with the following composition:

To define the relationship between the calculated data and the model data, a matrix b is used which contains the difference in the values of the two data augmented by a 0 matrix.
While W is a weighting matrix as a function of the density (ρ) and depth (z) of each hexagon.
P is a diagonal matrix containing the constant . Vatankhah et.al (2012) formulated the selection of depending on the initial guess density value (ρ0). For 0=0, the value of is usually 10-2. If the value of 0 is not equal to 0, the value of used is 1.
The Q matrix was first introduced by Li and Oldenburg (1996, 1998) and Pilkington (1997). While the matrix V is a weighting factor as a function of density (ρ). It was first introduced by Last and Kubik (1983). The constant is also a small value to avoid singularities. :
Where is a small constant to avoid singularity, and is a weighting constant as a function of depth (z). Under standard conditions the value of used is 1. The results of the minimization of the Lagrangian multiplier equation by Green (1975) as a function of and produce the following 2 equations:

3. EXPERIMENT
The research was conducted by dividing the subsurface into 261 hexagon-shaped grids. The value of the anomalous response (∆gcal) will be compared with the anomalous response of synthetic data made with a similar construction (∆gobs). The difference between the values of gcal and gobs will then be used to carry out the process of updating the parameter of each hexagon using Equation 10 and Equation 11. If the new value is less than the given error limit value or has exceeded the maximum iteration given, the program will stores the last value as the result of the inversion process.

Figure 3. Initial results of the experiment.

4. RESULTS
Testing with Synthetic Data
To test the inversion program that has been made, synthetic data obtained from synthetic models are used. The synthetic model used is a hexagon with a side length of 10 m, which is at a depth of 50 m with a horizontal position of 50 m (Figure 3). Here are the results of the program:
Figure 4. Test results with a) cH=0, b) cH=1, c) cH=10-3

The test was carried out by varying the cH. The value of cH was chosen based on the reference written by Vatankhah et.al ranging from 10-3 –10-2. The results displayed are the results of the program iteration 20 times. Although the resulting response has a small error of 2.84×10-12, ambiguity still occurs. For cH values ​​below 10-3, i.e. 0, the distribution pattern is centered in a position that corresponds to the synthetic model with the highest density value of 1.3 g/cm3. However, the resulting distribution pattern is still widespread. When cH=1 the resulting pattern is more and more irregular. In addition, the highest density value is only 0.7 gr/cm3, much smaller than the synthetic model given. This indicates that if the selected cH value is much greater than the reference cH value, the distribution pattern will be irregular and will not produce a unique solution. The best results were obtained with a value of cH=10-3 which resulted in a small error and the ability to approach the synthetic model well given.

Comparison Using Field Data
To apply the program to field data, data from a survey by Vatankhah and Ardestani (2012) in the Zereshlu Mining Camp area, Zanjan were used. Located at UTM [704296-704554] E and [4130627-4130990] this area is an area consisting mostly of red Andesite outcrops.

Figure 5. Comparison of inversion results using field data (a) Vatankanh et.al ,(b) our results.
The program created is able to produce a density distribution pattern that is similar to other similar programs. These results were obtained by applying a cH value of 1×10-7. However, the resulting density distribution value is lower (maximum 0.6 gr/cm3), this is because the resulting density distribution pattern is wider. If examined from the resulting error value, the value is very small, namely 1.68 × 10-19. This indicates that the program created is able to predict the shape and position of anomaly objects in general.

5. CONCLUSION
After conducting the research, it can be concluded that the 2D gravity inversion made with the Talwani hexagon model approach is able to follow the synthetic data pattern using a small cH range, namely 10-3-10-2. Although the resulting density values are different, the program can predict the position and shape of the anomaly in general terms. In order to get better inversion results, the authors suggest applying a more complex weighting technique.

REFERENCES
1. Backus, G. dan Gilbert, Numerical Applications of A Formalism for Geophysical Inverse Problems, Geophysical Journal of the Royal Astronomical Society, 247-276, 2001.
2. Boulanger, O. and Chouteau, M., Constraints in 3D gravity inversion, Geophysical Prospecting, 49, 265-280, 2001.
3. Green, W. R., Inversion of Gravity Profiles by Use of A Backus-Gilbert Approach, Geophysics, 40, 763-772, 1975.
4. Last, B. J. dan Kubik, K., Compact Gravity Inversion, Geophysics, 48, 713-721, , 1983.
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6. Li, Y. dan Oldenburg, D. W., 3D Inversion of Gravity Data, Geophysics, 63, 109-119, 1998.
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8. Talwani, M., Worzel, J.L., dan Landisman, M., Rapid Gravity Computations For Two-Dimensional Bodies With Application to The Mendicino Submarine Fracture Zone: J. Geophys. Res., 64, 49-59, 1959.
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