^{1,2}

^{1,3}

^{1}

^{2}

^{3}

We present an analysis of the error involved in the so-called low induction number approximation in the electromagnetic methods. In particular, we focus on the EM34 equipment settings and field configurations, widely used for geophysical prospecting of laterally electrical conductivity anomalies and shallow targets. We show the theoretical error for the conductivity in both vertical and horizontal dipole coil configurations within the low induction number regime and up to the maximum measuring limit of the equipment. A linear relationship may be adjusted until slightly beyond the point where the conductivity limit for low induction number (

The induction method consists basically in determining subsurface rock conductivities with the help of electromagnetic fields generated by a coil at the Earth’s surface and by catching the response to this field from the conducting media under surface by using a reception coil [

From the Maxwell equations, in particular the Faraday induction law applied to an infinite homogenous half-plane, the subsurface rock conductivity can be estimated through the ratio between the magnetic field measured in the receiving coil and the magnetic field produced at the transmission coil with both at surface. Then we can take laterally distributed measurements along a transect for identifying conductivity-related anomalies. We can also get information on the vertical conductivity structure by varying the coil’s dipole configurations (vertical dipole or horizontal dipole) as well as by increasing the instrument height. This information is very useful in several geophysical problems as, for example, water prospecting or mapping pollution plumes.

The basic model for both configurations is described in Figure

Representation of the transmission (Tx) and reception (Rx) coils for both the vertical and the horizontal dipole configurations.

In general, the secondary magnetic field

In order to understand the induction number we need to define the “electromagnetic skin depth”

The induction number

In this work we introduce an error analysis for the induction method taking as a starting point the set of equations given by McNeill [

The equations for the primary and secondary field ratios for vertical and horizontal dipole configurations as given by McNeill [

Expressions (

To do this we begin by writing

By substituting in (

By taking only the quadrature of

Similarly, for the horizontal dipole configuration we obtain

The imaginary part (quadrature term) is related to the conductivity measurements under low induction number conditions. The magnitude of the secondary magnetic field is now directly proportional to the conductivity and its phase leads the primary field to 90°. Under low induction number (

This is the low induction number approximation key expression which leads to the conductivity value from the equipment readings.

The induction number ^{1/2} * m^{1/2}). Since the low induction number condition (

We define the error associated with the low induction number approximation as the theoretical deviation from the exact model which is the electromagnetic response for a homogenous half-plane as follows:

From the induction number definition we have

In this way, the error associated with the approximation is

Substituting those

Relative percent error (%) as a function of conductivity (mS/m). It also shows error isolines for the error levels of 100%, 50%, and 25%. The vertical black lines mark two main limits: induction number equals to one (

From the studied lower limit through the whole low induction number regime (considering its limit as

Figure

Apparent conductivity versus true conductivity in mS/m until slightly above the EM34 instrument limit. The vertical black lines mark two main limits: induction number equals to one (

It is well known that the vertical dipole configuration is more sensitive to conductivity anomalies in deep, while the horizontal one is more sensitive to near surface conductivity variations [

An approximately linear tendency can be observed over the whole low induction number range and up to 400 mS/m. Above this value there are strong deviations from linearity indicating that we are close to singularities of the complex function (not only the quadrature term). Despite a correction procedure suggested by Beamish [

It is worth noting that the determination of these bounds depends on the configurations (operation frequency and intercoil spacing) of the considered instrument.

This work was supported by the Brazilian agency CNPq through a postdoctoral scholarship to George Caminha-Maciel. Thanks are due to E. La Terra for the valuable discussions and M. Ernesto for helping in improving the paper. Any use of product names is for descriptive purposes only and does not imply any endorsement.