Estimating depth and model type due finite step, thin and thick dikes using the continuous wavelet transform



The continuous wavelet transform has been proposed recently for the interpretation of potential field anomalies. Using Poisson wavelets, which are equivalent to an upward continuation of the analytic signal, this technique allows us to estimate the depth of the buried homogeneous field source and to determine the nature of the source in the form of a structural index.
The interpretation of potential field data is not a straightforward process because of the many models capable of explaining the observed field. For this reason, the interpretation method selected must relate to the geology. Furthermore, if we wish to design an automatic interpretation technique that is to be applied over a large area with many anomalies, we prefer the model to be as simple as possible.
We propose an alternative technique that estimates the depth and the structural index from the ratio of wavelets of successive orders. We show how to interpret these results over multiple (non homogeneous) sources, such as a finite step, and thin and thick dikes. This method is an extension of the technique of Hsu et al. (1998) and Sailhac et al. (2000), who both consider cases of extended sources with one finite dimension such as thin dikes of finite depth extent and thick dike models.
Summary of the implementation: We propose an implementation of this algorithm that includes the following steps:
1) Start with a magnetic field profile.
2) Compute the horizontal and vertical derivatives.
3) Compute the analytic signal of orders 0 and 1 and normalize by the dilations to obtain the wavelet transforms.
4) Search for the maxima along the wavelet coefficient profiles and find the associated maxima for successive heights.
5) Apply equations to estimate the depth and structural index from different pairs of dilations.
6) Select the best estimates for the variation of the estimates with dilations.
In this research, first, we use technique over multipole sources, such as a finite step thin and thick dikes. Then we test the proposed technique on a profile data in area located in Khoramabad in Iran. The anomaly on this profile has a structural index 0.1 which indicates a step in 1120 m depth. At the end this method was compared with the Euler Deconvolution method.
Both Euler and wavelet techniques give satisfactory results, but the wavelet approach is preferable for three reasons : (1) it allows the estimation of both depths zo and structural index N in a simple manner without the need primary information; (2) it includes an upward continuation procedure which provides useful symmetries between the wavelet transform domain and buried sources , and also reduces high-frequency noise present in the data which may produce strong artifacts in Euler Deconvolution; and (3) the wavelet approach simplifies the characterization of sources of finite extent.