Abstract

Resistivity soundings are often carried out using a Schlumberger array. With this configuration, a current I is fed into the ground using two current electrodes A and B placed a distance L apart. At the center of these two electrodes, the voltage U is measured between two potential electrodes M and N placed a distance a apart. The condition a < L is necessary to satisfy the assumption of a Schlumberger sounding curve, which is a log-log plot of apparent resistivitiesEquation (1)versus half the distance between the current electrodes r = L/2.For relatively large distances L, an increase in the distance a between the potential electrodes is generally necessary to obtain a sufficiently large voltage. Thereby, in most cases, the apparent resistivity curve is segmented. The displacement of each segment can be attributed to two effects: (1) the change in the geometry of the configuration, because the ratio a/L can no longer be assumed to be infinitely small (upon expanding the distance a between M and N), and/or (2) inhomogeneities of the resistivities near the potential electrodes. A precise interpretation of the sounding data requires a knowledge of the influence of the first effect, because a change in the geometry of the electrode array may change the form of the curve. The influence of this change decreases as L/a increases. The second effect can be eliminated by a parallel shift of one segment to the other. The construction of correction curves which incorporate the first effect are described below. These curves should provide sufficient accuracy even in field work.As was shown by Deppermann (1954), the apparent resistivity function rho (super (a)) a (L/2) for a symmetrical four-point AMNB configuration (finite a) for a horizontally stratified ground may be obtained by an integration of the apparent resistivities rho (super (0)) a for a Schlumberger configuration (a --> 0) as follows:Equation (2)where V is the potential created at the earth's surface. On the other hand, the resistivity function for an idealized Schlumberger configuration (a --> 0) isEquation (3)Solving this equation for dV/dr, integrating between L/2 - a/2 and L/2 + a/2, and inserting the result in equation (2) yieldsEquation (4)Deppermann (1954) used a Lagrange interpolation formula to solve this equation, thus obtaining the rho (super (a)) a values in the form of a weighted sum of four FIG. 1. Ratio of apparent resistivity rho (super (a)) a (distance a between potential electrodes M and N) to apparent resistivity rho (super (0)) a for a Schlumberger configuration (a --> 0) versus the ratio AB/MN for several slopes of the Schlumberger curve. rho (super (0)) a values. A small procedure can be applied by using linear filter theory (Koefoed, 1979). Here, a simpler integration procedure is used.To obtain a first approximation for rho (super (a)) a , the rho (super (0)) a curve on a log-log plot is approximated within the interval of integration by a straight line with slope gamma :Equation (5)With this approximation, the integration of equation (4) can be carried out immediately:Equation (6)This means that, to a first approximation, the ratio of the apparent resistivities is only a function of a/L and gamma . For a horizontally stratified ground, the sounding curve can have a maximum slope of 1. By taking the limit gamma --> 1 in expression (6), one obtainsEquation (6a)Formulas (6) and (6a) were used to construct the correction curves in Figure 1, which show the ratio rho (super (a)) a /rho (super (0)) a [for various values of the slope (in degrees) of the Schlumberger curve] as a function of the ratio AB/MN (ratio of current electrode spacing to potential electrode spacing). Additionally, the correction Delta (in millimeters) is shown for log paper with a log cycle of 83.33 mm, as is used in Germany. For a quick determination of the slope of the rho (super (0)) a curve during field work, a goniometer has been drawn within Figure 1. In practice, the rho (super (0)) a curve is taken as that portion of the curve segment that was measured at the largest AB/MN ratio prior to a change of the MN distance.Since, according to equation (4), the values for rho (super (0)) a are weighted with 1/r 2 , the first section of the integration interval has a larger influence on rho (super (a)) a than the latter one. Comparison of the rho (super (a)) a values obtained from the curves in Figure 1 with exact values shows as a rule of thumb that only the mean slope of the Schlumberger curve within the interval L/2 - a/2 to L/2 should be used in Figure 1. The exact values were obtained using a computer program based on the following equation for the n-layer case (Kunetz, 1966):FIG. 2. Schlumberger and Wenner sounding curve for the model in the top of the figure, and Wenner data (*) as obtained with AB/MN = 3 from Figure 1. Equation (7)where J 0 is a Bessel function of first kind, zero order, andEquationwhereEquation (7a)where h i is the thickness of the ith layer, k i = (rho (sub i + 1) - rho i )/(rho (sub i + 1) + rho i ) is the reflection coefficient of the ith layer, and rho i is the resistivity of the ith layer. As an example, the Schlumberger and the Wenner curves for a specific model are shown in Figure 2. The rho (super (a)) a values which were obtained from the Schlumberger curve with the help of Figure 1 and the above-mentioned rule of the rho (super (0)) a curve and with AB/MN = 3 are also shown (*). The good fit may be observed.

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