Abstract:
A method is discussed of obtaining a parameter of interest of an earth formation, typically a formation resistivity or a distance to a bed boundary, in conditions where an induction tool is using having a body with finite, non-zero conductivity. The method substantially removes the effects of the conductivity of the tool from the signal received from the earth formation. A Taylor series expansion in one half of odd integer powers of time is used to represent the received signal. At least one leading term of the Taylor series expansion can be subtracted from the second signal. A filtering operation is applied to the second signal to remove the terms most dominated by pipe effects. Typical filtering operations can be a differential filtering operation or an integral filtering operation.

Description:
BACKGROUND OF THE INVENTION  
       [0001]     1. Field of the Invention  
         [0002]     The invention is related to the field of electromagnetic induction well logging for determining the resistivity of earth formations penetrated by a wellbore. More specifically, the invention relates to measuring the transient signals in an induction tool having a metallic pipe with finite, non-zero and high conductivity.  
         [0003]     2. Description of the Related Art  
         [0004]     Electromagnetic induction resistivity instruments can be used to determine the electrical conductivity of earth formations surrounding a wellbore. An electromagnetic induction well logging instrument is described, for example, in U.S. Pat. No. 5,452,761 issued to Beard et al. The instrument described in the Beard &#39;761 patent includes a transmitter coil and a plurality of receiver coils positioned at axially spaced apart locations along the instrument housing. An alternating current is passed through the transmitter coil. Voltages which are induced in the receiver coils as a result of alternating magnetic fields induced in the earth formations are then measured. The magnitude of certain phase components of the induced receiver voltages are related to the conductivity of the media surrounding the instrument.  
         [0005]     The development of deep-looking electromagnetic tools has a long history. Such tools are used to achieve a variety of different objectives. Deep looking tools attempt to measure the reservoir properties between wells at distances ranging from tens to hundreds of meters (ultra-deep scale). There are single-well and cross-well approaches, most of which are rooted in the technologies of radar/seismic wave propagation physics. This group of tools is naturally limited by, among other things, their applicability to only high resistivity formations and the power available down-hole.  
         [0006]     At the ultra-deep scale, a technology may be employed based on transient field behavior. The transient electromagnetic field method is widely used in surface geophysics. Examples of transient technology are seen, for example, in Kaufman et al., 1983, “Frequency and transient soundings”, Elsevier Science.; Sidorov et al., 1969, “Geophysical surveys with near zone transient EM.” Published by NVIGG, Saratov, Russia.; and Rabinovich et al., 1981, “Formation of an immersed vertical magnetic dipole field”:  J. Geologiya I Geofizika , N 3. Typically, voltage or current pulses that are excited in a transmitter initiate the propagation of an electromagnetic signal in the earth formation. Electric currents diffuse outwards from the transmitter into the surrounding formation. At different times, information arrives at the measurement sensor from different investigation depths. Particularly, at a sufficiently late time, the transient electromagnetic field is sensitive only to remote formation zones and does not depend on the resistivity distribution in the vicinity of the transmitter (see Kaufman et al., 1983). This transient field is especially important for logging. A comprehensive study of deep-reading transient EM in wireline measurements has been conducted and discussed by Geldmacher et al., 1997 (“Single well (deep-reading) EM system.” Report on the pre-feasibility study, Western Atlas, Inc.). Use of a symmetric logging tool using transient field measurements for formation detection is discussed in U.S. Pat. No. 5,530,359, issued to Habashy et al.  
         [0007]     U.S. Pat. No. 5,955,884, issued to Payton et al. discusses methods for measuring transient electromagnetic fields in rock formations. Electromagnetic energy is applied to the formation at selected frequencies and waveforms that maximize the radial depth of penetration of the magnetic and electric energy. Payton &#39;884 comprises at least one electromagnetic transmitter and at least one electric transmitter for applying electric energy. The transmitter ands may be either single-axis or multi-axis electromagnetic and/or electric transmitter. In one embodiment the TEM transmitters and TEM receivers are separate modules that are spaced apart and interconnected by lengths of cable, with the TEM transmitter and TEM receiver modules being separated by an interval of one meter up to 200 meters as selected. Radial depth of investigation is related to the skin depth δ={square root}{square root over (2/σμω)} which in turn is related to frequency. Lower frequency signals can increase the skin depth. Similarly, the conductivity of the surrounding material inversely affects the skin depth. As conductivity increases the depth of investigation decreases. Finite conductivity casing of the apparatus therefore can reduce the depth of investigation.  
         [0008]     Rapidly emerging measurement-while-drilling (MWD) technology introduces a new, meso-deep (3-10 meters) scale for an electromagnetic logging application related to well navigation in thick reservoirs. The major problem associated with the MWD environment is the introduction of a metal drill pipe close to the area being measured. This pipe produces a very strong response and significantly reduces the sensitivity of the measured EM field to the effects of formation resistivities and remote boundaries. The only previous solution for this problem has been to create a large spacing (up to 20 meters) between transmitter and receiver (as discussed in U.S. Pat. No. 6,188,222 B1, issued to Seydoux et al.). The sensitivity of such a tool to remote boundaries is low. Currently, Stolar Horizon, Inc. is developing drill string radar, DSR, for CBM (Coal Bed Methane) wells. DSR provides 3-D imaging within a close range of the wellbore.  
         [0009]     There is a need for a method of processing data acquired with real MWD tools having finite non-zero conductivity in transient field studies. The present invention satisfies this need.  
       SUMMARY OF THE INVENTION  
       [0010]     The present invention is a method of obtaining a parameter of interest of an earth formation using a tool having a body with finite, non-zero conductivity. The parameter of interest is typically a resistivity of the formation or a distance to a bed boundary in the formation. The method obtains a signal from the earth formation that is substantially independent of the conductivity of the tool. A first signal is produced using a transmitter on the tool. An axially separated receiver receives a second signal that results from an interaction of the first signal with the earth formation. The second signal is dependent on the conductivity of the induction tool. This second signal can be represented using a. Taylor series expansion in one half of odd integer powers of time. The t −1/2  and t −3/2  terms are typically due to the finite-conductivity pipe. The t −5/2  term is dominated by the effects of the formation. At least one leading term of the Taylor series expansion can be subtracted from the second signal to reduce the effect of the metal pipe.  
         [0011]     A filtering operation is applied to the second signal. In one mode of the invention, a differential filtering operation is employed. A typical differential filter further comprises taking a time-derivative of the product of the second signal with t 1/2 . Such filter eliminates the main term of the pipe contribution to the signal. In an alternate mode of the invention, an integral filtering operation is employed. Such an integral filtering operation can further comprise defining a first and second specified time, t, and t 2 . An average of the measured field with a weight of t 1/2  can be made over two time intervals center on the first and second specified times, both intervals having a duration Δ. A difference produces an integration filter:  
           I   1     -     I   2       ≈       1   Δ     ⁢       ∫       -   Δ     /   2       Δ   /   2       ⁢         t     1   /   2       ⁡     [         H   z   0     ⁡     (     t   -     t   1       )       -       H   z   0     ⁡     (     t   -     t   2       )         ]       ⁢           ⁢     ⅆ   t               
 
 where H z  represents the second signal. Such a filter eliminates most of the influence of the pipe.
 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0013]      FIG. 1  (Prior Art) shows an induction logging instrument as it is typically used to make measurements suitable for use with the method of the invention.  
         [0014]      FIG. 2A  shows a typical cylindrical wellbore configuration for oil exploration.  
         [0015]      FIG. 2B  shows the wellbore from  FIG. 2A  introducing a perfectly conductive pipe.  
         [0016]      FIG. 3  shows a transmitter-receiver assembly positioned near a formation boundary.  
         [0017]      FIG. 4  shows transient responses due to a remote plane boundary of the configuration of  FIG. 3 .  
         [0018]      FIG. 5  shows a cylindrical model of the borehole configuration.  
         [0019]      FIG. 6  shows transient responses due to the remote cylindrical boundary in the configuration of  FIG. 5 .  
         [0020]      FIG. 7  shows transient responses in the presence of a remote boundary and a perfectly conductive metal pipe.  
         [0021]      FIG. 8A , B show transient responses using pipes of differing conductivities.  
         [0022]      FIG. 9  shows an unfiltered transient responses of data obtained in the presence of a conductive pipe.  
         [0023]      FIG. 10  shows the data are of  FIG. 9  subjected to least squares processing.  
         [0024]      FIG. 11  shows the results of differential processing of the data of  FIG. 9 .  
         [0025]      FIG. 12  shows the results of integral processing of the data of  FIG. 9 .  
         [0026]      FIGS. 13A and 13B  show the results of integral processing for pipes of varying conductivities.  
         [0027]      FIG. 13C  shows the transient responses in a conductive benchmark.  
         [0028]      FIG. 13D  shows the results of the integral processing of the responses shown in  FIG. 13C . 
     
    
     DESCRIPTION OF PREFERRED EMBODIMENT  
       [0029]      FIG. 1  shows a schematic diagram of a drilling system  10  with a drillstring  20  carrying a drilling assembly  90  (also referred to as the bottom hole assembly, or “BHA”) conveyed in a “wellbore” or “borehole”  26  for drilling the wellbore. The drilling system  10  includes a conventional derrick  11  erected on a floor  12  which supports a rotary table  14  that is rotated by a prime mover such as an electric motor (not shown) at a desired rotational speed. The drillstring  20  includes a tubing such as a drill pipe  22  or a coiled-tubing extending downward from the surface into the borehole  26 . The drillstring  20  is pushed into the wellbore  26  when a drill pipe  22  is used as the tubing. For coiled-tubing applications, a tubing injector, such as an injector (not shown), however, is used to move the tubing from a source thereof, such as a reel (not shown), to the wellbore  26 . The drill bit  50  attached to the end of the drillstring breaks up the geological formations when it is rotated to drill the borehole  26 . If a drill pipe  22  is used, the drillstring  20  is coupled to a drawworks  30  via a Kelly joint  21 , swivel  28 , and line  29  through a pulley  23 . During drilling operations, the drawworks  30  is operated to control the weight on bit, which is an important parameter that affects the rate of penetration. The operation of the drawworks is well known in the art and is thus not described in detail herein.  
         [0030]     During drilling operations, a suitable drilling fluid  31  from a mud pit (source)  32  is circulated under pressure through a channel in the drillstring  20  by a mud pump  34 . The drilling fluid passes from the mud pump  34  into the drillstring  20  via a desurger (not shown), fluid line  28  and Kelly joint  21 . The drilling fluid  31  is discharged at the borehole bottom  51  through an opening in the drill bit  50 . The drilling fluid  31  circulates uphole through the annular space  27  between the drillstring  20  and the borehole  26  and returns to the mud pit  32  via a return line  35 . The drilling fluid acts to lubricate the drill bit  50  and to carry borehole cutting or chips away from the drill bit  50 . A sensor Si preferably placed in the line  38  provides information about the fluid flow rate. A surface torque sensor S 2  and a sensor S 3  associated with the drillstring  20  respectively provide information about the torque and rotational speed of the drillstring. Additionally, a sensor (not shown) associated with line  29  is used to provide the hook load of the drillstring  20 .  
         [0031]     In one embodiment of the invention, the drill bit  50  is rotated by only rotating the drill pipe  22 . In another embodiment of the invention, a downhole motor  55  (mud motor) is disposed in the drilling assembly  90  to rotate the drill bit  50  and the drill pipe  22  is rotated usually to supplement the rotational power, if required, and to effect changes in the drilling direction.  
         [0032]     In the preferred embodiment of  FIG. 1 , the mud motor  55  is coupled to the drill bit  50  via a drive shaft (not shown) disposed in a bearing assembly  57 . The mud motor rotates the drill bit  50  when the drilling fluid  31  passes through the mud motor  55  under pressure. The bearing assembly  57  supports the radial and axial forces of the drill bit. A stabilizer  58  coupled to the bearing assembly  57  acts as a centralizer for the lowermost portion of the mud motor assembly.  
         [0033]     In one embodiment of the invention, a drilling sensor module  59  is placed near the drill bit  50 . The drilling sensor module contains sensors, circuitry and processing software and algorithms relating to the dynamic drilling parameters. Such parameters preferably include bit bounce, stick-slip of the drilling assembly, backward rotation, torque, shocks, borehole and annulus pressure, acceleration measurements and other measurements of the drill bit condition. A suitable telemetry or communication sub  72  using, for example, two-way telemetry, is also provided as illustrated in the drilling assembly  90 . The drilling sensor module processes the sensor information and transmits it to the surface control unit  40  via the telemetry system  72 .  
         [0034]     The communication sub  72 , a power unit  78  and an MWD tool  79  are all connected in tandem with the drillstring  20 . Flex subs, for example, are used in connecting the MWD tool  79  in the drilling assembly  90 . Such subs and tools form the bottom hole drilling assembly  90  between the drillstring  20  and the drill bit  50 . The drilling assembly  90  makes various measurements including the pulsed nuclear magnetic resonance measurements while the borehole  26  is being drilled. The communication sub  72  obtains the signals and measurements and transfers the signals, using two-way telemetry, for example, to be processed on the surface. Alternatively, the signals can be processed using a downhole processor in the drilling assembly  90 .  
         [0035]     The surface control unit or processor  40  also receives signals from other downhole sensors and devices and signals from sensors S 1 -S 3  and other sensors used in the system  10  and processes such signals according to programmed instructions provided to the surface control unit  40 . The surface control unit  40  displays desired drilling parameters and other information on a display/monitor  42  utilized by an operator to control the drilling operations. The surface control unit  40  preferably includes a computer or a microprocessor-based processing system, memory for storing programs or models and data, a recorder for recording data, and other peripherals. The control unit  40  is preferably adapted to activate alarms  44  when certain unsafe or undesirable operating conditions occur.  
         [0036]      FIG. 2A  shows a typical cylindrical structure, comprising three zones substantially defined by a metal pipe  200 , adjacent transition layers  203 , and remote infinite formation  205 . Conductivities of the three zones are σ 1 , σ 2 , and σ 3 , respectively. The magnetic permeability of the entire space is μ. As illustrated, the cylindrical boundary  202  separating the metal pipe from the transition layer and the cylindrical boundary  204  separating the regions of transition layer and remote formation share a common z-axis  210 . As measured from the z-axis, the radius of boundary  202  is labeled as r md , and the radius of boundary  204  is labeled as r tl . An electromagnetic field is excited by a transmitter current loop  215  of radius, r xt , and is measured by a receiver loop  220  of radius r xr . Transmitter loop and receiver loop are separated by distance L. The amplitude and frequency of the AC transmitter current are I and ω, respectively.  
         [0037]     There is only one component E φ  of the electric field in the considered model of  FIG. 2A , and it satisfies the Maxwell&#39;s equation detailed in Eq. (1) under the conditions of Eq. (2):  
                   1   r     ⁢       ∂               ∂   r       ⁢     (     r   ⁢       ∂     E   φ         ∂   r         )       -       E   φ       r   2       +         ∂   2     ⁢     E   φ         ∂     z   2           =       k   2     ⁢     E   φ               (   1   )             
 
 where 
 
k j   2   =−iωμσ   j , j=1, 2, 3  (2) 
 
 As is well known in the art, boundary conditions require a continuity of the tangential electric field E φ , and of the tangential magnetic field, H z , at boundaries  202  and  204 . These conditions may be expressed mathematically in the form:  
                   [             [     E   φ     ]     =   0                 [       ∂     E   φ         ∂   r       ]     =   0           }     ⁢           ⁢   at   ⁢           ⁢   r     =     r   mn       ,     r   tl             (   3   )             
 
 An analytical solution for the boundary value problem of Eqs. (1) and (3), may be found using known techniques of the Fourier transform and separation of variables. The resulting expressions for an electromotive force induced in the receiver, E f =2πr xt E φ , are shown below:  
               E   f     =         ∫   0   ∞     ⁢       (       F   a     +     F   b       )     ⁢     cos   ⁡     (     λ   ⁢           ⁢   L     )       ⁢           ⁢     ⅆ   λ         +         ⅈ   ⁢           ⁢     ωμ   ⁡     (     2   ⁢   π   ⁢           ⁢     r   xt       )       ⁢     (     2   ⁢   π   ⁢           ⁢     r   xr       )         2   ⁢   π       ⁢       ∫   0   ∞     ⁢       λ     2   ⁢     p   2         ⁢     ⅇ       -     p   2       ⁢   L       ⁢       J   1     ⁡     (     λ   ⁢           ⁢     r   xt       )       ⁢       J   1     ⁡     (     λ   ⁢           ⁢     r   xr       )       ⁢           ⁢     ⅆ   λ                     (   4   )                 F   a     =           ⅈωμ   ⁡     (     2   ⁢   π   ⁢           ⁢     r   xt       )       ⁢     (     2   ⁢   π   ⁢           ⁢     r   xr       )         2   ⁢     π   2     ⁢   Det       ⁡     [                 I   1     ⁡     (       p   2     *     r   md       )           K   1     ⁡     (       p   2     *     r   md       )         ⁢         K   1     ⁡     (       p   2     *     r   tl       )           I   1     ⁡     (       p   2     *     r   tl       )         ⁢       K   1     ⁡     (       p   2     *     r   xt       )       ⁢       I   1     ⁡     (       p   2     *     r   xt       )       ⁢         α   11     ⁢     α   22           α   12     ⁢     α   21                       -         K   1     ⁡     (       p   2     *     r   tl       )           I   1     ⁡     (       p   2     *     r   tl       )           ⁢       I   1     ⁡     (       p   2     *     r   xt       )       ⁢       I   1     ⁡     (       p   2     *     r   xr       )       ⁢       α   22       α   21               ]               (   5   )                 F   b     =         ⅈ   ⁢           ⁢     ωμ   ⁡     (     2   ⁢   π   ⁢           ⁢     r   xt       )       ⁢     (     2   ⁢   π   ⁢           ⁢     r   xr       )         2   ⁢     π   2     ⁢   Det       ⁡     [                 I   1     ⁡     (       p   2     *     r   md       )           K   1     ⁡     (       p   2     *     r   md       )         ⁢         K   1     ⁡     (       p   2     *     r   tl       )           I   1     ⁡     (       p   2     *     r   tl       )         ⁢       I   1     ⁡     (       p   2     *     r   xt       )       ⁢       K   1     ⁡     (       p   2     *     r   xt       )       ⁢         α   11     ⁢     α   22           α   12     ⁢     α   21                     -     (           I   1     ⁡     (       p   2     *     r   md       )           K   1     ⁡     (       p   2     *     r   md       )         ⁢       K   1     ⁡     (       p   2     *     r   xt       )       ⁢       K   1     ⁡     (       p   2     *     r   xr       )       ⁢       α   11       α   12                   ]               (   6   )                 p   j   2     =       k   j   2     +     λ   j   2               (   7   )                 α   11     =     1   -         p   2       p   1       ⁢           I   1     ⁡     (       p   1     *     r   md       )       ⁢       I   0     ⁡     (       p   2     *     r   md       )               I   0     ⁡     (       p   1     *     r   md       )       ⁢       I   1     ⁡     (       p   2     *     r   md       )                       (   8   )                 α   12     =     1   +         p   2       p   1       ⁢           I   1     ⁡     (       p   1     *     r   md       )       ⁢       K   0     ⁡     (       p   2     *     r   md       )               I   0     ⁡     (       p   1     *     r   md       )       ⁢       K   1     ⁡     (       p   2     *     r   md       )                       (   9   )                 α   21     =     1   +         p   2       p   3       ⁢           K   1     ⁡     (       p   3     *     r   tl       )       ⁢       I   0     ⁡     (       p   2     *     r   tl       )               K   0     ⁡     (       p   3     *     r   tl       )       ⁢       I   1     ⁡     (       p   2     *     r   tl       )                       (   10   )                 α   22     =     1   -         p   2       p   3       ⁢           K   1     ⁡     (       p   3     *     r   tl       )       ⁢       K   0     ⁡     (       p   2     *     r   tl       )               K   0     ⁡     (       p   3     *     r   tl       )       ⁢       K   1     ⁡     (       p   2     *     r   tl       )                       (   11   )               Det   =     1   -             I   1     ⁡     (       p   2     *     r   md       )       ⁢       K   1     ⁡     (       p   2     *     r   tl       )               K   1     ⁡     (       p   2     *     r   md       )       ⁢       I   1     ⁡     (       p   2     *     r   tl       )           ⁢         α   11     ⁢     α   22           α   12     ⁢     α   21                     (   12   )             
 
 The second term of Eq. (4) describes the field generated by a transmitter in an infinite uniform space with conductivity, σ 2 . The term F a  in Eq. (4) describes the field reflected from the boundary  204 . The coefficient, F b , describes the field reflected from the inner boundary  202 . 
 
         [0042]      FIG. 3  shows a transmitter-receiver assembly lying next to a formation boundary  305 . Transmitter and receiver lie along a common axis  301  that is horizontally positioned in the upper half-space  302  parallel to the formation boundary and is separated by a distance d from said boundary. The upper half-space  302  has a resistivity of 50 Ohm-m and the remote formation (lower half-space)  303  has a resistivity of 2 Ohm-m.  FIG. 4  shows transient responses of the transmitter-receiver assembly operated in the presence of a remote plane boundary without the presence of a metal pipe. Responses are measured in Volts along the vertical axis and the time is shown in seconds along the horizontal axis. The transmitter-receiver space is 2 m and transient responses are shown for several distances. Responses at distances of 1, 2, 4, 6, 8, and 10 meters are shown as  401 ,  402 ,  403 ,  404 ,  405 , and  406 , respectively. The response for infinite separation distance is shown as  407 . Due to the skin-effect, at a very early times (t&lt;100 nanoseconds) there is no sensitivity to remote boundaries. In the time interval 100 nsec&lt;t&lt;100 μsec, the responses depend significantly on the distance to the boundary  305 . It is during this time interval that the time when the currents diffuse deeper into formation and reach this boundary. Later, t&gt;100 μsec, the responses once again show a lack of dependence on the distance to the boundary, because the diffusing currents have passed the boundary and are flowing predominantly in the conductive lower half-space  303 .  
         [0043]      FIG. 5  introduces a cylindrical pipe into the model of  FIG. 3 . Introducing a cylindrical pipe into the considered formation introduces  3 D effects. However, as in  FIG. 5 , one can consider a ID cylindrically layer model where the inner layer  502  represents the 50 Ohm-m inner layer  302  (of  FIG. 3 ) and the outer formation  504  has a resistivity of 2 Ohm-m. A more intense influence of the outer cylinder compared to the outer space in a plane model of  FIG. 3  is expected. Nevertheless, any approach developed for a cancellation of the pipe effect in a cylindrically layered formation will remain valid in the model with a plane boundary.  FIG. 6  shows the transient response in the presence of a remote cylindrical boundary without a metal pipe. The responses at a distance of 1, 2, 4, 6, 8, and 10 meters are shown as  601 ,  602 ,  603 ,  604 ,  605 , and  606 , respectively. The response for an infinite separation distance is shown as  607 . Comparison of the responses in  FIG. 6  to those is  FIG. 4  indicate that the resolution in a cylindrical geometry is larger than the resolution obtained from a plane model. Such resolution is, however, sufficient for boundary detection. A greater separation can be found between curves  601  through  606  than are found between curves  401  through  407 . The separation of response curves is most pronounced at earlier times in  FIG. 6 .  
         [0044]     The ability of the method of the present invention to obtaining a temporal signal substantially independent of conductivity enables for a reduced distance between transmitter and receiver. Typical prior art methods requires transmitter-receiver spacing in a range of 20-30 m in order to obtain a measurement at a radial depth of about 10 m. In the method of present invention, the same depth can be achieved with a spacing of 1-2 m. The sensitivity to remote formation volumes is substantially independent of transmitter-receiver spacing.  
         [0045]     It is possible to use the general model of  FIG. 2A  to obtain equations under more specific conditions. For example,  FIG. 2B  shows the model from  FIG. 2A  under the condition of a perfectly conductive pipe σ 1 =∞ and with σ 2 =σ 3 . When σ 2 =σ 3 , it follows from Eq. (11) that the coefficient α 22 , vanishes and, consequently F α =0 (see Eq. 5). The generalized equations above are subsequently simplified to Eqs. (13) and (14) shown below:  
                 E   f     =         ∫   0   ∞     ⁢       (     F   b     )     ⁢     cos   ⁡     (     λ   ⁢           ⁢   L     )       ⁢           ⁢     ⅆ   λ         +           ⅈωμ   ⁡     (     2   ⁢   π   ⁢           ⁢     r   xt       )       ⁢     (     2   ⁢   π   ⁢           ⁢     r   xr       )         2   ⁢   π       ⁢       ∫   0   ∞     ⁢       λ     2   ⁢     p   2         ⁢     ⅇ       -     p   2       ⁢   L       ⁢       J   1     ⁡     (     λ   ⁢           ⁢     r   xt       )       ⁢       J   1     ⁡     (     λ   ⁢           ⁢     r   xr       )       ⁢           ⁢     ⅆ   λ               ⁢     
     ⁢   where           (   13   )                 F   b     =         ⅈ   ⁢           ⁢     ωμ   ⁡     (     2   ⁢   π   ⁢           ⁢     r   xt       )       ⁢     (     2   ⁢   π   ⁢           ⁢     r   xr       )         2   ⁢     π   2         ⁡     [           I   1     ⁡     (       p   2     *     r   md       )           K   1     ⁡     (       p   2     *     r   md       )         ⁢       K   1     ⁡     (       p   2     *     r   xt       )       ⁢       I   1     ⁡     (       p   2     *     r   xr       )         ]               (   14   )             
 
 These equations describe the frequency dependence of the measured field for all frequencies of the transmitter current (for I=1 Ampere). The second term of the Eq. (13) describes the electromotive force in a uniform space with conductivity, σ 2  (primary field). The first term represents the effect of the pipe (secondary field). The transient response of this perfectly conducting pipe with a remote boundary is shown in  FIG. 7 . The time duration is measured on the horizontal axis from 10 −8  seconds to 10 −2  seconds. The response is measured in Volts. Response curves obtained distances d=1, 2, 4, 6, 8, and 10 meters are shown as  701 ,  702 ,  703 ,  704 ,  705 , and  706 . The response obtained at an infinite separation distance is shown as  707 . The separation of the curves indicates that the measured field retains a high sensitivity to remote boundaries even in the presence of a perfectly conductive pipe. Comparison of  FIGS. 6 and 7  shows that they are substantially identical. 
 
         [0047]     An expansion of the local fields can be made in the frequency domain. The electric field, E, may be expanded in the following Taylor series with respect to ½ powers of the frequency ω.  
               E   =       ∑     k   =   2       k   =   ∞       ⁢           ⁢         u     k   /   2       ⁡     (     -   ⅈω     )         k   /   2           ⁢     
     ⁢       u     3   /   2       =   0             (   15   )             
 
 The coefficient u 5/2  corresponding to the term ω 5/2  is independent of the properties of a near borehole zone, thus u 5/2 =u 5/2   0 . Rather this term is sensitive only to the conductivity distribution in the undisturbed formation. 
 
         [0049]     The magnetic field can similarly be expanded in a Taylor series (16):  
               H   =       ∑     k   =   0       k   =   ∞       ⁢           ⁢         s     k   /   2       ⁡     (       -   ⅈ     ⁢           ⁢   ω     )         k   /   2           ⁢     
     ⁢       s     1   /   2       =   0             (   16   )             
 
 The coefficient s 3/2  of the term corresponding to ω 3/2  can be shown to depend only on the properties of the background formation, in other words s 3/2 =s 3/2   0 . 
 
         [0051]     Referring to Eq. (A3.9) of the Appendix, in the frequency domain, Eq. (17), below, accounts for the finite conductivity of the pipe if the shape of the applied current is assumed to be the Dirac function, δ(t):  
               H   ⁡     (   ω   )       =         H   0     ⁡     (   ω   )       +       1       σ   c         ⁢     (         b   0         (       -   ⅈ     ⁢           ⁢   ωμ     )       1   /   2         +         (       -   ⅈ     ⁢           ⁢   ωμ     )       1   /   2       ⁢     b   1       +       (       -   ⅈ     ⁢           ⁢   ωμ     )     ⁢     b     3   /   2         +         (       -   ⅈ     ⁢           ⁢   ωμ     )       3   /   2       ⁢     b   2       +   …     )                 (   17   )             
 
 where H is the measured magnetic field in the presence of a finite conductivity pipe; H 0  is the magnetic field that would be measured if the pipe were perfectly conductive; σ c  is the conductivity of the pipe; ω=2πf, where f is the frequency; and μ is the magnetic permeability of the free space. This expansion can be transferred to the time domain to obtain an expression of the transient field response. Transformation of the Eq. (17) to the time domain using the Fourier transform yields:  
               H   ⁡     (   t   )       =         H   0     ⁡     (   t   )       +       1       σ   c         ⁢     1     π       ⁢     (         1       (     t   ⁢           ⁢   μ     )       1   /   2         ⁢     b   0       -         μ     1   /   2         2   ⁢     t     3   /   2           ⁢     b   1       +         3   ⁢     μ     3   /   2           4   ⁢     t     5   /   2           ⁢     b   2       +   …     )                 (   18   )             
 
 In the absence of a pipe, the measured electromotive force for step-function switch-on/switch-off modes, or the equivalent H z   0  component of the magnetic field for the Dirac current shape, have the following expansion  
               H   z   0     ≈           (     H   z   0     )       5   /   2         t     5   /   2         +         (     H   z   0     )       7   /   2         t     7   /   2         +   …             (   19   )             
 
 A comparison of Eq. (17) and Eq. (18) shows that the following correspondence exists between the frequency series and the time domain expansion of the field at late times (for step-function switch-on/switch-off modes): 
 
 t   −n/2           ω n/2−1 , n=1, 3, 5, . . .   (20) 
 
 The integer powers of the frequency expansions do not contribute to the late stage of the transient process. 
 
         [0056]     Terms that are on the order of ω 3/2  correspond to the terms that are on the order of t −5/2  in the transient domain. Collecting these terms leads to Eq. (21) below:  
                   (     H   z     )       3   /   2         t     5   /   2         ≈           (     H   z   0     )       3   /   2         t     5   /   2         +       1       σ   c         ⁢       3   ⁢     μ     3   /   2           4   ⁢     π         ⁢       b   2       t     5   /   2                     (   21   )             
 
 In order to obtain time domain results comparable with multi-frequency focusing (MFF), an elimination of the terms in t −1/2  and t −3/2  in Eq. (18) is practical. 
 
         [0058]     The structure of the expansion (18) enables the following general trends in the behavior of the transient response when the conductivity of the pipe is finite but large: 
        As time increases, the signal due to the pipe dominates over the signal generated by the formation. The dominance occurs because the formation response decays much faster (t −5/2 ) compared to the pipe response (t −1/2 ). Increasing the pipe contribution to the measured field results in a reduced sensitivity to remote boundaries.     The higher the conductivity of the pipe, the less the effect of the pipe on the resolution, because the terms describing the pipe signal are inversely proportional to σ 1/2 .        
 
         [0061]     The present invention removes the effects of term t −1/2  and t −3/2 . Typically, coefficient b 0  can be determined by examining the signal at a time late in the transient behavior (in the range of 10 −4  to 10 −2 . As terms of t −5/2  and higher decay at earlier times, data at all varying transmitter-receiver distances exhibit the same tail end behavior.  
         [0062]      FIG. 8A  shows the transient responses obtained in the presence of a typical conductive pipe. The conductivity is σ=1.4*10 6  S/m. Curves  801 ,  802 , and  803 , indicate responses at distances of 1, 2, and 4 meters to a remote boundary. Response curve  804  represents the response to a remote boundary at an infinite distance. Response curve  804  is nearly indistinguishable from and overlaps response curves at a distance of 6 m, 8 m, and 10 m. Increasing the conductivity of the metal pipe improves the sensitivity of the tool to a remote boundary.  FIG. 8B  shows the transient responses for a tool with an increased conductivity of the pipe, σ=1.4*10 10  S/m, taken at various distances to the remote boundary. The a of the pipe of  FIG. 8B  is 4 orders of magnitude higher than the typical conductivity of a metal pipe of  FIG. 8A . The response curves of  FIG. 8B  reflects an improvement in sensitivity. Curves  811 ,  812 ,  813 ,  814 ,  815 , and  816  represent the response to a remote boundary at a distance of 1, 2, 4, 6, 8, and 10 meters, respectively. Curve  817  represents the response to a remote boundary at an infinite distance. Comparing the curves of  FIG. 8B  to the curves of  FIG. 8A , it is possible to observe a significant increase in the spread in the magnitude of the response curves  811  through  816  in  FIG. 8B  to that of the response curves  801  through  804  in  FIG. 8A .  
         [0063]     The present invention outlines several numerical methods for correcting measurements obtained in the presence of a pipe with finite conductivity. The asymptotic Eq. (18) indicates that, within certain time intervals, the electromotive transient magnetic field may be expanded in the following Taylor series expansion:  
               (             H   z     ⁡     (     t   1     )                   H   z     ⁡     (     t   2     )               ⋮               H   z     ⁡     (     t     m   -   1       )                   H   z     ⁡     (     t   m     )             )     =       (           t   1       -   1     /   2             t   1       -   3     /   2             t   1       -   5     /   2           …         t   1     n   /   2                 t   2       -   1     /   2             t   2       -   3     /   2             t   1       -   5     /   2           …         t   2     n   /   2               ⋮       ⋮       ⋮       …       ⋮             t     m   -   1         -   1     /   2             t     m   -   1         -   3     /   2             t     m   -   1         -   5     /   2           …         t     m   -   1       n   /   2                 t   m       -   1     /   2             t   m       -   3     /   2             t   m       -   3     /   2           …         t   m     n   /   2             )     ⁢     (           s     1   /   2                 s     3   /   2                 s     5   /   2               ⋮             s       (       2   ⁢   n     -   1     )     /   2             )               (   22   )             
 
 where to t 1 , . . . , t m  is the measurement time, and s 1/2 , . . . , s n/ 2 are the expansion coefficients. 
 
         [0065]     A comparison of Eqs. (18), (19) and (22), reveals that coefficients, S 1/2  and S 3/2 , are mostly dependent on the pipe conductivity. Only the third expansion coefficient S 5/2  is directly related to the formation conductivity (see Eq. 21):  
               S     5   /   2       =         (     H   z   0     )       5   /   2       +       1       σ   c         ⁢       3   ⁢     μ     3   /   2           4   ⁢     π         ⁢     b   2                 (   23   )             
 
 Given a sufficient measurement time (m&gt;n), one can find the least squares solution for the system of Eq. (22) and preferably subtract the terms corresponding to S 1/2  from the measured field to remove effects of pipe conductivity. Numerical modeling confirms that, for all practical purposes, it is satisfactory to cancel only the S 1/2  term. Alternatively, and depending on the pipe and formation conductivity, one can calculate both the effects of S 1/2  and S 3/2  in Eq. (22) and Eq. (24):  
                 H   ~     ⁡     (   t   )       =         H   z     ⁡     (   t   )       -       S     1   /   2         t     1   /   2         -       S     3   /   2         t     3   /   2                   (   24   )             
 
 where H z  is the measured response.  FIG. 9  shows a transient behavior of data prior to processing. Reference numerals  901 ,  902 , and  903 , indicate responses at distances of 1, 2, and 4 meters to a remote boundary. Response curve  904  represents the response to a remote boundary at distances of 6 m, 8 m, 10 m, or an infinite distance. These response curves are nearly indistinguishable from each other. In  FIG. 10 , the data of  FIG. 9  are subjected to the least square processing of Eq. (22). The resolution is thereby significantly improved. Separation can be seen between response curves  1001 ,  1002  and  1003  representing response to a remote boundary at 1, 2, and 4 meters. Furthermore response curves  1004 ,  1005 , and  1006 , representing responses to a remote boundary at 6, 8, and 10 meters, are distinguishable from each other and from curve  1007 , which represents a boundary at infinite distance. 
 
         [0068]     Alternatively, Eq. (18) enables the construction of several localized filters for eliminating the pipe influence. Typically, these filters comprise small sliding windows. Filtering can be performed using differentiation or integration methods. The following filter of Eq. (25) eliminates the main term bo, of the pipe contribution:  
               H   z   dif     =       ∂     (       t     1   /   2       ⁢     H   z       )         ∂   t               (   25   )             
 
 where H z  is the measured response.  FIG. 11  shows the results of differential processing of the curves of  FIG. 9 . For time durations less than 10 −4  seconds, response curves can be distinguished from each other. Curves  1101 ,  1102 ,  1103 ,  1104 ,  1105  and  1106  represent responses to a remote boundary at a distance of 1, 2, 4, 6, 8, and 10 meters. Curve  1107  represents a response to a remote boundary at an infinite distance. Comparison with  FIG. 9  indicated an improved resolution. 
 
         [0070]     In order to filter using integration, it is practical to consider two times, t 1  and t 2 . An average of the measured field with a weight of t 1/2  can be made over two time intervals, both having a duration Δ. The first and the second intervals are centered at the points, t 1  and t 2 , respectively. Integrating over the time interval, one obtains:  
               I   1     =         1   Δ     ⁢       ∫       -   Δ     /   2       Δ   /   2       ⁢       t     1   /   2       ⁢       H   z     ⁡     (     t   -     t   1       )       ⁢     ⅆ   t           ≈         1   Δ     ⁢       ∫       -   Δ     /   2       Δ   /   2       ⁢       t     1   /   2       ⁢       H   z   0     ⁡     (     t   -     t   1       )       ⁢     ⅆ   t           +       b   0         π   ⁢           ⁢   μ   ⁢           ⁢     σ   c                       (   26   )                 I   2     =         1   Δ     ⁢       ∫       -   Δ     /   2       Δ   /   2       ⁢       t     1   /   2       ⁢       H   z     ⁡     (     t   -     t   2       )       ⁢     ⅆ   t           ≈         1   Δ     ⁢       ∫       -   Δ     /   2       Δ   /   2       ⁢       t     1   /   2       ⁢       H   z   0     ⁡     (     t   -     t   2       )       ⁢     ⅆ   t           +       b   0         π   ⁢           ⁢   μ   ⁢           ⁢     σ   c                       (   27   )             
 
 The difference of Eq. (26) and (27) produces an integration filter Eq. (28) that eliminates the influence of the pipe:  
                 I   1     -     I   2       ≈       1   Δ     ⁢       ∫       -   Δ     /   2       Δ   /   2       ⁢         t     1   /   2       ⁡     [         H   z   0     ⁡     (     t   -     t   1       )       -       H   z   0     ⁡     (     t   -     t   2       )         ]       ⁢     ⅆ   t                   (   28   )             
 
  FIG. 12  shows the results of integral processing of the data of  FIG. 9 . Pipe conductivity is 1.4*10 6  S/m. Curves  1201 ,  1202 ,  1203 ,  1204 ,  1205  and  1206  represent responses to a remote boundary at a distance of 1, 2, 4, 6, 8, and 10 meters. Curve  1207  represents a response to a remote boundary at an infinite distance. Both differential and integration processing result in a resolution improvement. Processing via integration is typically more stable. 
 
         [0073]      FIGS. 13A and 13B  show the results of integral processing for more conductive pipes.  FIG. 13A  shows a pipe conductivity equal to 3*10 7  S/m. Curves  1301 ,  1302 ,  1303 ,  1304 ,  1305  and  1306  represent responses to a remote boundary at a distance of 1, 2, 4, 6, 8, and 10 meters. Curve  1307  represents a response to a remote boundary at an infinite distance.  FIG. 13B  shows a pipe conductivity equal to 1.4*10 10  S/m. Curves  1311 ,  1312 ,  1313 ,  1314 ,  1315  and  1316  represent responses to a remote boundary at a distance of 1, 2, 4, 6, 8, and 10 meters. Curve  1317  represents a response to a remote boundary at an infinite distance. The filtered signal does not depend on the pipe conductivity at earlier times. At later times, the pipe makes a dominant contribution to the measured signal, and this is reflected in the loss of resolution.  
         [0074]      FIG. 13C  shows the response for 2-meter spacing of a more conductive formation. Curves  1321 ,  1322 , and  1323  represent responses to a remote boundary at a distance of 1, 2, and 4 meters. Response curve  1327  represents the response to a remote boundary at an infinite distance. Response curves at distances of 6 m, 8 m, and 10 m are indistinguishable from curve  1327 . These response curves are nearly indistinguishable from each other. Results shown in the figure ensure that, on the low resistivity end, the response retains the necessary sensitivity. The formation resistivity is changed to 10 Ohm-m and the shale/water resistivity to 1 Ohm-m. Comparing  FIG. 13C  to  FIG. 9 , one can observe the skin-effect in  FIG. 13C  at very early times (i.e. 10 −8  sec) and the shift of maximum sensitivity (˜3*10 −7  sec in  FIG. 9 ) to later times (˜10 −6  sec in  FIG. 13C ). Both facts can be explained due to an increase of the formation conductivity.  
         [0075]     In  FIG. 13D , results of the integral processing are presented. Curves  1331 ,  1332 ,  1333 ,  1334 ,  1335  and  1336  represent responses to a remote boundary at a distance of 1, 2, 4, 6, 8, and 10 meters. Curve  1337  represents a response to a remote boundary at an infinite distance. Comparison of  FIG. 13D  with  FIG. 12  shows a time period of maximum sensitivity of the processed data being shifted to later times. (See, for example, curves  1201  through  1204  in  FIG. 12  and curves  1331  through  1334  in  FIG. 13D ) This maximum sensitivity has the same order of magnitude as the sensitivity of the processed data in the resistive benchmark.  
         [0076]     The invention has been described above with reference to a MWD apparatus carried on a drillstring. The method of the invention can also be used on other types of MWD apparatus conveyed on a drilling tubular, and may also be used on a logging tool carried on a wireline. The last such method is of relatively minor importance since on wireline devices, it is possible to have a housing of very high conductivity so that the correction methods described herein may not be necessary. Such means of conveyance would be known to those versed in the art and are not discussed further.  
         [0077]     While the foregoing disclosure is directed to the preferred embodiments of the invention, various modifications will be apparent to those skilled in the art. It is intended that all such variations within the scope and spirit of the appended claims be embraced by the foregoing disclosure.  
         [heading-0078]     Appendix: Taylor&#39;s Frequency Series for MWD Electromagnetic Tool  
         [0079]     We intend to evaluate the asymptotic behavior of magnetic field on the surface of a metal mandrel as described in Eq. (6):  
                 H   α     ⁡     (   P   )       =         H   a   0     ⁡     (   P   )       +     β   ⁢       ∫   S     ⁢       {       H   -&gt;     ⁢             M   ⁢           ⁢   α       ⁢     h   →         }     ⁢     ⅆ   S                     (     A3   ⁢   .1     )             
 
 where H α (P) is the magnetic field measure along the direction α (α-component), P is the point of measurement, H α   0 (P) is the α-component of the measured magnetic field given a perfectly conducting mandrel, S is the surface of the tool mandrel, β=1/{square root}{square root over (−iωμσ c )}, where ω and μ are frequency and magnetic permeability, and  ma h is the magnetic field of an auxiliary magnetic dipole in a formation where the mandrel of a finite conductivity is replaced by an identical body with a perfect conductivity. The dipole is oriented along α-direction. At high conductivity, βis small. The primary and auxiliary magnetic fields, H α   0  and  Mα {right arrow over (h)}, depend only on formation parameters. The total magnetic filed, H α , depends on both formation parameters and mandrel conductivity. The dependence on mandrel conductivity, σ c , is reflected only in parameter β:  
             β   =       1     k   c       =     1         -   ⅈ     ⁢           ⁢   ω   ⁢           ⁢   μ   ⁢           ⁢     σ   c                     (     A3   ⁢   .2     )             
 
 The perturbation method applied to Eq.(A3.1) leads to the following result:  
               H   α     =       ∑     i   =   0       i   =   ∞       ⁢     H   α                 (   i   )                 (     A3   ⁢   .3     )                 H   α                 (   0   )       =     H   α   0             (       A3     ⁢   .4     )                   H   α                 (   i   )       =     β   ⁢       ∫   S     ⁢       {                     (     i   -   1     )       ⁢     H   →         ⁢             M   ⁢           ⁢   α       ⁢     h   →         }     ⁢     ⅆ   S             ⁢     
     ⁢           ⁢       i   =   1     ,   …   ⁢           ,   ∞             (       A3     ⁢   .5     )             
 
         [0082]     Let us consider the first order approximation that is proportional to the parameter β:  
                         (   1   )       ⁢     H   α     =       β   ⁢       ∫   S     ⁢         {     (   0   )       ⁢         H   ⇀       M   ⁢           ⁢   α       ⁢     h   ⇀       }     ⁢           ⁢     ⅆ   S           =     β   ⁢       ∫   S     ⁢       {           H   ⇀     0       M   ⁢           ⁢   α       ⁢     h   ⇀       }     ⁢     ⅆ   S                     (       A   ⁢   3     ⁢   .6     )             
 
 The integrand in Eq. (A3.6) does not depend on mandrel conductivity. Therefore, the integral in right-hand side, Eq. (A3.6), may be expanded in wireline-like Taylor series with respect to the frequency:  
                 ∫   S     ⁢       {           H   ⇀     0       M   ⁢           ⁢   α       ⁢     h   ⇀       }     ⁢     ⅆ   S         ≈       b   0     +       (       -   ⅈ     ⁢           ⁢   ωμ     )     ⁢     b   1       +         (     -   ⅈωμ     )       3   /   2       ⁢     b     3   /   2         +         (     -   ⅈωμ     )     2     ⁢     b   2       +   …             (       A   ⁢   3     ⁢   .7     )             
 
 In axially symmetric models, coefficients b j  have the following properties: 
        b 0  does not depend on formation parameters. It is related to so called ‘direct field’;     b 1  is linear with respect to formation conductivity. It is related to Doll&#39;s approximation;     b 3/2  depends only on background conductivity and does not depend on near borehole parameters;     b 2  includes dependence on borehole and invasion.        
 
         [0089]     Let us substitute Eq.(A3.7) into Eq.(A3.6):  
                         (   1   )       ⁢       H   α     =       1       σ   c         ⁢     (         b   0         (     -   ⅈωμ     )       1   /   2         +         (       -   ⅈ     ⁢           ⁢   ωμ     )       1   /   2       ⁢     b   1       +       (     -   ⅈωμ     )     ⁢     b     3   /   2         +         (       -   ⅈ     ⁢           ⁢   ωμ     )       3   /   2       ⁢     b   2       +   …     )                 (       A   ⁢   3     ⁢   .8     )             
 
 Eq.(A3.3), (A3.4), and (A3.8) yield:  
               H   α     ≈       H   α   0     +       1       σ   c         ⁢     (         b   0         (     -   ⅈωμ     )       1   /   2         +         (       -   ⅈ     ⁢           ⁢   ωμ     )       1   /   2       ⁢     b   1       +       (     -   ⅈωμ     )     ⁢     b     3   /   2         +         (       -   ⅈ     ⁢           ⁢   ωμ     )       3   /   2       ⁢     b   2       +   …     )                 (       A   ⁢   3     ⁢   .9     )             
 
 Collecting traditionally measured in MFF terms ˜ω 3/2 , we obtain:  
                   (     -   ⅈωμ     )       3   /   2       ⁢       (     H   α     )       3   /   2         ≈           (       -   ⅈ     ⁢           ⁢   ωμ     )       3   /   2       ⁢       (     H   α   0     )       3   /   2         +           (       -   ⅈ     ⁢           ⁢   ωμ     )       3   /   2       ⁢     b   2           σ   c                   (       A   ⁢   3     ⁢   .10     )             
 
 The first term in the right hand side, Eq.(A3.10), depends only on background formation. The presence of imperfectly conducting mandrel makes the MFF measurement dependent also on a near borehole zone parameters (second term, coefficient b 2 ) and mandrel conductivity, σ c . This dependence, obviously, disappears for a perfect conductor (σ c →∞). We should expect a small contribution from the second term since conductivity a, is very large. 
 
         [0093]     To measure the term ˜ω 3/2 , we can modify MFF transformation in such a way that contributions proportional to I/(−iωμ) 1/2  and (−iωμ) 1/2 , Eq. (A3.9), are cancelled. We also can achieve the goal by compensating the term ˜1/(−iωμ) 1/2  in the air and applying MFF to the residual signal. The latter approach is preferable because it improves the MFF stability (less number of terms needs to be compensated). Let us consider a combination of compensation in the air and MFF in more detail. It follows from Eq. (A3.9) that the response in the air, H α (σ=0), may be expressed in the following form:  
                 H   α     ⁡     (     σ   =   0     )       ≈         H   α   0     ⁡     (     σ   =   0     )       +       1       σ   c         ⁢     (       b   0         (       -   ⅈ     ⁢           ⁢   ωμ     )       1   /   2         )                 (       A   ⁢   3     ⁢   .11     )             
 
 Compensation of the term ˜b 0 , Eq.(A3.11), is critical. Physically, this term is due to strong currents on the conductor surface and its contribution (not relating to formation parameters) may be very significant. Equations (A3.9) and (A3.11) yield the following compensation scheme:  
                 H   α     -       H   α     ⁡     (     σ   =   0     )         ≈         (       -   ⅈ     ⁢           ⁢   ωμ     )     ⁢       (     H   α     )     1       +         (       -   ⅈ     ⁢           ⁢   ωμ     )       3   /   2       ⁢       (     H   α     )       3   /   2         +       1       σ   c         ⁢     (           (       -   ⅈ     ⁢           ⁢   ωμ     )       1   /   2       ⁢     b   1       +       (     -   ⅈωμ     )     ⁢     b     3   /   2         +         (     -   ⅈωμ     )       3   /   2       ⁢     b   2       +   …     )                 (       A   ⁢   3     ⁢   .12     )             
 
 Considering measurement of imaginary component of the magnetic field, we obtain:  
               Im   ⁡     [       H   α     -       H   α     ⁡     (     σ   =   0     )         ]       ≈     -     {         1       σ   c         ⁢       (     ωμ     2       )       1   /   2       ⁢     b   1       +       ωμ   ⁡     (     H   α     )       1     +         (     ωμ     2       )       3   /   2       ⁢     (         (     H   α     )       3   /   2       +       b   2         σ   c           )         }               (       A   ⁢   3     ⁢   .13     )             
 
         [0096]     Equation (A3.13) indicates that in MWD applications, two frequency terms must be cancelled as opposed to only one term in wireline. Equation, (A1.4), modified for MWD applications has the following form:  
               (           H   ⁡     (     ω   1     )                 H   ⁡     (     ω   2     )               ⋮             H   ⁡     (     ω     m   -   1       )                 H   ⁡     (     ω   m     )             )     =       (           ω   1     1   /   2             ω   1   1           ω   1     3   /   2             ω   1     5   /   2           ⋯         ω   1     n   /   2                 ω   2     1   /   2             ω   2   1           ω   2     3   /   2             ω   2     5   /   2           ⋯         ω   2     n   /   2               ⋮       ⋮       ⋮       ⋮       ⋯       ⋮             ω     m   -   1       1   /   2             ω     m   -   1     1           ω     m   -   1       3   /   2             ω     m   -   1       5   /   2           ⋯         ω     m   -   1       n   /   2                 ω   m     1   /   2             ω   m   1           ω   m     3   /   2             ω   m     5   /   2           ⋯         ω   m     n   /   2             )     ⁢     (           s     1   /   2                 s   1               s     3   /   2                 s     5   /   2               ⋮             s     n   /   2             )               (       A   ⁢   3     ⁢   .14     )             
 
 The residual signal (third term) depends on the mandrel conductivity but the examples considered in the report illustrate that this dependence is negligible due to very large conductivity of the mandrel. Similar approaches may be considered for the voltage measurements.