#### Resistivity Method, Physics tutorial

Survey Fundamentals:

The technique of measuring ground resistivity by concurrently passing current and measuring voltage between single pair of grounded electrodes doesn't work, due to contact resistances which depend on such things as ground moisture and contact area and that may amount to thousands of ohms. Problem can be avoided if voltage measurements are made between the second pair of electrodes utilizing the high-impedance voltmeter. Such a voltmeter draws virtually no current, and voltage drop through electrodes is thus negligible. Resistances at current electrodes restrict current flow but don't affect resistivity calculations. Geometric factor is required to convert readings achieved with the four-electrode arrays to resistivity.

Result of any single measurement with any array could be interpreted as because of homogeneous ground with constant resistivity. Geometric factors utilized to compute this apparent resistivity, Ρα, can be derived from formula:

V = ΡI/2πa

It is for electric potential V at the distance a from a point electrode at surface of a uniform half-space (homogeneous ground) of resistivity Ρ (referenced to a zero potential at infinity). Current I may be positive (if into ground) or negative. For arrays, potential at any voltage electrode is equivalent to sum of contributions from individual current electrodes. In four-electrode survey over homogeneous ground:

V = IΡ(I/[Pp]-1/[Np] -1/[Pn] + 1/[Nn])/2π

Here V is voltage difference between electrodes P and N because of a current I flowing between electrodes p and n, and quantities in square brackets represent inter-electrode distances.

Electrode arrays:

Some common electrode arrays and their geometric factors are given below. Names are those generally utilize and may upset pedants. A dipole, for instance, must comprise of two electrodes separated by the distance which is negligible compared to distance to any other electrode. Application of term to dipole-dipole and pole- dipole arrays, where distance to next electrode is generally from 1 to 6 times the dipole spacing, is therefore officially incorrect. Long cables required can impede field work and may also serve as aerials, picking up stray electromagnetic signals (inductive noise) which can affect readings.

Array descriptions:

Wenner array: It is very extensively utilized, and supported by huge amount of interpretational literature and computer packages. Standard array against which others are frequently estimated.

Two-electrode (pole-pole) array: Theoretically interesting as it is possible to compute from readings taken along traverse the results which would be achieved from any other kind of array, if coverage is sufficient. Though, noise which accumulates when large numbers of results achieved with closely spaced electrodes are added prevents any practical use being made of this fact. Array is very popular in archaeological work as it lends itself to rapid one-person operation. As normal array, it is one of the standards in electrical well logging.

Schlumberger array: Only array to rival Wenner in availability of interpretational material, all of which relates to ideal array with insignificant distance between inner electrodes. Preferred along with Wenner for electrical depth-sounding work.

Gradient array: Extensively utilized for reconnaissance. Large numbers of readings can be taken on parallel traverses without moving current electrodes if powerful generators are available. Figure shows how geometrical factor differs with position of voltage dipole.

Dipole-dipole (Eltran) array: Popular in induced polarization (IP) work as complete separation of current and voltage circuits decreases vulnerability to inductive noise. Significant body of interpretational material is available. Information from various depths is achieved by changing n. In principle, larger the value of n, the deeper the penetration of current path sampled. Results are generally plotted as pseudo-sections.

Pole-dipole array: Generates asymmetric anomalies which are therefore more difficult to interpret than those generated by symmetric arrays. Peaks are displaced from centres of conductive or chargeable bodies and electrode positions have to be recorded with especial care. Values are generally plotted at point mid-way between moving voltage electrodes.

Square array: Four electrodes positioned at corners of square are variously combined in voltage and current pairs. Depth soundings are made by expanding square. In traversing, whole array is moved laterally. Inconvenient, but can give experienced interpreter with important information about ground anisotropy and inhomogeneity.

Lee array: Looks likes Wenner array but has extra central electrode. Voltage differences from centre to two normal voltage electrodes give the measure of ground in-homogeneity. Two values can be summed for application of Wenner formula.

Offset Wenner: Like Lee array but with all five electrodes same distance apart. Measurements made using four right-hand and four left-hand electrodes discretely as standard Wenner arrays are averaged to provide apparent resistivity and differenced to give measure of ground variability.

Focused arrays: Multi-electrode arrays have been developed that allegedly focus current in ground and give deep penetration without large expansion. Perhaps, this is an attempt to do impossible, and arrays must be utilized only under direction of experienced interpreter.

Signal-contribution sections:

Near-surface in-homogeneities strongly influence choice of array. Their effects are graphically shown by contours of signal contributions which are made by each unit volume of ground to measured voltage, and therefore to apparent resistivity. For linear arrays contours have same look in any plane, whether vertical, horizontal or dipping, through line of electrodes (i.e. they are semicircles when array is viewed end on).

First reaction is that helpful resistivity surveys are not possible, as contributions from regions close to electrodes are very large. Though, variations in sign mean those conductive near-surface layers will in some places increase and in other places decrease apparent resistivity.

In homogeneous ground such effects can cancel quite accurately. When the Wenner or dipole-dipole array is expanded, all electrodes are moved and contributions from near-surface bodies differ from reading to reading. With Schlumberger array, near-surface effects differ much less, if only outer electrodes are moved, and therefore array is frequently preferred for depth sounding. Though, offset methods permit excellent results to be obtained with Wenner.

Depth penetration:

Arrays are generally selected at least partially for their depth penetration that is almost impossible to define as depth to which given fraction of current penetrates depends on layering and on separation between current electrodes. Voltage electrode positions determine which part of current field is sampled, and penetrations of Wenner and Schlumberger arrays are therefore probable to be very similar for similar total array lengths. For either array, expansion at which existence of the deep interface first becomes apparent depends on resistivity contrast (and levels of background noise) but is of order of half the spacing between outer electrodes. Quantitative determination of resistivity change would, of course, need much greater expansion. For any array, there is also the expansion at which effect of thin horizontal layer of different resistivity in otherwise homogeneous ground is maximum. It is, maybe, to be expected that much greater expansion is required in this case than is needed just to detect interface. By this criterion, dipole-dipole is most and Wenner is the least penetrative array. Wenner peak takes place when array is 10 times as broad as conductor is deep, and Schlumberger is only little better. Wenner curve to be the most sharply peaked, signifying superior vertical resolving power. This is confirmed by signal contribution contours that are slightly flatter at depth for Wenner than for Schlumberger, signifying that Wenner locates flat lying interfaces more correctly. Signal-contribution contours for dipole-dipole array are near vertical in some places at considerable depths, signifying poor vertical resolution and suggesting that array is best suitable to mapping lateral changes.

Noise in electrical surveys:

Electrodes may in principle be positioned on ground surface to any preferred degree of accuracy (though errors are always possible and become more probable as separations increase). Most modern instruments give current at one of a number of preset levels and fluctuations in supply are usually small and insignificant. Noise thus enters apparent resistivity values almost completely via voltage measurements, ultimate limit being determined by voltmeter sensitivity. There may also be noise because of induction in cables and also to natural voltages that may differ with time and so be partly cancelled by reversing current flow and averaging. Large separations and long cables must be avoided if possible, but most effective technique of enhancing signal/noise ratio is to increase signal strength. Modern instruments frequently give observers with direct readings of V/I, estimated in ohms, and so tend to cover voltage magnitudes. Small ohm values signify small voltages but current levels also have to be taken into consideration.

For the given input current, voltages estimated using Schlumberger array are always less than those for Wenner array of same overall length, as separation between voltage electrodes is always smaller. For dipole-dipole array, comparison depends on n parameter but even for n = 1 (i.e. for array very like Wenner in appearance), signal strength is smaller than for Wenner by factor of three. Differences between gradient and two electrode reconnaissance arrays are even more striking. If distances to fixed electrodes are 30 times dipole separation, two-electrode voltage signal is more than150 times gradient array signal for same current. However, gradient array voltage cable is shorter and easier to handle, and less vulnerable to inductive noise. Much larger currents can safely be utilized as current electrodes are not moved.

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