Role of Euler Deconvolution in Near Surface Gravity and Magnetic Applications

Abstract

Euler deconvolution method builds a part of depth estimation methods in semi-automated interpretation of potential fields in applied geophysics. It is suitable for the interpretation of well developed and separated anomalies from isolated sources, which often occur in near surface applications (detection of iron bodies in magnetometry, cavities in gravimetry, etc.). This contribution describes theoretical background and algorithm of the method. Important part is also the stabilization of the method by means of regularized derivatives introduction and selection of correct solution by means of several statistical criteria. Results on synthetic data from simple and complex models show possibilities and limits of this method. Finally, several real world examples from near surface potential fields applications are given (microgravity detection of sub-surface cavities, geomagnetic search for UneXploded Ordnance).

Appendices

Appendix 1: Derivation of the Euler’s Equation

This is a very simple derivation of the Euler equation from the homogeneity theorem. As it cannot be found in common text-books of mathematical analysis, it is given here. We work with homogeneous function f(x), where x has components (x₁, x₂, x₃) (in Cartesian coordinates can be written x₁ = x, x₂ = y, x₃ = z).

Euler homogeneity theorem can be written:

$${\text{t}}^{\text{n}} {\text{f}}\left( {\mathbf{x}} \right) = {\text{f}}\left( {{\text{t}}{\mathbf{x}}} \right),$$

(9.20)

where t and n are real numbers (n is the homogeneity degree).

Both sides of Eq. (9.20) are differentiated with respect to the component x_i, and we get:

$$\begin{aligned} & \frac{{\partial \left[ {{\text{t}}^{\text{n}} {\text{f}}\left( {\mathbf{x}} \right)} \right]}}{{\partial {\text{x}}_{\text{i}} }} = \frac{{\partial \left[ {{\text{f}}\left( {{\text{t}}{\mathbf{x}}} \right)} \right]}}{{\partial {\text{x}}_{\text{i}} }}, \\ & \frac{{\partial \left[ {{\text{t}}^{\text{n}} {\text{f}}\left( {\mathbf{x}} \right)} \right]}}{{\partial {\text{f}}\left( {\mathbf{x}} \right)}}\frac{{\partial {\text{f}}\left( {\mathbf{x}} \right)}}{{\partial {\text{x}}_{\text{i}} }} = \frac{{\partial \left[ {{\text{f}}\left( {{\text{t}}{\mathbf{x}}} \right)} \right]}}{{\partial \left( {{\text{tx}}_{\text{i}} } \right)}}\frac{{\partial \left( {{\text{tx}}_{\text{i}} } \right)}}{{\partial {\text{x}}_{\text{i}} }}, \\ & {\text{t}}^{\text{n}} \frac{{\partial {\text{f}}\left( {\mathbf{x}} \right)}}{{\partial {\text{x}}_{\text{i}} }} = \frac{{\partial \left[ {{\text{f}}\left( {{\text{t}}{\mathbf{x}}} \right)} \right]}}{{\partial \left( {{\text{tx}}_{\text{i}} } \right)}}{\text{t}}, \\ & {\text{t}}^{{{\text{n}} - 1}} \frac{{\partial {\text{f}}\left( {\mathbf{x}} \right)}}{{\partial {\text{x}}_{\text{i}} }} = \frac{{\partial \left[ {{\text{f}}\left( {{\text{t}}{\mathbf{x}}} \right)} \right]}}{{\partial \left( {{\text{tx}}_{\text{i}} } \right)}}. \\ \end{aligned}$$

(9.21)

Now both sides of Eq. (9.20) will be differentiated with respect to the real number t:

$$\begin{aligned} & \frac{{\partial \left[ {{\text{t}}^{\text{n}} {\text{f}}\left( {\mathbf{x}} \right)} \right]}}{{\partial {\text{t}}}} = \frac{{\partial \left[ {{\text{f}}\left( {{\text{t}}{\mathbf{x}}} \right)} \right]}}{{\partial {\text{t}}}}, \\ & {\text{nt}}^{{{\text{n}} - 1}} {\text{f}}\left( {\mathbf{x}} \right) = \frac{{\partial \left[ {{\text{f}}\left( {{\text{t}}{\mathbf{x}}} \right)} \right]}}{{\partial \left( {{\text{tx}}_{1} } \right)}}\frac{{\partial \left( {{\text{tx}}_{1} } \right)}}{{\partial {\text{t}}}} + \frac{{\partial \left[ {{\text{f}}\left( {{\text{t}}{\mathbf{x}}} \right)} \right]}}{{\partial \left( {{\text{tx}}_{2} } \right)}}\frac{{\partial \left( {{\text{tx}}_{2} } \right)}}{{\partial {\text{t}}}} + \cdots + \frac{{\partial \left[ {{\text{f}}\left( {{\text{t}}{\mathbf{x}}} \right)} \right]}}{{\partial \left( {{\text{tx}}_{\text{M}} } \right)}}\frac{{\partial \left( {{\text{tx}}_{\text{M}} } \right)}}{{\partial {\text{t}}}}, \\ & {\text{nt}}^{{{\text{n}} - 1}} {\text{f}}\left( {\mathbf{x}} \right) = \sum\limits_{{{\text{i}} = 1}}^{\text{M}} {\frac{{\partial \left[ {{\text{f}}\left( {{\text{t}}{\mathbf{x}}} \right)} \right]}}{{\partial \left( {{\text{tx}}_{\text{i}} } \right)}}\frac{{\partial \left( {{\text{tx}}_{i} } \right)}}{{\partial {\text{t}}}}} ,{\text{where we usually take M}} = 3, \\ & {\text{nt}}^{{{\text{n}} - 1}} {\text{f}}\left( {\mathbf{x}} \right) = \sum\limits_{{{\text{i}} = 1}}^{\text{M}} {\frac{{\partial \left[ {{\text{f}}\left( {{\text{t}}{\mathbf{x}}} \right)} \right]}}{{\partial \left( {{\text{tx}}_{\text{i}} } \right)}}{\text{x}}_{\text{i}} } . \\ \end{aligned}$$

(9.22)

Now in the right-hand side of Eq. (9.22) (in the sum) the expression from the left-hand side of Eq. (9.21) will be entered:

$$\begin{aligned} & {\text{nt}}^{{{\text{n}} - 1}} {\text{f}}\left( {\mathbf{x}} \right) = \sum\limits_{{{\text{i}} = 1}}^{\text{M}} {{\text{t}}^{{{\text{n}} - 1}} \frac{{\partial \left[ {{\text{f}}\left( {\mathbf{x}} \right)} \right]}}{{\partial {\text{x}}_{\text{i}} }}{\text{x}}_{\text{i}} } , \\ & {\text{nf}}\left( {\mathbf{x}} \right) = \sum\limits_{{{\text{i}} = 1}}^{\text{M}} {\frac{{\partial \left[ {{\text{f}}\left( {\mathbf{x}} \right)} \right]}}{{\partial {\text{x}}_{\text{i}} }}{\text{x}}_{\text{i}} } . \\ \end{aligned}$$

(9.23)

And this is the Euler’s equation, which is solved in the Euler deconvolution method. For M = 3 and x₁ = x, x₂ = y, x₃ = z, Eq. (9.23) can be written in the more common form:

$${\text{nf}}\left( {{\text{x}},{\text{y}},{\text{z}}} \right) = {\text{x}}\frac{{\partial {\text{f}}}}{{\partial {\text{x}}}} + {\text{y}}\frac{{\partial {\text{f}}}}{{\partial {\text{y}}}} + {\text{z}}\frac{{\partial {\text{f}}}}{{\partial {\text{z}}}}.$$

(9.24)

Appendix 2: Derivation of Structural Index Values for Selected Bodies

For bodies with elementary shape (Table 9.1 in the main text), values of structural index (SI) can be derived analytically. This was done by several authors (Reid et al. 1990; Pašteka 2001; Reid 2003; Stavrev and Reid 2010, etc.). There is no space to give all of these derivations here for each of the source type. We give here few typical examples, so reader can have an idea how it works and can derive other types of bodies by themselves. Simplest derivations are of course for simple bodies, like poles/dipoles and line of poles/dipoles. For tabular bodies (dike, sill), derivations are little bit more complicated, but can be done with tools of simple mathematical analysis. The most complicated source is the step (contact) in gravimetry, which has a value of −1. Such a negative value doesn’t work well with the algorithm of classical Euler deconvolution and method has to use either modified equations (Stavrev and Reid 2010) or a non-constant background term in Euler equation (Pašteka 2006). This negative value was also hardly accepted by the community, because negative indices contradict the idea of SI as a fall of rate of interpreted function (as it is mentioned in the main text). It seems that the anomalous field of this structure is not homogeneous in Euler’s sense (at least that from vertically limited step) and several authors (e.g. Marson and Klinglele 1993) recommend to apply firstly a vertical derivative evaluation and then the application of the Euler deconvolution itself (with SI value around 0, which is valid for this type of structure in magnetometry—see Table 9.1 in main text).

In the analytical derivation of SI values we can follow two concepts—based on Eqs. 9.1 and 9.2 from the main text. From Eq. 9.1 (homogeneity equation) it follows after few simple adjustments (Stavrev and Reid 2007, p. L2) that for SI value we get a simple formula:

$${\text{N}} = - \frac{{\ln \left[ {{{{\text{f}}\left( {{\text{tx}},{\text{ty}},{\text{tz}}} \right)} \mathord{\left/ {\vphantom {{{\text{f}}\left( {{\text{tx}},{\text{ty}},{\text{tz}}} \right)} {{\text{f}}\left( {{\text{x}},{\text{y}},{\text{z}}} \right)}}} \right. \kern-0pt} {{\text{f}}\left( {{\text{x}},{\text{y}},{\text{z}}} \right)}}} \right]}}{{\ln \left( {\text{t}} \right)}},$$

(9.25)

which is valid for t > 0 and t ≠ 1.

From Eq. 9.2 (Euler equation) we get after simple adjustments:

$${\text{N}} = - \frac{{{\text{x}}\frac{{\partial {\text{f}}}}{{\partial {\text{x}}}} + {\text{y}}\frac{{\partial {\text{f}}}}{{\partial {\text{y}}}} + {\text{z}}\frac{{\partial {\text{f}}}}{{\partial {\text{z}}}}}}{\text{f}},$$

(9.26)

which is valid for f(x, y, z) ≠ 0.

Entering with specific functions and their derivatives into these two equations, they should give identical results for identical source structures. Further we show few examples of their application.

9.2.1 2D-Horizontal Cylinder, Gravimetry

We take a 2D-horizontal cylinder (rod) with the length density λ, (e.g. Telford et al., 1990, p. 37), which central axis is identical with the y-axis (crossing the xz-plane in point [0, 0]). For the vertical component of gravitational acceleration V_z and derivatives with respect to x- and z-direction (derivative with respect to y-direction is equal to zero) we can write:

$${\text{V}}_{{\text{z}}} \left( {{\text{x}},{\text{z}}} \right) = 2{\text{G}}\uplambda\frac{z}{{{\text{x}}^{2} + {\text{z}}^{2} }}{ ,}\frac{{\partial {\text{V}}_{{\text{z}}} \left( {{\text{x}},{\text{z}}} \right)}}{{\partial {\text{x}}}} = - 4{\text{G}}\uplambda\frac{{\text{xz}}}{{\left( {{\text{x}}^{2} + {\text{z}}^{2} } \right)^{2} }} ,\frac{{\partial {\text{V}}_{{\text{z}}} \left( {{\text{x}},{\text{z}}} \right)}}{{\partial {\text{z}}}} = - 2{\text{G}}\uplambda\frac{{{\text{z}}^{2} - {\text{x}}^{2} }}{{\left( {{\text{x}}^{2} + {\text{z}}^{2} } \right)^{2} }}.$$

(9.27)

Using Eq. 9.25 we get:

$$\begin{aligned} {\text{N}} & = - \frac{{{ \ln }\left[ {{{{\text{V}}_{\text{z}} \left( {{\text{tx}},{\text{ty}}} \right)} \mathord{\left/ {\vphantom {{{\text{V}}_{\text{z}} \left( {{\text{tx}},{\text{ty}}} \right)} {{\text{V}}_{\text{z}} \left( {{\text{x}},{\text{y}}} \right)}}} \right. \kern-0pt} {{\text{V}}_{\text{z}} \left( {{\text{x}},{\text{y}}} \right)}}} \right]}}{{\ln \left( {\text{t}} \right)}} = - \frac{{\ln \left[ {\frac{{2{\text{G}}\uplambda\frac{\text{tz}}{{{\text{t}}^{2} {\text{x}}^{2} + {\text{t}}^{2} {\text{z}}^{2} }}}}{{2{\text{G}}\uplambda\frac{\text{z}}{{{\text{x}}^{2} + {\text{z}}^{2} }}}}} \right]}}{{\ln \left( {\text{t}} \right)}} = - \frac{{\ln \left[ {\frac{{2{\text{G}}\uplambda\frac{\text{z}}{{{\text{t}}\left( {{\text{x}}^{2} + {\text{z}}^{2} } \right)}}}}{{2{\text{G}}\uplambda\frac{\text{z}}{{{\text{x}}^{2} + {\text{z}}^{2} }}}}} \right]}}{{\ln \left( {\text{t}} \right)}} = \\ & = - \frac{{\ln \left[ {{\text{t}}^{ - 1} \frac{{2{\text{G}}\uplambda\frac{\text{z}}{{\left( {{\text{x}}^{2} + {\text{z}}^{2} } \right)}}}}{{2{\text{G}}\uplambda\frac{\text{z}}{{{\text{x}}^{2} + {\text{z}}^{2} }}}}} \right]}}{{\ln \left( {\text{t}} \right)}} = - \frac{{\ln \left( {{\text{t}}^{ - 1} } \right)}}{{\ln \left( {\text{t}} \right)}} = - \frac{{\left( { - 1} \right)\ln \left( {\text{t}} \right)}}{{\ln \left( {\text{t}} \right)}} = 1. \\ \end{aligned}$$

(9.28)

Entering expressions 9.27 into Eq. 9.26 we get:

$${\text{N}} = - \frac{{ - 4{\text{G}}\uplambda\frac{{{\text{x}}^{2} {\text{z}}}}{{\left( {{\text{x}}^{2} + {\text{z}}^{2} } \right)^{2} }} - 2{\text{G}}\uplambda\frac{{{\text{z}}^{3} - {\text{x}}^{2} {\text{z}}}}{{\left( {{\text{x}}^{2} + {\text{z}}^{2} } \right)^{2} }}}}{{2{\text{G}}\uplambda\frac{\text{z}}{{{\text{x}}^{2} + {\text{z}}^{2} }}}} = - \frac{{ - 2{\text{G}}\uplambda\frac{{2{\text{x}}^{2} {\text{z}} + {\text{z}}^{3} - {\text{x}}^{2} {\text{z}}}}{{\left( {{\text{x}}^{2} + {\text{z}}^{2} } \right)^{2} }}}}{{2{\text{G}}\uplambda\frac{\text{z}}{{{\text{x}}^{2} + {\text{z}}^{2} }}}} = - \frac{{ - 2{\text{G}}\uplambda\frac{\text{z}}{{{\text{x}}^{2} + {\text{z}}^{2} }}}}{{2{\text{G}}\uplambda\frac{\text{z}}{{{\text{x}}^{2} + {\text{z}}^{2} }}}} = 1.$$

(9.29)

Both ways came to an identical result (SI = N = 1) for the 2D-horizontal cylinder (rod) in gravimetry. In magnetometry it would be N = 2.

9.2.2 2D-Inclined Sheet, Magnetometry

In the case of magnetized inclined sheet (dike), positioned with its edge along the y-axis) we can write for the ΔT field following formula (Werner 1953):

$$\Delta {\text{T}}\left( {{\text{x}},{\text{z}}} \right) = \frac{{{\text{Ax}} + {\text{Bz}}}}{{{\text{x}}^{2} + {\text{z}}^{2} }} ,$$

(9.30)

where: A = −2b (M_x sin I + M_z cos I sin α) and B = 2b (−M_x cos I sin α + M_z sin I). M_x and M_z are components of the magnetization vector, 2b is the thickness of the sheet, I is inclination of the magnetization vector and α is the azimuth of the profile. For the derivatives with respect to x and z (derivative with respect to y-direction is equal to zero) we can write:

$$\frac{{\partial \Delta {\text{T}}\left( {{\text{x}},{\text{z}}} \right)}}{{\partial {\text{x}}}} = \frac{{ - {\text{Ax}}^{2} + {\text{Az}}^{2} - 2{\text{Bzx}}}}{{\left( {{\text{x}}^{2} + {\text{z}}^{2} } \right)^{2} }} ,\frac{{\partial \Delta {\text{T}}\left( {{\text{x}},{\text{z}}} \right)}}{{\partial {\text{z}}}} = \frac{{{\text{Bx}}^{2} - {\text{Bz}}^{2} - 2{\text{Axz}}}}{{\left( {{\text{x}}^{2} + {\text{z}}^{2} } \right)^{2} }} \, .$$

(9.31)

Using the Eq. 9.25 we get:

$${\text{N}} = - \frac{{\ln \left[ {{{\Delta {\text{T}}\left( {{\text{tx}},{\text{ty}}} \right)} \mathord{\left/ {\vphantom {{\Delta {\text{T}}\left( {{\text{tx}},{\text{ty}}} \right)} {\Delta {\text{T}}\left( {{\text{x}},{\text{y}}} \right)}}} \right. \kern-0pt} {\Delta {\text{T}}\left( {{\text{x}},{\text{y}}} \right)}}} \right]}}{{\ln \left( {\text{t}} \right)}} = - \frac{{\ln \left[ {\frac{{\frac{{{\text{Atx}} + {\text{Btz}}}}{{{\text{t}}^{2} {\text{x}}^{2} + {\text{t}}^{2} {\text{z}}^{2} }}}}{{\frac{{{\text{Ax}} + {\text{Bz}}}}{{{\text{x}}^{2} + {\text{z}}^{2} }}}}} \right]}}{{\ln \left( {\text{t}} \right)}} = - \frac{{\ln \left[ {{\text{t}}^{ - 1} \frac{{\frac{{{\text{Ax}} + {\text{Bz}}}}{{{\text{x}}^{2} + {\text{z}}^{2} }}}}{{\frac{{{\text{Ax}} + {\text{Bz}}}}{{{\text{x}}^{2} + {\text{z}}^{2} }}}}} \right]}}{{\ln \left( {\text{t}} \right)}} = - \frac{{\left( { - 1} \right)\ln \left( {\text{t}} \right)}}{{\ln \left( {\text{t}} \right)}} = 1$$

(9.32)

Entering expressions 9.31 into Eq. 9.26 we get:

$${\text{N}} = - \frac{{\frac{{{\text{x}}\left( { - {\text{Ax}}^{2} + {\text{Az}}^{2} - 2{\text{Bzx}}} \right)}}{{\left( {{\text{x}}^{2} + {\text{z}}^{2} } \right)^{2} }} + \frac{{{\text{z}}\left( {{\text{Bx}}^{2} - {\text{Bz}}^{2} - 2{\text{Axz}}} \right)}}{{\left( {{\text{x}}^{2} + {\text{z}}^{2} } \right)^{2} }}}}{{\frac{{{\text{Ax}} + {\text{Bz}}}}{{{\text{x}}^{2} + {\text{z}}^{2} }}}} = - \frac{{\frac{{ - \left( {{\text{x}}^{2} + {\text{z}}^{2} } \right)\left( {{\text{Ax}} + {\text{Bz}}} \right)}}{{\left( {{\text{x}}^{2} + {\text{z}}^{2} } \right)^{2} }}}}{{\frac{{{\text{Ax}} + {\text{Bz}}}}{{{\text{x}}^{2} + {\text{z}}^{2} }}}} = 1.$$

(9.33)

Both ways came to an identical result (SI = N = 1) for the 2D-inclined sheet (dike) in magnetometry. In gravimetry it would be N = 0.

9.2.3 2D-Semiinfinite Horizontal Sheet, Gravimetry

We take a 2D-semiinfinite horizontal sheet (sill) with the surface m (e.g. Telford et al. 1990, p. 40), which starts in the begin of the coordinate system [0, 0] and runs along the positive part of x-axis-axis for the vertical component of gravitational acceleration V_z and derivatives with respect to x- and z-direction (derivative with respect to y-direction is equal to zero) we can write:

$${\text{V}}_{\text{z}} \left( {{\text{x}},{\text{z}}} \right) = 2{\text{G}}\upmu\left( {\frac{\uppi}{2} + {\text{arctg}}\frac{{\text{x}}}{{\text{z}}}} \right) ,\quad \frac{{\partial {\text{V}}_{{\text{z}}} \left( {{\text{x}},{\text{z}}} \right)}}{{\partial {\text{x}}}} = 2{\text{G}}\upmu\frac{{\text{z}}}{{{\text{x}}^{2} + {\text{z}}^{2} }} ,\;\;\frac{{\partial {\text{V}}_{{\text{z}}} \left( {{\text{x}},{\text{z}}} \right)}}{{\partial {\text{z}}}} = - 2{\text{G}}\upmu\frac{{\text{x}}}{{{\text{x}}^{2} + {\text{z}}^{2} }} .$$

(9.34)

In this case, we can use only Eq. 9.26:

$${\text{N}} = - \frac{{2{\text{G}}\upmu\frac{{\text{xz}}}{{{\text{x}}^{2} + {\text{z}}^{2} }} \, - 2{\text{G}}\upmu\frac{{\text{zx}}}{{{\text{x}}^{2} + {\text{z}}^{2} }}}}{{2{\text{G}}\upmu\left( {\frac{\uppi}{2} + {\text{arctg}}\frac{{\text{x}}}{{\text{z}}}} \right)}} = - \frac{0}{{2{\text{G}}\upmu\left( {\frac{\uppi}{2} + {\text{arctg}}\frac{{\text{x}}}{{\text{z}}}} \right)}} = 0.$$

(9.35)

From it follow that SI = N = 0 for the 2D-semiinfinite horizontal sheet (sill) in gravimetry. In magnetometry it would be N = 1.

There are of course many different approaches and ways for the derivation and expression of SI value for various sources types (e.g. Stavrev 1997; Stavrev and Reid 2007; Fedi 2016).