Induction logging method and apparatus including means for combining on-phase and quadrature components of signals received at varying frequencies and including use of multiple receiver means associated with a single transmitter
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention generally relates to a method and apparatus for investigating earth formations traversed by a borehole, and more particularly, pertains to a new and improved method and apparatus for electromagnetic well logging. Still more particularly this invention relates to a new and improved method and apparatus for acquiring multiple signals during induction logging, combining these signals in software, and creating a depth tagged representation of resistivity or conductivity from a weighted sum of signals from that depth or from that depth and neighboring depths. 
     2. Description of the Prior Art 
     Generally speaking, in electromagnetic well logging, commonly referred to as &#34;induction logging,&#34; a transmitter coil energized by alternating current is lowered into a well or borehole and indications are obtained of the influence of surrounding formations on the electromagnetic field established by the coil. Usually such indications are obtained by observing the voltage induced in a receiver coil lowered into the borehole in coaxial relationship with the transmitter coil and axially spaced apart therefrom. 
     Commonly available induction logging tools include a transmitter array and a receiver array each consisting of a set of coils connected together in series. Such a transmitter/receiver array is illustrated in U.S. Pat. No. 3,067,383 issued Dec. 4, 1962 to Dennis R. Tanguy. The transmitter is driven by a constant amplitude current at 20 kHz. The induction measurement consists of recording the total voltage induced in the receiver array. 
     Such commonly available commercial logging tools in service today record only the component of the total voltage which is 180 degrees out of phase (called the &#34;in-phase&#34; or &#34;real voltage, R&#34;) with the transmitter current. Such an induction sonde is sensitive to a spatial average of the conductivity of the surrounding formation. The averaging function is determined by the placement and winding of the transmitter and receiver coils. 
     Logging methods and apparatus are described in U.S. Pat. No. 4,513,376 issued Apr. 23, 1985 to Thomas Barber where not only is determined the real &#34;R&#34; or in-phase of the receiver voltage component, but also the quadrature or 90° out-of-phase &#34;X&#34; component of the receiver voltage. The R and X components are used to estimate formation resistivity. 
     Only minimum processing of the receiver data is currently used with commercially available tools to enhance receiver voltage signals. Tools designed for medium radial measurement of conductivity use a single point by point boost method to account for skin-effect in the formation. Such a &#34;boosting&#34; of the receiver signal is described in U.S. Pat. No. 3,226,633 issued Dec. 28, 1965 to W. P. Schneider. 
     Tools designed for deep radial measurement of conductivity use a three depth deconvolution process followed by a skin effect boost. Such a deconvolution process is described in U.S. Pat. No. 3,166,709 issued Jan. 19, 1965 to Henri-Georges Doll. The purpose of the three point deconvolution is to reduce the tool response sensitivity to high conductivity shoulders which are far away. 
     Another approach to induction logging is described in a paper by D. W. Martin presented to an SPWLA Symposium in 1984 entitled, &#34;The Digital Induction--A New Approach to Improving the Response of the Induction Measurement&#34;. Martin describes a &#34;Digital Induction Sonde&#34; which includes four receiver coils spaced vertically above a transmitter coil on a sonde mandrel The signal induced in each of the four receiver coils is amplified, digitized and transmitted direct to the surface for computer processing before the resistivity is determined for the borehole log. 
     All the &#34;raw data&#34; from the receiver coils is recorded and stored for computer processing. Specifically, the depth of investigation can, during post job processing be varied continuously, thereby allowing an invasion profile to be produced. 
     Identification of Objects of the Invention 
     It is generally an object of the invention to provide an improved induction method and apparatus having advantages and features superior to the induction logging methods and systems of the prior art. 
     It is an object of the invention to provide a new and improved induction sonde arrangement for induction logging. 
     It is another object of the invention to provide multiple receiver arrays having different array lengths for surface multi-channel deconvolution processing at a measure depth and adjacent depths. 
     It is another object of the invention to provide induction logging apparatus in which multiple frequencies are used to excite the transmitter, thereby inducing multiple receiver signals, each with a different associated spatial response. 
     It is another object of the invention to provide an induction logging system with multiple frequency excitation in which both in-phase and quadrature components of each receiver signal are acquired. 
     It is another object of the invention to provide an accurate model for the response functions of the arrays as a function of coil placement, conductivity and frequency. 
     SUMMARY 
     The objects identified above as well as other advantages and features of the invention are incorporated in a new and improved multi-channel induction sonde having a single transmitter coil and at least one receiver coil array. A source of electrical energy is coupled to the transmitter coil. The electrical energy source includes at least two distinct oscillating signals, the frequency of each signal being substantially constant. Electrical receivers including an amplifier, A/D converter and multi-frequency phase sensitive detector are coupled to the receiver arrays for producing receiver signals at the frequencies of the oscillator source. A surface instrumentation unit digital processor combines the signals and produces an output signal representative of a characteristic of the formation. 
     In a preferred embodiment of the invention, a plurality of receiver arrays, R 1 , R 2 , R 3 , etc. are coaxially mounted with the transmitter coil. Each of the receiver coils has a primary receiver coil disposed at increasing longitudinal distances, L 1 , L 2 , L 3 , etc., from the transmitter coil. The odd numbered receiver arrays, R 1 , R 3 , R 5 , etc., are mounted on one longitudinal side of the transmitter coil, and the even numbered receiver arrays, R 2 , R 4 , R 6 , etc., are mounted on the opposite longitudinal side of the transmitter coil. Electrical receivers produce an in-phase (R) signal and an out-of-phase (X) signal for each array at each of the frequencies of the source of electrical energy. Such signals are stored for each of the measurement locations in the borehole as the induction sonde is transported through it. For each measurement location, such stored signals are combined by using a predetermined weighting matrix to produce a signal representative of the conductivity or resistivity of the formation at that measurement location. Such signals are recorded as a function of measurement depth to produce a log of the characteristic. 
     A method for obtaining a radial profile at a measurement location is also provided whereby a transmitter coil is driven with at least two oscillating currents of different frequencies. A first signal from a receiver coil at frequency f 1  and a second signal f 2  from such receiver coil are obtained and a difference signal between such signals is produced. The difference signal is representative of a characteristic of the formation at a greater radial depth than the first signal or the second signal. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The objects, advantages and features of the invention will become more apparent by reference to the drawings which are appended hereto and wherein like numerals indicate like parts and wherein an illustrative embodiment of the invention is shown, of which: 
     FIG. 1 illustrates a balanced symmetrical induction logging sonde with a single transmitter and a pair of two coil receiver arrays, where the transmitter is excited with simultaneous multiple frequencies and the received induced voltages at individual frequencies are separated into in-phase and quadrature components; 
     FIG. 2 illustrates a balanced four receiver induction design; 
     FIG. 3 illustrates individual radial response functions for three coil induction arrays with main coil spacings of 1, 2, 3, 5 and 8 feet; 
     FIGS. 4-6 illustrate composite radial response functions of the outputs of three coil arrays with coil spacings 1, 2, 3, 5 and 8 feet, optimally weighted to produce respectively a zero output inside 6 inches (FIG. 4), inside 24 inches (FIG. 5) and inside 48 inches (FIG. 6); 
     FIG. 7 illustrates composite radial response functions of the outputs of three coil arrays with coil spacings of 0.5, 1, 2, 4 and 8 feet optimally weighted to produce combined radial geometrical factors which are non-zero inside six inches, non-zero from six to twenty-four inches, and zero within twenty four inches (equivalently, non-zero outside 24 inches); 
     FIGS. 8 and 9 illustrate respectively the composite vertical response function and radial response function of a symmetrical balanced sonde with three foot arrays operating at multiple frequencies; 
     FIG. 10 illustrates a computed log of the logging apparatus according to the invention having vertical and radial characteristics of FIG. 8 and 9 showing resolution of a high conductivity one foot bed; 
     FIG. 11 illustrates a computed log from a sonde with array spacings 1, 2, and 3 feet, with each array operating at different combinations of multiple frequencies and shows the resolution of a very thin invaded bed; 
     FIG. 12 illustrates the frequency dependence of a radial response function; 
     FIG. 13 illustrates the frequency dependence of differences of radial response functions; 
     FIG. 14 illustrates a preferred embodiment of a logging sonde according to the invention; and 
     FIG. 15 schematically illustrates a system for multi-channel processing with multiple receivers showing down hole components for producing signals at each measurement location corresponding to the in-phase (R) and out-of-phase (X) components at each array for each driving frequency. 
    
    
     DESCRIPTION OF THE INVENTION 
     FIG. 1 illustrates a sonde arrangement 10 according to the invention. Simple individual arrays are preferred generally consisting of a single transmitter coil and two receiver coils 16, 14 (16&#39;, 14&#39;). The basic three-coil array includes a primary receiver coil 16 located at a distance L from the transmitter. A secondary or bucking coil 14 acts as a mutual inductance balancing coil. The secondary coil is connected in series with the primary coil, but is wound in an opposite sense to the primary coil. The placement of the secondary coil between the transmitter coil 12 and the primary receiver coil 16 is a matter of choice, but once its placement is fixed, the number of its windings may be selected so as substantially to balance or null the direct mutual coupling between the transmitter and the receiver array. If the position of the secondary coil is selected to be L/2, or half-way between the transmitter coil 12 and the receiver coil 16, it requires one-eighth as many turns as for the primary receiver coil 16. 
     Preferably the multi-channel sonde 10 is constructed of a number of these simple arrays by placing a single transmitter 12 at the center of the tool and placing pairs of receiver coils such as pairs 16,14 and 16&#39;,14&#39; on either side of it. Amplifiers 20, 20&#39; and phase-sensitive detectors 22, 22&#39; (PSD) may be constructed of conventional analog induction electronics but preferably digital PSD are used as will be discussed below with regard to a particular preferred embodiment. As illustrated, a multi-frequency oscillator 26, operating at frequencies of 20 and 40, kHz excite transmitter 12. The receiver arrays, spaced, for example, three feet from transmitter 12, respond with voltage signals, v 1 , v 1  &#39; which depend on formation characteristics. Such voltage signals are amplified and resolved into in-phase (R) and out-of-phase or quadrature (X) components at the two excitation frequencies 20 and 40 kHz. As a result eight channels of information are produced from two arrays, two oscillation frequencies, and two components of the response signal (in-phase and quadrature). 
     FIG. 2 illustrates schematically a four receiver array sonde 10&#39; including receiver arrays placed at three feet on both sides of the transmitter. Receiver arrays are also provided having their main receiver coils spaced at five feet on both sides of the transmitter. Many other receiver arrays may be placed above and/or below transmitter coil 26, depending on the preferred design of the induction equipment. 
     As the description of the deconvolution processors will show below, more information channels allow more precise determination of formation characteristics, but adding more and more receiver arrays ultimately undesirably lengthens the sonde, making it more difficult to use practically in field operations. It should also be mentioned here that more than two frequencies may be used with the logging system. Oscillators 26 may, for example, include current signals of 20, 40, 80 and 160 or 25, 50, 100 and 200 kHz. Certain frequencies may be used with certain receiver arrays. For example, the five foot receivers of FIG. 2 may be sensed at 20 and 40 kHz while the three foot receivers may be sensed at 20 and 160 kHz, etc. 
     At the heart of the invention is a method and apparatus for deconvolving or &#34;weighting&#34; in a summation the various received signal channels of sondes such as illustrated in FIGS. 1 and 2. The invention involves a multi-channel sonde having a plurality of arrays (not necessarily balanced and symmetrical as shown in FIGS. 1 and 2) with different spatial responses (e.g., the arrays are of different lengths from the transmitter coil) and uses deconvolution to combine the output of these arrays in digital processing under software control at a surface instrumentation system. 
     Deconvolution is used here as defined by Doll to mean that the output measurement (for example, conductivity or resistivity) at a particular depth is computed from a weighted combination of the sonde response at the measure depth and adjacent depths. Multi-channel deconvolution is a generalization of this induction deconvolution in that data from a number of depth points from several arrays are weighted and summed to determine the output measurement at a given depth. 
     The description below describes the design of a multi-channel deconvolution filter assuming that there is no skin effect. The technique is then extended to include skin effect. 
     Multi-channel Deconvolution Filter Assuming No Skin Effect 
     When the formation conductivity is low, the currents in the formation are small; consequently the fields in the formation are not significantly different from those which would exist in a vacuum. In other words, skin effect is small and may be ignored. In that case, each of the n arrays produces a spatial average of the formation conductivity 
     
         σ.sub.n (z)=δg.sub.n (ρ,z-z&#39;) σ(ρ,z&#39;)dρdz&#39;. (1) 
    
     The variables of equation (1) are 
     σ n  =conductivity as measured by the nth channel in the array, 
     ρ=radial distance from the center line of the borehole, 
     z&#39;=distance from the measure point z of the sonde, 
     σ (ρ,z&#39;)=the conductivity pattern of the formations adjacent the borehole, 
     g n  (ρ,z&#39;)=the response function of the nth channel in the array. 
     The response function g(ρ,z) is independent of the formation and can be calculated using Maxwell&#39;s equations in free space. The result is simply the well known Doll geometrical factor. Filters described immediately below are based on linear theory. 
     For formations which have no radial variation, the ρ integration may be done to obtain: 
     
         σ.sub.n (z)=δg.sub.n (z-z&#39;) σ(z&#39;)dz&#39;     (2) 
    
     
         g.sub.n (z)=δg.sub.n (ρ,z)dρ                 (3) 
    
     Similarly, in formations where there is no vertical variation, the z integration may be performed to obtain: 
     
         σ.sub.n =δg.sub.n (ρ) σ(ρ)dρ (4) 
    
     where 
     
         g.sub.n (ρ)=δg.sub.n (ρ,z)dz                 (5) 
    
     The relationship is known as g(ρ,z), the geometrical factor of the tool, and g(ρ) and g(z) are called radial and vertical geometrical factor respectively. The shape of the geometrical factor depends upon the placement and winding of the transmitter and receiver coils. 
     As described above, a multi-channel tool may be constructed of a number of simple arrays by placing a single transmitter at the center of the tool and placing pairs of receiver coils on one or the opposite side of it or on both sides of it. FIG. 2 illustrates the configuration for example of a possible four channel &#34;balanced&#34; sonde. The main coil spacings are three and five feet with receiver coils located above and below the transmitter. The response function for the sonde with the receiver coils below the transmitter is simply the reversal of the response function for the sonde with the receiver coils above the transmitter. That is: 
     
         g.sub.3cx (ρ,z)=g.sub.3cxR (ρ,-z).                 (6) 
    
     The radial geometrical factors of the two sondes is the same, and so having them both does not provide any additional radial information. There is, however, additional vertical and two-dimensional information. There is also the additional benefit of ease in making the response function symmetric, and possibly a benefit in improving borehole effect. 
     Deconvolution filters may be designed which are optimal in the sense that the response minimizes some error norm when compared with the desired response. 
     Additionally there are desirable characteristics of the response of an induction tool. Often these characteristics are contradictory. Some of the most common of the characteristics are: 
     Vertical Response--The vertical response g(z) should be sharp to allow the determination of resistivity of a thin bed and should be symmetric. The tails of the response should die off quickly away from the peak. A smooth response without overshoot may be desired although some log analysts seem to prefer a sharper response with overshoot. 
     Radial Response--The Radial response g(ρ) should be well localized to a known range of radial values. Often this means that the radial response should be small inside of or outside of a given radius. 
     Two-dimensional Effects--The response of the tool should not be overly sensitive to caves or to very high conductivity invaded regions, when the total integrated radial response is not sensitive to this region. 
     Noise Effects--The deconvolution process should not introduce excessive unwanted noise into the log. 
     It is impossible to achieve all of these characteristics simultaneously, and one of the primary advantages of this invention is that the trade-off between the various properties of the response may be varied and hence studied systematically. The basic equation of multi-channel deconvolution is: ##EQU1## where the subscript n refers to the channel number and σ n  (z) is the conductivity measured by the nth array at position z. The summation goes from channel 1 to channel N and from z nmin  to z nmax , the minimum and maximum depth for each channel. The a&#39;s are the filter coefficients which are to be determined. 
     This equation implies that the signal from a set of N arrays will be obtained at a number of positions and will be weighted and added to obtain a log value. The values of the filter coefficients (a&#39;s) as well as the choice of independent arrays will determine the tool response. If each of the arrays has a response which can be written in the form of Equation 1, then the total response of the multi-channel sonde can be written in the same general form, ##EQU2## where ##EQU3## 
     The vertical and radial geometrical factors can also be written in terms of the filter weights and the vertical and radial geometrical factors of the individual arrays, ##EQU4## 
     The two-dimensional geometrical factor, need not be considered, but rather only the vertical and radial geometrical factors of the individual subsondes. Equations 9, 10, and 11 produce the various response functions from the individual response functions and the filter weights. In order to determine the filter weights, the desired characteristics of the system must be expressed in terms of g(ρ,z), g(z), or g(ρ). The relative importance of each of the desired characteristics must also be set which in turn will determine how the trade-off between the conflicting criteria occur. 
     These equations can be written more compactly in matrix form ##EQU5## G is a m×n matrix composed of response functions, a is a n×1 column vector of filter weights, and d is an m×1 column vector of desired characteristics. There are m equations (corresponding to m desired characteristics), in n unknowns (corresponding to n filter weights). In general, there will be more equations than unknowns, and it will not be possible to solve them exactly but the vector a will have to be determined to minimize some error. The column vector a is all the separate filter weights in order ##EQU6## Each row of G and the corresponding value of d correspond to one of the Equations 9-11. As an example, suppose that it was desired to define a target function for the vertical geometrical factor. For instance, the vertical geometrical factor might be desired to look like a gaussian with some width 
     
         g(z)=g.sub.target (z).                                     (14) 
    
     In matrix notation this would correspond to ##EQU7## If there were only a single channel to deconvolve, this would reduce to the normal formula for deconvolution and the matrix G would have a Toeplitz form. That is 
     
         g.sub.ij =G(i-j)                                           (16) 
    
     Instead of a target function for the vertical geometrical factor, a target function for the radial geometrical factory may be desired. For instance, the radial geometrical factor could be specified to be zero for some range of radius, e.g., 
     
         g(ρ)=g.sub.target (ρ).                             (17) 
    
     This would result in a matrix equation of the form ##EQU8## 
     All the columns of G corresponding to a single array are identical. This is due to the fact that the radial geometrical factor is calculated by integrating the two-dimensional geometrical factor over z. 
     Certain aspects of the induction tool behavior are due to the two-dimensional response. Certain characteristics of the two-dimensional response may be specified. This leads to a matrix equation similar to Equation 15. ##EQU9## 
     An example of a two-dimensional condition might be to specify the response to be zero at radii of 4, 8, and 12 inches. Such a set of conditions contributes to controlling the tool&#39;s cave effect. 
     Where characteristics of the vertical, radial and two-dimensional geometrical factors are to be specified simultaneously, then the form of Equation (12) is still applicable. In this case, G and d will be the vertical catenation of the corresponding matricies presented in Equations 15, 18 and 19. 
     In general there are more design criteria than there are unknown filter weights. Consequently Equation (12) can not be solved exactly. Instead, a combination of filter weights is to be found which minimizes some norm of the error between the actual response achieved and the desired response. 
     Typical norms are often written in the form: ##EQU10## If p=2, the norm is of the common Euclidian form, while as p→∞, the norm corresponds to the Chebychev norm. Minimizing the Euclidian norm in Equation 9 corresponds to finding the least squares solution to the equation. Minimizing the Chebychev norm corresponds to finding the solution which minimizes the maximum error between the actual and the desired solution. 
     In a real situation, it is important to achieve all of the design criteria by equal amounts. This problem is handled by weighting the rows in G and d of Equation 9 by various amounts. This results in a larger contribution to the error norm (be it Euclidian or Chebychev) from those equations corresponding to conditions which are more important to approximate well and a smaller contribution from those equations corresponding to conditions which are less important. It is always necessary to make such a decision on priorities. The weight does not correspond precisely with a subjective decision on relative importance, since the magnitudes of the elements of Ga may differ greatly. In particular, the g(ρ,z) has the dimension of 1/(length **  2) while g(ρ) and g(z) have the dimension of 1/length. Thus the choice of the length scale will alter the relative importance of these equations without fundamentally changing the problem. 
     Equation 11 shows that the radial geometrical factor of a multi-channel sonde can be expressed as a sum of the raidal geometrical factors of the individual arrays. If the summation is performed over depth this dependence can be written more explicitly as ##EQU11## where ##EQU12## 
     A number of radial functions may be combined to achieve certain radial criteria. These criteria consist of the following types of conditions for the example radial focusing: 
     1. Set the radial geometrical factor to a certain value for a range of r. 
     2. Normalize the radial geometrical factor to one. 
     3. Zero the mutual induction of the array. If the sonde is built up of sondes which are not mutually balanced, then the sum Σ1/L n   3  must be zero, where L n  is the effective length of each of the sub-arrays. 
     FIGS. 3 through 6 show the results of this focusing process. In each case, the radial response of five three-coil, mutually balanced sondes with main coil spacings of 1, 2, 3, 5 and 8 feet have been combined. Each sonde includes a one turn transmitter located at z=0 and two receivers located at L and L/2. FIG. 3 shows the five basis functions. They are identical except for scaling. Each is normalized to unit area. FIGS. 4 through 6 show the best linear combination of the basis functions in order to make the radial response function zero out to some limit. This limit varies from 6 to 48 inches. For each attempt, the &#34;best&#34; combination is defined in either a least squares sense or a Chebychev sense. 
     The Chebychev norm is equivalent to minimizing the maximum deviation between the actual and the desired response. The minimax solution always reaches equal deviations from the desired solution at a multiple of points, and that these maxima appear to be larger than the least squares solution. This multiple equal maxima deviation is a property of this norm. The apparent excess of the minimax norm is not true. In every case the least squares solution reaches a higher value at the boundary of the region where it is desired to achieve a value of zero. 
     FIG. 7 shows three responses which are formed from the basis functions associated with array lengths of 0.5, 1, 2, 4, and 8 feet. In these cases, it is attempted to achieve responses which are nonzero only inside of six inches, only outside of 24 inches, and only between 6 and 24 inches. For this display, the curves are normalized to the same maximum vertical height. The software combination of these elemental sondes achieves far more separation of radial response that the currently available commercial tools. In addition, the responses are almost orthogonal, that is, they have little signal coming from the same radial part of the formation. This makes it far easier to interpret the invasion of the borehole. Using this method, a true conductivity profile can be produced by combining the separate sub-sondes in software to concentrate the response from a particular part of the formation. 
     These curves are achieved by combining radial response functions only. In a real tool it is important to achieve a balance between the radial performance and the other aspects of sonde response. In particular, the achievement of a very deep response requires a larger weighting on the longer arrays, while a shallow response requires a larger weighting on the shorter ones. Good vertical resolution is, of course, much easier with short sondes than with long ones. Thus there is the unavoidable problem of conflicting criteria. That is, it is difficult to achieve a good depth of investigation with a sonde which also has good vertical resolution. This means that the possibilities of obtaining a radial profile will be greatest for thick beds where there need not be as much concern with a sharp vertical response. 
     Multi-channel Deconvolution Filter Assuming Presence of Skin Effect 
     In high-conductivity formations, the response of an array depends not only upon its coil configuration but also upon the skin depth of the formation. This skin depth is dependent upon the product of the formation conductivity and the frequency of operation of the induction sonde. Since the frequency may be specified, and since the skin depth is very often an important length scale in the problem and is often the dominant one, the skin effect can be used to advantage in producing a localized measurement of conductivity. 
     As above, very simple sondes are used, e.g., a symmetrical sonde with a single transmitter and a pair of two-coil receiver arrays. Multiple channels of information are obtained by acquiring both in-phase and quadrature components of each array signal at several frequencies. The simple sonde described above operating at 20, 40, 100, and 200 kHz provides 16 channels, each with a different associated spatial response. 
     The inversion of this data requires an accurate model for the response functions of the arrays as a function of coil placement, conductivity and frequency. The Born approximation to the array response, which is the first term in an expansion of the response in powers of deviations in the conductivity from some average background conductivity provides the model. 
     The use of the Born approximation starts from a slightly different point of view than the deconvolution filter described above assuming low conductivity formations. If the formation may reasonably be determined to have an &#34;average&#34; or &#34;background&#34; conductivity σ, then it can be determined as to how the received voltage varies with changes in the formation conductivity. 
     This can be written in a form similar to the geometrical factor 
     
         V(z)=V(σ)+δgb(ρ,z;σ) (σ(ρ,z-z&#39;)-σ)dρdz                     (23) 
    
     Here gb (ρ,z : σ) is the Born response function or the Born kernal. The Born kernal is now complex. Following as above, two apparent conductivities are defined as σ aR  (z)=K R  v R  (z) and σ aX  (z)=K XR  v X  (z) based upon the normal resistive or upon the reactive signal. Thus these signals are now put on the same footing. The geometrical factor theory can be viewed as a special case of the more general Born approximation where the average conductivity is chosen to be zero. 
     Multi-channel deconvolution now looks just like the above analysis for low conductivity formations except for the inclusion of both R and X signals. 
     
         σ.sub.d (z)=σ+Σ.sub.n Σ.sub.z&#34; a.sub.n (z&#34;) (σ.sub.a,n (z-z&#34;)-σ.sub.a,n (σ)         (24) 
    
     The index n now runs over all arrays and also over R and X signals. 
     As an example of the use of multiple frequencies, the simple symmetric sonde shown in FIG. 1 may be considered. Each main coil spacing is 3 feet, making a total sonde length of 6 feet. Frequencies of operation are 20 and 40 kHz, yielding 8 channels (2 frequencies, in-phase and quadrature signal components, 2 arrays). The resulting response functions are shown in FIGS. 8 and 9. The vertical resolution is 5.5 inches and depth of investigation is 73 inches. There is virtually no response within a radius of 10 inches. A computed log of a layered formation (FIG. 10) shows this sonde capable of resolving a 1-ft high conductivity (low resistivity) bed. The technique works because the signals at 20 and 40 kHz for each array are weighted nearly equally and opposite in sign in the deconvolution filter. Thus, the total response is small near the borehole where skin-effect is small and is nearly frequency-independent, and numerically significant further into the formation where skin-effect results in a difference between the responses at the two frequencies. Small frequency differences result in large depths of investigation; as the frequency difference is made larger, the radial response of the sonde moves in toward the borehole axis. 
     Very thin invaded beds (where the bed thickness is on order or even less than the invasion diameter) represent a very difficult class of two-dimensional heterogeneities that often degrade the apparent resolution of a sonde as seen on a log. Additional formation information may be obtained by a sonde design using both multiple array spacings and multiple frequencies. FIG. 11 illustrates the log obtained from such a sonde with array spacings of 1, 2, and 3 feet. The one foot array pair operates at 20 and 160 kHz, the two foot sonde at 20 and 80 kHz, and the three foot array pair at 20 and 40 kHz. The one foot thick bed with a radius of invasion of 10 inches is well resolved. 
     Formation Radial Conductivity Profile From Multiple Frequency Responses 
     As discussed above, the response of a transmitter-receiver coil array is dependent on its transmitter frequency due to skin effect of the induced loop of current in the formation. At high frequencies, the loop current tends to flow in a radial band closer to the borehole. Conversely, at lower frequencies the induced loop current in the formation tends not to be so much influenced by skin effect and current flows in a radial based relatively further from the borehole than when a higher frequency is used. 
     FIG. 12 illustrates the effect of frequency on radial depth of investigation. FIG. 12 shows relative response functions for a single three coil, three foot array in a formation of 2 S/M conductivity. The three coil, three foot array is one with a single transmitter with a two coil receiver array, where the receiver array has a primary coil receiver spaced three feet away from the transmitter and having a secondary receiver coil disposed between the primary receiver coil and the transmitter coil and is wound in the opposite direction from that of the primary receiver coil. As indicated above, the secondary receiver coil is in electrical series with the primary receiver coil and the mutual coupling signal induced in the secondary receiver coil tends to balance the mutual coupling signal induced in the primary receiver coil. 
     FIG. 13 shows the relative response of the difference between several responses of FIG. 12. Specifically three different response functions are shown: the first is the difference between the 25 kHz response and the 50 kHz response; the second is the difference between the 50 kHz response and the 75 kHz response; the third is the difference between the 75 kHz response and the 100 kHz response. The first difference curve has the deepest depth of investigation, the second difference curve has not so deep a depth of investigation as the first, and the third difference curve has a more shallow depth of investigation than the other two curves. 
     Consequently, as shown above, operating a single induction coil array at multiple frequencies and taking the difference between responses at different frequencies provides a way advantageously to use &#34;skin effect&#34; to obtain a radial profile of the formation. By using the difference curves of FIG. 13, at each logging depth a plurality of resistivity or conductivity values for different radial distances from the borehole may be obtained. An inspection of FIG. 13 reveals that the depth of investigation of the &#34;difference&#34; response curves of FIG. 13 is each greater than any one of the individual frequency response curves of FIG. 12. 
     Preferred Sonde and Induction Logging System 
     FIGS. 14 and 15 illustrate respectively the preferred downhole sonde and system of the invention. The sonde 50 includes a transmitter and receiver coil array having a single transmitter coil T and nine receiver arrays R1-R9. Each receiver array includes a pair of coils including a primary receiver coil and a secondary receiver coil placed between the primary receiver coil and the transmitter coil. For example, receiver coil R1 of sonde 50 includes a primary coil spaced a distance L 1  from transmitter coil T. The receiver coil array also includes, as indicated above, a series connected, oppositely would secondary receiver coil spaced between the transmitter coil and the primary receiver coil and has a number of turns on it to substantially eliminate or balance the mutual signal induced by mutual coupling between the transmitter and the receiver array. 
     The coil array of the sonde 50 uniquely includes an interweaving pattern, such that the receiver coil array having the next largest spacing from receiver array R 1  is placed below the transmitter coil T a distance L 2  which is greater in absolute terms than the distance L 1 . Likewise, receiver array R 3  is placed above array R 1  at a distance L 3  from transmitter coil T. The distance L 3  is greater absolutely than is the distance L 2  Receiver coils R 4 , R 5 , R 6 , R 7 , R 8  and R 9  are placed above and below the transmitter coil T in the same pattern. That is, L 4  &gt;L 3  ; L 5  &gt;L 4  ; L 6  &gt;L 5  ; L 7  &gt;L 6  ; L 8  &gt;L 7  ; and L 9  &gt;L 8 . The transmitter and receiver coils are preferably disposed on a metallic support 69 according to the construction described in U.S. Pat. No. 4,651,101, assigned to the assignee of this application. Such patent is incorporated herein. 
     The coil array and electrical leads or conductors connecting them to receiver electronics section 64 or transmitter electronics 63 are subject to the drilling fluid pressure of the well. The receiver electronics 64 and transmitter electronics 63 are mechanically isolated from the drilling fluid pressure. Pressure heads 66 and 65 and pressure compensator 67 provide electrical conductor paths protected from pressure between the electronics sections 64 and 63 and the transmitter and receiver coils. Lower head 61 serves as a connector to other logging tools to be placed below sonde 50 or as an end cap for the sonde 50. Upper head 62 serves as a connector to other logging tools which may be connected to sonde 50 or to the logging cable 54. 
     FIG. 15 is a schematic illustration of the sonde 50 of FIG. 14 suspended in a borehole 52 by a multiconductor wireline cable 54. The sonde 50 communicates with a surface instrumentation system 80 which includes an uphole telemetry system 82, a signal processing computer 84 and an output device such as recorder 86. A winch and a depth signal encoder 88 are provided for transporting the sonde 50 in the borehole and providing depth information to computer 84. 
     The transmitter electronics include a master clock 90 and a four frequency transmitter 91. The output of transmitter 91 drives transmitter coil T at four separate frequencies, preferably at about 25 kHz, 50 kHz, 100 kHz and 200 kHz. The receiver electronics preferably includes a dual frequency amplifier associated with a particular receiver coil array R 1  . . . R 9 . Amplifier 101 is associated with receiver R 1  ; . . . Amplifier 109 is associated with receiver R 9 . Preferably, each amplifier is tuned so as to be responsive to at least two frequencies of the transmitter 91. The preferred frequency tuning for each of the amplifiers 101-109 is presented in Table I. 
     
                       TABLE I                                                     
______________________________________                                    
Amplifier/Receiver Array                                                  
                Spacing   Frequency(s) (kHz)                              
______________________________________                                    
101/R.sub.1     +L.sub.1  100, 200                                        
102/R.sub.2     -L.sub.2  100, 200                                        
103/R.sub.3     +L.sub.3   50, 100                                        
104/R.sub.4     -L.sub.4   50, 100                                        
105/R.sub.5     +L.sub.5  25, 50                                          
106/R.sub.6     -L.sub.6  25, 50                                          
107/R.sub.7     +L.sub.7  25, 50                                          
108/R.sub.8     -L.sub.8  25, 50                                          
109/R.sub.9     +L.sub.9  25, 50                                          
______________________________________                                    
 The output of each amplifier 101 . . . 109 is applied to an individual
 analog to digital (A/D) converter 111 . . . 119. At each three inch
 increment in borehole depth, the digital output of each A/D converter is
 applied to a digital multiple frequency phase sensitive detector 120 where
 the in-phase signal (R) and quadrature signal (X) of each receiver at each
 of the frequencies specified in Table I is detected. For example, the
 output of A/D converter 115, responsive to the receiver R.sub.5, is
 resolved into four digital signals representing the received voltage v:
 
    
     v[R 5 , 25, R] 
     v[R 5 , 25, X] 
     v[R 5 , 50, R] 
     v[R 5 , 50, X]. 
     The signal at each of the receivers is resolved into separate signals at two frequencies, each further being resolved into its in-phase (R) component and out-of-phase (X) component. Consequently, at each three inch depth increment for each receiver, four signals are captured. Such signals are applied to downhole telemetry interface circuit 121 and transmitted for storage in Surface Instrumentation System 80 computer 84 via cable 54 and uphole telemetry interface 82. Such signals are depth tagged by means of encoder 88 which provides a simultaneous depth signal to computer 14. During logging operations, such depth tagged signals are stored in computer 84 for deconvolution processing according to a filter weighted sum of all the signals stored during the logging run. Such filter weights are preferably predetermined generally according to the techniques described above in the section labeled &#34;multi-channel deconvolution filter assuming no skin effect&#34; and in particular the section labeled &#34;multi-channel deconvolution filter assuming pressure of skin effect&#34;. 
     One set of preferred filter weights for the preferred sonde of FIG. 12 is shown in Table II below. Such weights apply for the transmitter-primary receiver coil spacings shown below in Table III. 
     
         TABLE II
  OLFIN R5R R5X R4R R4X R4R R4X R3R R2R R1R R9X R8X R8X INCHES 50 KMZ 50
 KMZ 50 KMZ 50 KMZ 100 KMZ 100 KMZ 100 KMZ 200 KMZ 200 KMZ 25 KMZ 25 KMZ
 50 KMZ
   144.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0167 0.0000 0.0000 141.0000  0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0124 0.0000 0.0000 138.0000  0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0092 0.0000 0.0000
 135.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0063 0.0000 0.0000 132.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0056 0.0000 0.0000 129.0000  0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0043 0.0000 0.0000 124.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0035
 0.0000 0.0000 123.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 -0.0045  0.0000 0.0000 120.0000  0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0014  0.0000 0.0000
 117.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 -0.0144   0.0000 0.0000 114.0000  0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 -0.0304  0.0000 0.0000 111.0000  0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0480  0.0000
 0.0000 108.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 -0.0532  0.0000 0.0000 105.0000  0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0496  0.0000 0.0000 102.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0785
 0.0000 0.0000 99.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 -0.0881  0.0000 0.0000 94.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0870  0.0000 0.0000 93.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0802  0.0000
 0.0000 90.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 -0.0769  0.0000 0.0000 87.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 -0.0485  0.0000 0.0000 84.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0119  0.0000 0.0000
 81.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0247 0.0000 0.0000 78.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0252 0.0000 0.0000 75.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1245 0.0000 0.0010 72.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1703
 0.0000 0.0000 69.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.1970 0.0000 0.0000 66.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.2135 0.0000 0.0000 63.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2189 0.0000
 0.0000 60.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.2048 0.0000 0.0000 57.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.1253 0.0000 0.0000 54.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1172 0.0219 0.0180
 51.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0473 0.0165 0.0100 48.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0378 0.0130 0.0059 45.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.003  0.0123 0.0042 42.0000
 0.0108 -0.1149  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0144
  0.0127 0.0045 39.0000 0.0244 -0.1191  0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 -0.0194  0.0134 0.0115 34.0000 -0.0003  -0.1001
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0017  0.0114 0.0047
 33.0000 0.0387 - 0.0700  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0160 0.0054 0.0004 30.0000 0.0239 -0.0327  0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0355 0.0004 -0.0055  27.0000 -0.0130
 0.0092 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0418 -0.0070
 -0.0172  24.0000 -0.0160  0.0532 0.0000 0.0000 0.0000 0.0000 -0.0430
 0.0000 0.0000 0.0404 -0.0114  -0.0257  21.0000 -0.0357  0.0942 0.0000
 0.0000 0.0000 0.0000 -0.0224  0.0000 0.0000 0.0434 -0.0108  -0.0271
 16.0000 -0.0413  0.1304 0.0000 0.0000 0.0000 0.0000 -0.0294  0.0000
 0.0000 0.0790 -0.0108  -0.0305  15.0000 -0.0421  0.1544 -0.0333  -0.0932
  0.0563 -0.1419  0.0719 0.0000 0.0000 0.1125 0.0154 0.0094 12.0000
 -0.0419  0.1610 -0.0162  -0.0412  0.0555 -0.1451  0.0402 0.0000 0.0264
 0.1220 0.0232 0.0134  9.0000 -0.0598  0.1426 0.0044 -0.0512  0.0500
 -0.1180  0.0715 0.0147 0.0059 0.1247 0.0420 0.0334  4.0000 -0.0709
 0.1093 0.0019 -0.0127  0.0127 -0.0755  0.0198 0.0109 -0.0077  0.1326
 0.0426 0.0541  3.0000 -0.0560  0.0459 -0.0181  0.0257 -0.0399  -0.0312
 -0.0047  -0.0412  -0.0094  0.1319 0.0746 0.0604  0.0000 -0.0217  0.0244
 -0.0371  0.0443 -0.0912  0.0197 -0.0444  -0.0352  -0.0149  0.1007 0.0272
 -0.0232  -3.0000  0.0044 -0.0134  0.0040 0.1056 -0.0784  0.0728 -0.0347
 -0.0160  -0.0144  0.0475 0.0418 -0.0045  -4.0000  0.0103 -0.0547  0.0328
 0.1307 -0.0480  0.1045 -0.0377  -0.0024  -0.0094  0.0409 0.0474 0.0020
 -9.0000  0.0243 -0.0904  0.0499 0.1419 -0.0381  0.1222 -0.0230  0.0145
 0.0000 0.0491 0.0513 0.0115 -12.0000  0.0424 -0.1080  0.0767 0.1294
 -0.0253  0.1029 -0.0312  0.0388 0.0000 0.0458 0.0492 0.0160 -15.0000
 0.0774 -0.1104  0.0419 0.1043 0.0002 0.0713 0.0000 0.0451 0.0000 0.0439
 0.0380 0.0037 -18.0000  0.0471 -0.1133  0.0341 0.0705 -0.0180  0.0226
 0.0000 0.0144 0.0000 0.0249 0.0344 0.0005 -21.0000  0.0769 -0.1224
 0.0000 0.0334 -0.0219  -0.0239  0.0000 0.0000 0.0000 0.0135 0.0279
 0.0034 -24.0000  0.0000 0.0000 -0.0244  -0.0075  -0.0150  -0.0742
 0.0000 0.0000 0.0000 0.0044 0.0302 0.0123 -27.0000  0.0000 0.0000
 -0.0472  -0.0551  0.0028 -0.1334  0.0000 0.0000 0.0000 -0.0144  0.0347
 0.0232 -30.0000  0.0000 0.0000 -0.0809  -0.0454  0.0012 -0.1403  0.0000
 0.0000 0.0000 -0.0234  0.0545 0.0572 -33.0000  0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0222  0.0811 0.0915
 -34.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 -0.0147  0.1049 0.1268 -39.0000  0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 -0.0103  0.1341 0.1407 -42.0000  0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0040  0.1425
 0.1444 -45.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 -0.0045  0.1744 0.1496 -48.0000  0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0048  0.1734 0.1489 -51.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0039
 0.1590 0.1310 -54.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 -0.0025  0.1344 0.0424 -57.0000  0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0008  0.1144 0.0371
 -60.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0002 0.0254 -0.0274  -43.0000  0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0004 0.0204 -0.1115  -44.0000  0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0014 -0.0257
 -0.1739  -49.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0024 -0.0474  -0.2255  -72.0000  0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0024 -0.1035  -0.2434
 -75.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 -0.1234  -0.2752  -78.0000  0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 -0.1141  -0.2400  -81.0000  0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0805
 -0.1724  -84.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 -0.0541  -0.1144  -87.0000  0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0276  -0.0421
 -90.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 -0.0097  -0.0254  -93.0000  0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0019  -0.0080  -96.0000  0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0035
 -0.0087  -99.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0044 0.0054 -102.0000   0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0049 0.0109 -105.0000
   0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0132 0.0219 -108.0000   0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 0.0193 0.0319
   OLFIN R7R R7X R7X R6R R6X R6R R6X R5R R5X INCHES 25 KMZ 25 KMZ 50 KMZ
 25 KMZ 25 KMZ 50 KMZ 50 KMZ 25 KMZ 25 KMZ
   144.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 141.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 138.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 135.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 132.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 129.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 124.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 123.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 120.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 117.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 114.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 111.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 108.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 105.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 102.0000  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 99.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 94.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 93.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 90.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 87.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 84.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 81.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 78.0000 0.0259 0.0091 0.0320 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 75.0000 -0.0047  0.0045 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 72.0000 0.0023 -0.0230  -0.0233  0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 69.0000 -0.0033  -0.0482  -0.0639  0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000 66.0000 0.0168 -0.0451  -0.1048  0.0000 0.0000 0.0000
 0.0000 0.0000 0.0000 63.0000 -0.0009  -0.0764  -0.1341  0.0000 0.0000
 0.0000 0.0000 0.0000 0.0000 60.0000 0.0037 -0.0744  -0.1475  0.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 57.0000 0.0029 -0.0712  -0.1545
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 54.0000 -0.0025  -0.0881
 -0.1954  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 51.0000 -0.0454
 -0.0491  -0.1441  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 48.0000
 -0.1814  -0.0135  -0.0981  0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 45.0000 -0.1982  0.0244 -0.0412  0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 42.0000 -0.1954  0.0594 0.0138 0.0000 0.0000 0.0000 0.0000
 -0.0345  -0.0409  39.0000 -0.1447  0.0490 0.0429 0.0000 0.0000 0.0000
 0.0000 -0.0140  -0.0540  34.0000 -0.0441  0.1172 0.1124 0.0000 0.0000
 0.0000 0.0000 -0.0224  -0.1390  33.0000 -0.0202  0.1329 0.1442 0.0000
 0.0000 0.0000 0.0000 0.0342 -0.0144  30.0000 0.0558 0.1412 0.1444 0.0000
 0.0000 0.0000 0.0000 0.0344 0.0097 27.0000 0.1091 0.1380 0.1449 -0.0160
 -0.0687  0.0412 -0.0942  0.0205 0.0374 24.0000 0.1345 0.1294 0.1542
 -0.0113  -0.0549  0.0542 -0.0411  0.0351 0.0454 21.0000 0.1224 0.1148
 0.1310 -0.0117  -0.0580  0.0518 -0.0965  0.0310 0.0915 16.0000 0.0021
 0.1004 0.1057 -0.0047  -0.0441  0.0511 -0.1129  0.0378 0.1137 15.0000
 0.0854 0.0935 0.0983 -0.0076  -0.0419  0.0439 -0.1255  0.0443 0.1283
 12.0000 0.0134 0.0321 0.0298 0.0407 -0.0140  0.0232 -0.1211  0.0440
 0.1321  9.0000 0.0334 0.0701 0.0382 -0.0029  -0.0325  0.0189 -0.0941
 0.0224 0.1216  4.0000 -0.0322  0.0595 0.0143 -0.0094  -0.0151  -0.0034
 -0.0254  0.0012 0.1020  3.0000 -0.0948  0.0445 -0.0155  -0.0029  0.0034
 -0.0125  -0.0499  0.0019 0.0743  0.0000 -0.1341  0.0251 -0.0491  -0.0429
  0.0241 -0.0440  -0.0200  0.0204 0.0510 -3.0000  -0.2224  -0.0100
 -0.1054  -0.0299  0.0523 -0.0723  0.0250 0.0348 0.0270 -4.0000  0.0238
 -0.0045  -0.0027  0.0074 0.0403 -0.0509  0.0710 0.0143 -0.0004  -9.0000
 0.0441 -0.0344  -0.1224  0.0535 0.1057 -0.0143  0.1135 0.0175 -0.0232
 -12.0000  0.0458 -0.0531  -0.1553  0.0434 0.1241 0.0008 0.1474 0.0195
 -0.0325  -15.0000  0.0948 -0.0553  -0.1501  0.0442 0.1393 -0.0017
 0.1481 0.0435 -0.0441  -18.0000  0.0117 -0.0418  -0.1107  0.0712 0.1435
 -0.0217  0.1721 0.0027 -0.0504  -21.0000  0.0116 -0.0258  -0.0427
 0.0304 0.1385 -0.0412  0.1411 0.0235 -0.0411  -24.0000  0.0234 -0.0117
 -0.0492  -0.0172  0.1232 -0.1021  0.01320 0.0000 0.0000 -27.0000  0.0493
 -0.0014  -0.0227  -0.0527  0.1010 -0.1243  0.0917 0.0000 0.0000 -30.0000
  0.0044 0.0000 0.0013 -0.0526  0.0743 -0.1121  0.0487 0.0000 0.0000
 -33.0000  0.0037 0.0149 0.0200 -0.0512  0.0488 -0.0947  0.0020 0.0000
 0.0000 -34.0000  0.0094 0.0217 0.0318 -0.0433  0.0143 -0.0477  -0.0549
 0.0000 0.0000 -39.0000  0.0308 0.0234 0.0375 0.0489 -0.0251  0.0501
 -0.1178  0.0000 0.0000 -42.0000  0.0000 0.0000 0.0000 0.0351 -0.0493
 0.0441 -0.1843  0.0000 0.0000 -45.0000  0.0000 0.0000 0.0000 -0.0528
 -0.0805  -0.0093  -0.1934  0.0000 0.0000 -48.0000  0.0000 0.0000 0.0000
 -0.0005  -0.0877  0.0541 -0.1935  0.0000 0.0000 -51.0000  0.0000 0.0000
 0.0000 -0.0322  -0.0868  0.0298 -0.1815  0.0000 0.0000 -54.0000  0.0000
 0.0000 0.0000 -0.0124  -0.0295  0.0517 -0.1597  0.0000 0.0000 -57.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -60.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -43.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -44.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -49.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -72.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -75.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -78.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -81.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -84.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -87.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -90.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -93.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -96.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -99.0000
 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -102.0000
   0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 -105.0000   0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 -108.0000   0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
 0.0000 0.0000
 
    
     
                       TABLE III                                                   
______________________________________                                    
Preferred Transmitter - Primary Receiver Spacings                         
Receiver Coil                                                             
             Spacing from Transmitter                                     
______________________________________                                    
R.sub.1      L.sub.1 = 6&#34;                                                 
R.sub.2      L.sub.2 = 9&#34;                                                 
R.sub.3      L.sub.3 = 12&#34;                                                
R.sub.4      L.sub.4 = 15&#34;                                                
R.sub.5      L.sub.5 = 21&#34;                                                
R.sub.6      L.sub.6 = 27&#34;                                                
R.sub.7      L.sub.7 = 39&#34;                                                
R.sub.8      L.sub.8 = 54&#34;                                                
R.sub.9      L.sub.9 = 72&#34;                                                
______________________________________                                    
 
    
     The filter weights of Table II, stored in computer 84 are applied to all of the measurement signals at the three inch borehole measurement increments. For example, in determining the conductivity at a particular depth (corresponding to 0.00 inches in Table II), a weighted sum of the measured voltages is made for the v[R 1 , 200, R] signals at three inch measurement points above and below the particular depth, depth=0, plus the weighted sum for the v(R 2 , 200, R) signals and so on. For example, from Table II, and equation 24, ##EQU13## where σ is a prior determined average conductivity of the formation and K R  is a constant of proportionality. Note that for the preferred embodiment of the invention certain signals are not included (or in other words they are weighted with zero weights): [R 9  25, R], [R 8 , 25, R], [R 8 , 50, R], [R 7 , 50, R], [R 3 , 100, X], [R 2 , 200, X] and [R 1 , 200, X]. 
     A log of conductivity versus depth may be produced from the conductivity determination at each depth on recorder 86. 
     Various modifications and alterations in the described structures will be apparent to those skilled in the art of the foregoing description which does not depart from the spirit of the invention. For example, the tool illustrated in FIG. 14 may be placed in a measurement sub in a drilling string for measurement-while-drilling (MWD) applications. Communication with surface instrumentation may be by means other than a logging cable. A pulse monitoring system known in the MWD art may be used. For this reason, these changes are desired to be included in the appended claims. The appended claims recite the only limitation to the present invention and the descriptive manner which is employed for setting forth the embodiments and is to be interpreted as illustrative and not limitative.