Estimating depth-to-source

The large contribution that the upper region of a source

(Fig. 3.65) makes to the gravity and magnetic responses

above the ground surface means that the data are sensitive to the depth to the top of the source. Depth-to-source is most accurately determined by modelling the anomaly (see

Section 2.11). Often though, it is necessary to make an

estimate of the depth of the source of an anomaly quickly and easily, or depth estimates of a large number of anom- alies in a dataset may be required. Depth-to-source estima- tion is the simplest form of inverse modelling (seeSection 2.11.2).

Manual depth-to-source techniques, now largely obso- lete, involve identifying certain characteristics (zero cross- over points, gradients, distance between inflection points etc.) of the central or principal profile of the anomaly which, ideally, passes through its maximum and minimum peaks, and usually trends perpendicular to its gradients. Details for various source geometries are given by Am (1972), Atchuta Rao and Ram Babu (1984), and Blakely (1995) provides a compilation of the various methods; see also Salem et al. (2007). Modelling techniques that work with all the data points to find a ‘best-fit’ to the whole profile provide more reliable results.

3.10.4.1 Euler deconvolution

Euler (pronounced‘oiler’) deconvolution (Reid et al.,1990; Zhang et al., 2000) is a commonly used semi-automated depth-to-source method useful for quickly analysing a large number of responses in a dataset. The method is based on anomaly gradients for selected source geometry and is sequentially applied to all the points along the anomaly profile.

Euler’s equation represents the strength (f) of the poten- tialfield at a point (x, y, z) in space, due to a source located

at (x0, y0, z0), in terms of thefirst-order derivatives (∂f =∂x

etc.) of thefield in the following form: ðx  x0Þ_∂x∂f+ðy  y0Þ

∂f

∂y+ðz  z0Þ∂f_∂z¼ NðB  f Þ ð3:28Þ

which includes a background (regional) component (B). Note that for magnetic data, information about the direc- tion of the magnetism is not required, so remanent mag- netism does not present a problem.

The structural index (N) accounts for the rate of decrease in the amplitude of the response with distance from the source (see Section 3.10.1.1). This affects the measured gradients and depends on the source geometry. For the case of a spherical source, N is equal to 2 for gravity data and 3 for magnetics. Indices have been derived for a variety of source types, and they fall in the range 0 to 3 (for 0 the equation has to be modified slightly).

The source position (x0, y0, z0) and the backgroundfield

are obtained by solving the Euler equation (Eq. (3.28)). If N is too low the depth estimate (z0) will be too shallow, and if

N is too high, the depth will be overestimated. The hori- zontal coordinates are much less affected. An effective strategy is to work with all values of N between 0 and 3, in increments of, say, 0.5. This will account for the geology not being properly represented by any one of the idealised model shapes, and also it has been shown that for more realistic models N varies with depth and location.

The derivatives/gradients are usually calculated but, as discussed in Gradients and curvature in Section 2.7.4.4, gradients are susceptible to noise, especially as their order increases, so the quality of the results will be affected accordingly. Note that in magnetic data the gradient in the across-line can be poorly constrained owing to spatial asymmetry in the sampling, and is potentially a major cause of error in Euler deconvolution.

Figure 3.67shows the implementation for profile gravity

(g) data. In this case the across-line (Y) component of the field is assumed to be symmetric about the profile and a 2D result is obtained (x0, y0). A window of predefined length

(n) is progressively moved along the gravity profile and the profiles of its vertical and horizontal derivatives, and the background field and source coordinates are obtained for each measurement location. Where there are three unknowns (x0, z0 and B), the window must span a min-

imum of three points; in practice at least double this number are used, which allows the reliability of the result to be estimated. For simplicity,Fig. 3.67shows afive-point window (n ¼ 5, centred at x0). An expression for the

unknowns is set up for each location in the window, forming a set of n simultaneous equations of the form:

ðxi x0Þ∂g

∂xi+ðzi z0Þ

∂g

∂zi¼ NðB  giÞ ð3:29Þ

where i identifies the data points in the window i ¼ 1 to n. The equations are solved for the three unknowns.

The window is then moved to the next point along the profile and the process repeated. Each window position provides an estimate of source location. For 3D implemen- tation, the window is moved across the gridded data and a 3D (x0, y0, z0) solution for the source position is obtained.

Window size in either case represents a trade-off between resolution and reliability; increasing the size reduces the former and increases the latter. The window size must be large enough to include significant variations in the field being analysed, and is usually set to the wavelength of the anomalies of interest. It can be used as a crude means of

Moving window Sample # 1 2 3 4 5 a) b) c) Gravity g x₁ x₂ x₃ x₄ x₅ x0 z0 1stvertical derivative g z 1sthorizontal derivative g x X d) X Source X X Z

Solution (current window) B

Figure 3.67 Schematic illustration of 2D Euler deconvolution using a

focusing the process on sources at different depths. How- ever, the larger the window the more likely that more than one source will influence the data within it, which will create spurious results. The quality of the data, i.e. sam- pling interval and noise, affects the result.

Euler deconvolution produces many solutions for an anomaly, most of which are spurious. Various techniques are used to identify the best solutions. These include assessing how well the solutions fit the data in each window; assessing the clustering of solutions from different windows; and comparing results for various values of N and only providing the solutions for the best-fit value. Various improvements have been proposed such as solving for several sources simultaneously (to address anomaly overlap), extracting additional information about source characteristics, and more sophisticated means of choosing the best solution from the large number produced. See, for example, Mushayandebvu et al. (2001).

Euler solutions are usefully analysed in terms of depths and horizontal positions. Three-dimensional results are presented in map form with different source depths and geometries (different N) distinguished using variations in colour and/or symbol type (Fig. 3.71). The edges of source bodies can be mapped in this way. Two-dimensional data are presented as points on a cross-section, with different N represented by different symbols.

Euler deconvolution can be applied to gradient data, in which case second-order derivatives are required. The advantage of using gradients is that they are more localised to the source and have greater immunity to neighbouring sources resulting in better spatial resolution in the Euler solutions. A disadvantage is that gradients have poorer signal-to-noise ratios than normal field data, especially higher-order derivatives, so the quality of the results may be significantly degraded.