Abstract:
The present disclosure relates to a method to determine a formation property of a subsurface formation. A downhole logging tool having two or more antennas, at least two of the antennas having a transversely-sensitive element and an axially-sensitive element is provided. Azimuthally-sensitive measurements are obtained using the antennas of the downhole logging tool. The measurements are fitted to a Fourier series having Fourier coefficients that include channel gains, if any. A DC component, a first harmonic component, and a second harmonic component are determined from the Fourier series, a measurement type is determined using the DC component, the first harmonic component, and/or the second harmonic component, and the formation property of the subsurface formation is determined using the determined measurement type.

Description:
CROSS-REFERENCE TO OTHER APPLICATIONS 
       [0001]    This application claims, under 35 U.S.C. 119(e), priority to and the benefit of U.S. Provisional Application No. 61/307,327, filed Feb. 23, 2010. This application is, under 35 U.S.C. 120, a continuation-in-part application of co-pending U.S. application Ser. No. 12/434,888, filed May 4, 2009. 
     
    
     BACKGROUND 
       [0002]    1. Technical Field 
         [0003]    The present disclosure relates generally to the logging of subsurface formations surrounding a wellbore using a downhole logging tool, and particularly to making directional measurements for well placement and formation evaluation. 
         [0004]    2. Background Art 
         [0005]    Logging tools have long been used in wellbores to make, for example, formation evaluation measurements to infer properties of the formations surrounding the borehole and the fluids in the formations. Common logging tools include electromagnetic tools, nuclear tools, and nuclear magnetic resonance (NMR) tools, though various other tool types are also used. 
         [0006]    Early logging tools were run into a wellbore on a wireline cable, after the wellbore had been drilled. Modern versions of such wireline tools are still used extensively. However, the need for information while drilling the borehole gave rise to measurement-while-drilling (MWD) tools and logging-while-drilling (LWD) tools. MWD tools typically provide drilling parameter information such as weight on the bit, torque, temperature, pressure, direction, and inclination. LWD tools typically provide formation evaluation measurements such as resistivity, porosity, and NMR distributions. MWD and LWD tools often have components common to wireline tools (e.g., transmitting and receiving antennas), but MWD and LWD tools must be constructed to not only endure but to operate in the harsh environment of drilling. 
         [0007]    In one manner of making deep LWD electromagnetic measurements, triaxial antennas carried on a transmitter sub are used in conjunction with a directional receiver (carried on a receiver sub). The relative azimuthal angle between the sub containing the transmitters and the sub containing the receiver usually cannot be controlled when the two subs are made up. Thus, the azimuthal offset between the transmitter and receiver antennas is not normally known. Also, due to technical limitations, the triaxial antennas are not co-located. Such is the current design for tools used in the LWD environment to make deep directional electromagnetic measurements for well placement and electromagnetic look-ahead applications. 
       SUMMARY 
       [0008]    The present disclosure relates to a method to determine a formation property of a subsurface formation. A downhole logging tool having two or more antennas, at least two of the antennas having a transversely-sensitive element and an axially-sensitive element is provided. Azimuthally-sensitive measurements are obtained using the antennas of the downhole logging tool. The measurements are fitted to a Fourier series having Fourier coefficients that include channel gains, if any. A DC component, a first harmonic component, and a second harmonic component are determined from the Fourier series, a measurement type is determined using the DC component, the first harmonic component, and/or the second harmonic component, and the formation property of the subsurface formation is determined using the determined measurement type. 
         [0009]    Other aspects and advantages will become apparent from the following description and the attached claims. 
     
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         [0010]      FIG. 1  illustrates an exemplary well site system. 
           [0011]      FIG. 2  shows a prior art electromagnetic logging tool. 
           [0012]      FIG. 3  schematically shows antennas of an exemplary resistivity logging tool having co-located antennas with two saddle coil antennas and an axial coil antenna, in accordance with the present disclosure. 
           [0013]      FIG. 4  schematically shows a shield design for a co-located axial and dual transverse coil antenna, in accordance with the present disclosure. 
           [0014]      FIG. 5  is a magnetic dipole representation of an exemplary embodiment of a single transmitter, single receiver, two-antenna group, in accordance with the present disclosure. 
           [0015]      FIG. 6  is a magnetic dipole representation of an alternate exemplary embodiment of a single transmitter, single receiver, two-antenna group, in accordance with the present disclosure. 
           [0016]      FIG. 7  is a magnetic dipole representation of an alternate exemplary embodiment of a single transmitter, single receiver, two-antenna group, in accordance with the present disclosure. 
           [0017]      FIG. 8  is a magnetic dipole representation of an alternate exemplary embodiment of a single transmitter, single receiver, two-antenna group having tilted antennas, in accordance with the present disclosure. 
           [0018]      FIG. 9  is a diagram of series-wired transverse and axial coils, in accordance with the present disclosure. A double throw, double switch relay allows switching the direction of the combined coil magnetic moment azimuthally by 180 degrees. 
           [0019]      FIG. 10  is a magnetic dipole representation of an alternate exemplary embodiment of a single transmitter, single receiver, two-antenna group having tilted antennas, in accordance with the present disclosure. 
           [0020]      FIG. 11  is a magnetic dipole representation of an alternate exemplary embodiment of a single transmitter, two receiver, three-antenna group, in accordance with the present disclosure. 
           [0021]      FIG. 12  is a magnetic dipole representation of an alternate exemplary embodiment of a single transmitter, two receiver, three-antenna group, in accordance with the present disclosure. 
           [0022]      FIG. 13  is a magnetic dipole representation of an alternate exemplary embodiment of a dual transmitter, dual receiver, four-antenna group, in accordance with the present disclosure. 
           [0023]      FIG. 14  is a magnetic dipole representation of an alternate exemplary embodiment of a dual transmitter, dual receiver, four-antenna group with coils wired in series, in accordance with the present disclosure. 
           [0024]      FIG. 15  is a schematic exemplary layout for a four-antenna group configuration, in accordance with the present disclosure. 
           [0025]      FIG. 16  shows an exemplary bottomhole assembly, in accordance with the present disclosure. 
           [0026]      FIG. 17  shows tilted antenna magnetic moment orientations corresponding to the bottomhole assembly of  FIG. 16 . 
           [0027]      FIG. 18  shows an exemplary bottomhole assembly, in accordance with the present disclosure. 
           [0028]      FIG. 19  shows an alternative exemplary bottomhole assembly, in accordance with the present disclosure. 
           [0029]      FIG. 20  shows antenna magnetic moment orientations corresponding to the bottomhole assembly of  FIG. 19 . 
           [0030]      FIG. 21  shows an exemplary implementation of a measurement device in a collar close to the bit with a four-antenna group (for borehole compensation), in accordance with the present disclosure. 
           [0031]      FIG. 22  shows an alternate implementation of a shallow saddle/axial coil system with a combination of propagation formation evaluation measurements and multiple two antenna tri-axial systems with different depths of investigation for deeper dip and formation evaluation anisotropy measurements, in accordance with the present disclosure. 
           [0032]      FIG. 23  show the antenna configuration used for simulation purposes, in accordance with the present disclosure. 
           [0033]      FIG. 24  is a set of plots showing the attenuation and phase shift for a borehole compensated transverse propagation measurement with an R 1 -R 2  distance of 1 ft and TX spacing of 3 ft (Dip=0), in accordance with the present disclosure. 
           [0034]      FIG. 25  is a set of plots showing the harmonic anisotropy measurement (two coils) with 2.5 ft spacing (Dip=0), in accordance with the present disclosure. 
           [0035]      FIG. 26  shows a dual antenna induction configuration, in accordance with the present disclosure. 
           [0036]      FIG. 27  shows an exemplary electronics layout for the induction type measurement device of  FIG. 26 . 
           [0037]      FIG. 28  shows an exemplary LWD tenser resistivity tool, in accordance with the present disclosure. 
           [0038]      FIG. 29  shows an exemplary tool response to a zero azimuth formation versus dip and anisotropy, in accordance with the present disclosure. 
       
    
    
     DETAILED DESCRIPTION 
       [0039]    Some embodiments will now be described with reference to the figures. Like elements in the various figures will be referenced with like numbers for consistency. In the following description, numerous details are set forth to provide an understanding of various embodiments and/or features. However, it will be understood by those skilled in the art that some embodiments may be practiced without many of these details and that numerous variations or modifications from the described embodiments are possible. As used here, the terms “above” and “below”, “up” and “down”, “upper” and “lower”, “upwardly” and “downwardly”, and other like terms indicating relative positions above or below a given point or element are used in this description to more clearly describe certain embodiments. However, when applied to equipment and methods for use in wells that are deviated or horizontal, such terms may refer to a left to right, right to left, or diagonal relationship as appropriate. 
         [0040]      FIG. 1  illustrates a well site system in which various embodiments can be employed. The well site can be onshore or offshore. In this exemplary system, a borehole  11  is formed in subsurface formations by rotary drilling in a manner that is well known. Some embodiments can also use directional drilling, as will be described hereinafter. 
         [0041]    A drill string  12  is suspended within the borehole  11  and has a bottom hole assembly  100  which includes a drill bit  105  at its lower end. The surface system includes platform and derrick assembly  10  positioned over the borehole  11 , the assembly  10  including a rotary table  16 , kelly  17 , hook  18  and rotary swivel  19 . The drill string  12  is rotated by the rotary table  16 , energized by means not shown, which engages the kelly  17  at the upper end of the drill string. The drill string  12  is suspended from a hook  18 , attached to a traveling block (also not shown), through the kelly  17  and a rotary swivel  19  which permits rotation of the drill string relative to the hook. As is well known, a top drive system could alternatively be used. 
         [0042]    In the example of this embodiment, the surface system further includes drilling fluid or mud  26  stored in a pit  27  formed at the well site. A pump  29  delivers the drilling fluid  26  to the interior of the drill string  12  via a port in the swivel  19 , causing the drilling fluid to flow downwardly through the drill string  12  as indicated by the directional arrow  8 . The drilling fluid exits the drill string  12  via ports in the drill bit  105 , and then circulates upwardly through the annulus region between the outside of the drill string and the wall of the borehole, as indicated by the directional arrows  9 . In this well known manner, the drilling fluid lubricates the drill bit  105  and carries formation cuttings up to the surface as it is returned to the pit  27  for recirculation. 
         [0043]    The bottom hole assembly  100  of the illustrated embodiment includes a logging-while-drilling (LWD) module  120 , a measuring-while-drilling (MWD) module  130 , a roto-steerable system and motor, and drill bit  105 . 
         [0044]    The LWD module  120  is housed in a special type of drill collar, as is known in the art, and can contain one or a plurality of known types of logging tools. It will also be understood that more than one LWD and/or MWD module can be employed, e.g. as represented at  120 A. (References, throughout, to a module at the position of  120  can alternatively mean a module at the position of  120 A as well.) The LWD module includes capabilities for measuring, processing, and storing information, as well as for communicating with the surface equipment. In the present embodiment, the LWD module includes a resistivity measuring device. 
         [0045]    The MWD module  130  is also housed in a special type of drill collar, as is known in the art, and can contain one or more devices for measuring characteristics of the drill string and drill bit. The MWD tool further includes an apparatus (not shown) for generating electrical power to the downhole system. This may typically include a mud turbine generator powered by the flow of the drilling fluid, it being understood that other power and/or battery systems may be employed. In the present embodiment, the MWD module includes one or more of the following types of measuring devices: a weight-on-bit measuring device, a torque measuring device, a vibration measuring device, a shock measuring device, a stick/slip measuring device, a direction measuring device, and an inclination measuring device. 
         [0046]    An example of a tool which can be the LWD tool  120 , or can be a part of an LWD tool suite  120 A, is shown in  FIG. 2 . As seen in  FIG. 2 , upper and lower transmitting antennas, T 1  and T 2 , have upper and lower receiving antennas, R 1  and R 2 , therebetween. The antennas are formed in recesses in a modified drill collar and mounted in insulating material. The phase shift of electromagnetic energy as between the receivers provides an indication of formation resistivity at a relatively shallow depth of investigation, and the attenuation of electromagnetic energy as between the receivers provides an indication of formation resistivity at a relatively deep depth of investigation. U.S. Pat. No. 4,899,112 can be referred to for further details. In operation, attenuation-representative signals and phase-representative signals are coupled to a processor, an output of which is coupleable to a telemetry circuit. 
         [0047]    Recent electromagnetic logging tools use one or more tilted or transverse antennas, with or without axial antennas. Those antennas may be transmitters or receivers. A tilted antenna is one whose dipole moment is neither parallel nor perpendicular to the longitudinal axis of the tool. A transverse antenna is one whose dipole moment is substantially perpendicular to the longitudinal axis of the tool, and an axial antenna is one whose dipole moment is substantially parallel to the longitudinal axis of the tool. A triaxial antenna is one in which three antennas (i.e., antenna coils) are arranged to be mutually independent. That is, the dipole moment of any one of the antennas does not lie in the plane formed by the dipole moments of the other two antennas. Three orthogonal antennas, with one antenna axial and the other two transverse, is one example of a triaxial antenna. Two antennas are said to have equal angles if their dipole moment vectors intersect the tool&#39;s longitudinal axis at the same angle. For example, two tilted antennas have the same tilt angle if their dipole moment vectors, having their tails conceptually fixed to a point on the tool&#39;s longitudinal axis, lie on the surface of a right circular cone centered on the tool&#39;s longitudinal axis and having its vertex at that reference point. Transverse antennas obviously have equal angles of 90 degrees, and that is true regardless of their azimuthal orientations relative to the tool. 
         [0048]    One possible antenna design includes axial and transverse antennas. The axial antennas are wound along the axis of the logging tool and produce a radiation pattern that is equivalent to a dipole along the axis of the tool (z direction). The transverse antennas are of saddle coil design, for example, as shown in  FIG. 3 . The transverse antennas generate a radiation pattern that is equivalent to a dipole that is perpendicular to the axis of the tool (x or y direction). The saddle coil design of the transverse antennas can be made by winding wires on the cylindrical surface of the logging tool as shown in the drawing and can include either one coil or two coils wound on the opposite sides of the tool surface. 
         [0049]    The transverse antennas can be made using individual wires or the pattern of wires can be made on a flexible printed circuit using printed circuit technology with traces of copper, silver, or other conductive material. The printed circuits can then be wrapped around the tool surface/antenna recess at the desired orientation as described in U.S. Pat. No. 6,690,170, the disclosure of which is incorporated herein by reference. Multiple saddle coils can be made using this technology on the same or different printed circuits for different orientations (e.g., x, y, or any other direction). The saddle-coils can also be melt-bonded into a thermal plastic or inlayed into mechanically cut grooves. 
         [0050]    In some applications, one may wish to surround the antennas with a shield that provides mechanical protection while allowing electromagnetic radiation to be transmitted or received with minimum attenuation. The shield can be a metallic cylinder with slots cut so as to be perpendicular to the wire (or printed circuit traces) to provide for the desired electromagnetic transparency. This design is shown in Figure for a triaxial implementation, wherein triaxial refers to an axial (z directed) antenna and two transverse (x and y) antennas. Note that the antennas do not have to be collocated. Alternatively, the shield can be a non-conductive cylinder with embedded metallic sections. 
         [0051]    In some applications one may want to use the transverse and axial antennas in series (i.e., excite them at the same time). In those cases the amount of electric power going into the different antennas and the relative phases can be varied to synthesize antenna patterns in desired directions. For example, equal excitation and phase of two orthogonal antennas generates an antenna pattern equivalent to that of a 45-degree tilted antenna. Other angular orientations can be achieved with a bi-axial antenna set if each antenna is energized with a different percentage of the total current. 
         [0052]    The saddle coil design is not limited to X and Y orientations. In fact, a design with more than four saddle coils can be constructed, each saddle coil being azimuthally distributed around the circumference of the collar (tool structural member). In that case, the shield would correspond to a pattern of slots resembling stars distributed in a manner similar to the coils. For the case of six evenly distributed coils (i.e., every 60 degrees), there would be generated three transverse antennas distributed azimuthally 120 degrees apart. Similarly, with eight evenly distributed coils (i.e., every 45 degrees), there would be produced four transverse antennas, two of which may be aligned along the axes while the other two are aligned between the axes. 
         [0053]    An exemplary two antenna group configuration is shown in  FIG. 5  with T 1 X and R 1 X as transverse saddle coils and T 1 Z and R 1 Z as standard axial coils. Each of the transmitter antenna group or receiver antenna group is an XZ group. However, it is clear that the group can be a YZ or XY group. Keeping the same angle between antennas, the axial coils will provide the standard ZZ coupling measurement while the transverse coils will provide the transverse (XX) measurement. The co-located transmitter and receiver antennas can also be wired in series to become a single tilted antenna (T 1 XZ and R 1 XZ). 
         [0054]    For an LWD tool rotating in a drilling environment, the voltage measurements as a function of the tool orientation (toolface) of the single transmitter-receiver antenna pairs can be expressed as functions of the elementary couplings as follows. Note, the transverse voltage DC component is directly proportional to (XX+YY)/2. 
         [0000]    
       
         
           
             
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         [0000]    Those formulas express the relationship between the elementary couplings and the single pair voltage measurements (amplitude and phase). The formulas also contain the channel gains that include electronics gain and antenna effective areas. When the gains of the individual channels are known (measured or estimated), all elementary couplings can be solved for. Having solved for the elementary couplings, ratios of elementary couplings as described in the table below can be measured. That is, ratios of elementary coupling with enhanced sensitivity to specific formation characteristics such as anisotropy or dip can be computed. The ratios in Table 1 are shown with their gain dependence. 
         [0000]    
       
         
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Measurement 
                   
                   
               
               
                 type 
                 Definition 
                 Generation 
               
               
                   
               
             
             
               
                 Symmetrized Directional 
                 
                   
                     
                       
                         
                           
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                               0 
                             
                             + 
                             
                               
                                 
                                   V 
                                    
                                   
                                     ( 
                                     
                                       
                                         T 
                                         i 
                                         Z 
                                       
                                       , 
                                       
                                         R 
                                         j 
                                         X 
                                       
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               · 
                               
                                 
                                   GR 
                                   j 
                                   Z 
                                 
                                 / 
                                 
                                   GR 
                                   j 
                                   X 
                                 
                               
                             
                           
                         
                         · 
                         
                           
                             
                               
                                 V 
                                  
                                 
                                   ( 
                                   
                                     
                                       T 
                                       i 
                                       Z 
                                     
                                     , 
                                     
                                       R 
                                       j 
                                       Z 
                                     
                                   
                                   ) 
                                 
                               
                               0 
                             
                             + 
                             
                               
                                 
                                   V 
                                    
                                   
                                     ( 
                                     
                                       
                                         T 
                                         i 
                                         X 
                                       
                                       , 
                                       
                                         R 
                                         j 
                                         Z 
                                       
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               · 
                               
                                 
                                   GT 
                                   i 
                                   Z 
                                 
                                 / 
                                 
                                   GT 
                                   i 
                                   X 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                  
                                 
                                   ( 
                                   
                                     
                                       T 
                                       i 
                                       Z 
                                     
                                     , 
                                     
                                       R 
                                       j 
                                       Z 
                                     
                                   
                                   ) 
                                 
                               
                               0 
                             
                             - 
                             
                               
                                 
                                   V 
                                    
                                   
                                     ( 
                                     
                                       
                                         T 
                                         i 
                                         X 
                                       
                                       , 
                                       
                                         R 
                                         j 
                                         Z 
                                       
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               · 
                               
                                 
                                   GT 
                                   i 
                                   Z 
                                 
                                 / 
                                 
                                   GT 
                                   i 
                                   X 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Anti- Symmetrized Directional 
                 
                   
                     
                       
                         
                           
                             ZZ 
                             - 
                             ZX 
                           
                           
                             ZZ 
                             + 
                             ZX 
                           
                         
                         · 
                         
                           
                             ZZ 
                             - 
                             XZ 
                           
                           
                             ZZ 
                             + 
                             XZ 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               
                                 V 
                                  
                                 
                                   ( 
                                   
                                     
                                       T 
                                       i 
                                       Z 
                                     
                                     , 
                                     
                                       R 
                                       j 
                                       Z 
                                     
                                   
                                   ) 
                                 
                               
                               0 
                             
                             - 
                             
                               
                                 
                                   V 
                                    
                                   
                                     ( 
                                     
                                       
                                         T 
                                         i 
                                         Z 
                                       
                                       , 
                                       
                                         R 
                                         j 
                                         X 
                                       
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               · 
                               
                                 
                                   GR 
                                   j 
                                   Z 
                                 
                                 / 
                                 
                                   GR 
                                   j 
                                   X 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                  
                                 
                                   ( 
                                   
                                     
                                       T 
                                       i 
                                       Z 
                                     
                                     , 
                                     
                                       R 
                                       j 
                                       Z 
                                     
                                   
                                   ) 
                                 
                               
                               0 
                             
                             + 
                             
                               
                                 
                                   V 
                                    
                                   
                                     ( 
                                     
                                       
                                         T 
                                         i 
                                         Z 
                                       
                                       , 
                                       
                                         R 
                                         j 
                                         X 
                                       
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               · 
                               
                                 
                                   GR 
                                   j 
                                   Z 
                                 
                                 / 
                                 
                                   GR 
                                   j 
                                   X 
                                 
                               
                             
                           
                         
                         · 
                         
                           
                             
                               
                                 V 
                                  
                                 
                                   ( 
                                   
                                     
                                       T 
                                       i 
                                       Z 
                                     
                                     , 
                                     
                                       R 
                                       j 
                                       Z 
                                     
                                   
                                   ) 
                                 
                               
                               0 
                             
                             - 
                             
                               
                                 
                                   V 
                                    
                                   
                                     ( 
                                     
                                       
                                         T 
                                         i 
                                         X 
                                       
                                       , 
                                       
                                         R 
                                         j 
                                         Z 
                                       
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               · 
                               
                                 
                                   GT 
                                   i 
                                   Z 
                                 
                                 / 
                                 
                                   GT 
                                   i 
                                   X 
                                 
                               
                             
                           
                           
                             
                               
                                 V 
                                  
                                 
                                   ( 
                                   
                                     
                                       T 
                                       i 
                                       Z 
                                     
                                     , 
                                     
                                       R 
                                       j 
                                       Z 
                                     
                                   
                                   ) 
                                 
                               
                               0 
                             
                             + 
                             
                               
                                 
                                   V 
                                    
                                   
                                     ( 
                                     
                                       
                                         T 
                                         i 
                                         X 
                                       
                                       , 
                                       
                                         R 
                                         j 
                                         Z 
                                       
                                     
                                     ) 
                                   
                                 
                                 1 
                               
                               · 
                               
                                 
                                   GT 
                                   i 
                                   Z 
                                 
                                 / 
                                 
                                   GT 
                                   i 
                                   X 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Resistivity Harmonic 
                 
                   
                     
                       
                         
                           2 
                            
                           
                               
                           
                            
                           ZZ 
                         
                         
                           XX 
                           + 
                           YY 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               V 
                                
                               
                                 ( 
                                 
                                   
                                     T 
                                     i 
                                     Z 
                                   
                                   , 
                                   
                                     R 
                                     j 
                                     Z 
                                   
                                 
                                 ) 
                               
                             
                             0 
                           
                           · 
                           
                             GT 
                             i 
                             X 
                           
                           · 
                           
                             GR 
                             j 
                             X 
                           
                         
                         
                           
                             
                               V 
                                
                               
                                 ( 
                                 
                                   
                                     T 
                                     i 
                                     X 
                                   
                                   , 
                                   
                                     R 
                                     j 
                                     X 
                                   
                                 
                                 ) 
                               
                             
                             0 
                           
                           · 
                           
                             GT 
                             i 
                             Z 
                           
                           · 
                           
                             GR 
                             j 
                             Z 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Resistivity Anisotropy 
                 
                   
                     
                       
                         XX 
                         YY 
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               V 
                                
                               
                                 ( 
                                 
                                   
                                     T 
                                     i 
                                     X 
                                   
                                   , 
                                   
                                     R 
                                     j 
                                     X 
                                   
                                 
                                 ) 
                               
                             
                             0 
                           
                           + 
                           
                             V 
                              
                             
                               
                                 ( 
                                 
                                   
                                     T 
                                     i 
                                     X 
                                   
                                   , 
                                   
                                     R 
                                     j 
                                     X 
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                         
                           
                             
                               V 
                                
                               
                                 ( 
                                 
                                   
                                     T 
                                     i 
                                     X 
                                   
                                   , 
                                   
                                     R 
                                     j 
                                     X 
                                   
                                 
                                 ) 
                               
                             
                             0 
                           
                           - 
                           
                             
                               V 
                                
                               
                                 ( 
                                 
                                   
                                     T 
                                     i 
                                     X 
                                   
                                   , 
                                   
                                     R 
                                     j 
                                     X 
                                   
                                 
                                 ) 
                               
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
         [0055]    For the above measurement types, only the resistivity anisotropy channel does not require gain ratio measurements between the transverse and axial antennas. All other measurement types require the transmitter and receiver ratio measurements. To measure these ratios, at least one of three options may be considered. A third single test loop antenna tilted at 45 degrees may be added to the dual antenna. To measure the receiver ratio, each channel is processed using this test loop as a “mini” transmitter. For the transmitter ratio, this test loop acts as a “mini” receiver. The test loop procedure can be either performed at the time of the signal voltage measurements or as a calibration procedure before the LWD tool embodiment is used downhole. 
         [0056]    Alternatively, a calibration signal that characterizes both receiver channels, but not the variations of moment direction (geometry) due to the antenna, may be used. Those variations need to be measured and need to be negligible with respect to the measurement specifications. The current passing through the TX antennas can be measured using a common electronic circuit, but, again, antenna variation needs to be characterized. 
         [0057]    A third option is to use an external tilted test loop as a standard that can be used to excite a tool response and can also be modeled. The ratio of the calculated measured response to the tilted test loop can be used as a gain correction. See, for example, U.S. Pat. No. 7,414,391, the disclosure of which is incorporated herein by reference. 
         [0058]    A further exemplary embodiment of a two antenna group configuration is shown in  FIG. 6  with T 1 X, T 1 Y, R 1 X, and R 1 Y as transverse saddle coils and T 1 Z and R 1 Z as standard axial coils. Keeping the same angle between antennas, the axial coils will provide the standard ZZ coupling measurement, while the transverse coils will provide the transverse (XX) and (YY) measurements. In this embodiment, the elementary couplings can be measured directly, without the need for tool rotation, Such a system is able to provide measurements while sliding. The ability to provide such measurements while sliding is important when positive displacement motors are used. In this embodiment, the gain of the individual channels is preferably measured or estimated though calibration using a combination of standard techniques such as test loops, temperature characterization, and electronics channel calibrations. 
         [0059]    A single transmitter-receiver group is not limited to an XZ antenna configuration. A configuration such as that shown in  FIG. 7  can also be used. This embodiment includes a single transmitter, single receiver configuration with X,Z transmitter coils, X,Y transverse receiver coils, and a Z receiver coil. On the right, the transmitter group is shown oriented azimuthally with respect to the receiver group by an alignment angle α. In the case of each antenna group residing on separate (modular) LWD collars, one group may be oriented (azimuthally) differently with respect to the other group determined by an alignment angle. In this specific case the dependence of the measured complex voltage with respect to the elementary couplings are as follows. 
         [0000]    
       
         
           
             
               V 
                
               
                 ( 
                 
                   
                     T 
                     i 
                     Z 
                   
                   , 
                   
                     R 
                     j 
                     Z 
                   
                 
                 ) 
               
             
             = 
             
               
                 
                   GT 
                   i 
                   Z 
                 
                 · 
                 
                   GR 
                   j 
                   Z 
                 
                 · 
                 
                   ZZ 
                   ij 
                 
               
               = 
               
                 
                   V 
                    
                   
                     ( 
                     
                       
                         T 
                         i 
                         Z 
                       
                       , 
                       
                         R 
                         j 
                         Z 
                       
                     
                     ) 
                   
                 
                 0 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     V 
                      
                     
                       ( 
                       
                         
                           T 
                           i 
                           
                             X 
                             ′ 
                           
                         
                         , 
                         
                           R 
                           j 
                           X 
                         
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       GT 
                       i 
                       X 
                     
                     · 
                     
                       GR 
                       j 
                       X 
                     
                     · 
                     
                       ( 
                       
                         
                           
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   · 
                                   
                                     
                                       
                                         XX 
                                         ij 
                                       
                                       + 
                                       
                                         YY 
                                         ij 
                                       
                                     
                                     2 
                                   
                                 
                               
                               - 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   α 
                                   · 
                                   
                                     
                                       
                                         XY 
                                         ij 
                                       
                                       + 
                                       
                                         YX 
                                         ij 
                                       
                                     
                                     2 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         XX 
                                         ij 
                                       
                                       - 
                                       
                                         YY 
                                         ij 
                                       
                                     
                                     2 
                                   
                                   · 
                                   cos 
                                 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 ϕ 
                               
                               + 
                               
                                 
                                   
                                     
                                       XY 
                                       ij 
                                     
                                     + 
                                     
                                       YX 
                                       ij 
                                     
                                   
                                   2 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 ϕ 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         V 
                          
                         
                           ( 
                           
                             
                               T 
                               i 
                               X 
                             
                             , 
                             
                               R 
                               j 
                               X 
                             
                           
                           ) 
                         
                       
                       0 
                     
                     + 
                     
                       
                         
                           V 
                            
                           
                             ( 
                             
                               
                                 T 
                                 i 
                                 X 
                               
                               , 
                               
                                 R 
                                 j 
                                 X 
                               
                             
                             ) 
                           
                         
                         2 
                       
                       · 
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               ϕ 
                             
                             + 
                             
                               ϕ 
                               0 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     V 
                      
                     
                       ( 
                       
                         
                           T 
                           i 
                           Z 
                         
                         , 
                         
                           R 
                           j 
                           X 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       GT 
                       i 
                       Z 
                     
                     · 
                     
                       GR 
                       j 
                       X 
                     
                     · 
                     
                       ( 
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             · 
                             
                               ZX 
                               ij 
                             
                           
                         
                         + 
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             · 
                             
                               ZY 
                               ij 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         V 
                          
                         
                           ( 
                           
                             
                               T 
                               i 
                               Z 
                             
                             , 
                             
                               R 
                               j 
                               X 
                             
                           
                           ) 
                         
                       
                       1 
                     
                     · 
                     
                       cos 
                        
                       
                         ( 
                         
                           ϕ 
                           + 
                           
                             ϕ 
                             0 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     V 
                      
                     
                       ( 
                       
                         
                           T 
                           i 
                           X 
                         
                         , 
                         
                           R 
                           j 
                           Z 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       GT 
                       i 
                       X 
                     
                     · 
                     
                       GR 
                       j 
                       Z 
                     
                     · 
                     
                       ( 
                       
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             · 
                             
                               XZ 
                               ij 
                             
                           
                         
                         + 
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ϕ 
                             · 
                             
                               YZ 
                               ij 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         V 
                          
                         
                           ( 
                           
                             
                               T 
                               i 
                               X 
                             
                             , 
                             
                               R 
                               j 
                               Z 
                             
                           
                           ) 
                         
                       
                       1 
                     
                     · 
                     
                       cos 
                        
                       
                         ( 
                         
                           ϕ 
                           + 
                           
                             ϕ 
                             0 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               where 
                
               
                   
               
                
               
                 GT 
                 
                   i 
                   , 
                   j 
                 
                 
                   X 
                   , 
                   Z 
                 
               
                
               
                   
               
                
               and 
                
               
                   
               
                
               
                 GR 
                 
                   i 
                   , 
                   j 
                 
                 
                   X 
                   , 
                   Z 
                 
               
                
               
                   
               
                
               are 
                
               
                   
               
                
               the 
                
               
                   
               
                
               channel 
                
               
                   
               
                
               complex 
                
               
                   
               
                
               gains 
                
               
                   
               
                
               for 
                
               
                   
               
                
               transmitter 
                
               
                   
               
                
               and 
                
               
                   
               
                
               receiver 
             
             , 
             
                 
             
              
             
               
 
             
              
             
               ϕ 
                
               
                   
               
                
               is 
                
               
                   
               
                
               the 
                
               
                   
               
                
               tool 
                
               
                   
               
                
               orientation 
             
             , 
             
               
 
             
              
             
               
                 ϕ 
                 0 
               
                
               
                   
               
                
               is 
                
               
                   
               
                
               the 
                
               
                   
               
                
               directional 
                
               
                   
               
                
               angle 
             
             , 
             and 
           
         
       
       
         
           
             
               V 
                
               
                 
                   ( 
                   
                     
                       T 
                       i 
                       
                         Z 
                         , 
                         X 
                       
                     
                     , 
                     
                       R 
                       j 
                       
                         Z 
                         , 
                         X 
                       
                     
                   
                   ) 
                 
                 
                   0 
                   , 
                   1 
                   , 
                   2 
                 
               
                
               
                   
               
                
               is 
                
               
                   
               
                
               the 
                
               
                   
               
                
               DC 
             
             , 
             
               first 
                
               
                   
               
                
               harmonic 
             
             , 
             
               and 
                
               
                   
               
                
               second 
                
               
                   
               
                
               harmonic 
                
               
                   
               
                
               component 
                
               
                   
               
                
               from 
                
               
                   
               
                
               
                 fitting 
                 . 
               
             
           
         
       
     
         [0000]    In this case, the rotation of the tool can be used to extract the alignment angle. If there is no rotation, the alignment angle will need to be measured. 
         [0060]    The transmitter and receiver dual antennas can also be used in series and act as a tilted antenna at any direction, as shown in  FIG. 8 . A single transmitter and receiver group can have their antennas combined (hardware or software) in series to yield any desired direction. The combination can also extend to a 3 coil system (XYZ) where YZ or XY could also be combined in series. A hardware implementation for a 45 degree tilted antenna would require that both X and Z magnetic moments be of same amplitude (i.e., the ratio of number of turns between antennas needs to be determined to achieve a desired orientation). Also, the receiver dual antennas can be wired in reverse series to yield a tilted moment with 180 degree azimuthal rotation, as represented by R 1 XZ down  (see  FIGS. 8 and 9 ). The received voltages can be expressed as follows for tilted antennas in series. 
         [0000]    
       
         
           
             
               V 
                
               
                 ( 
                 
                   
                     T 
                     i 
                     XZ 
                   
                   , 
                   
                     R 
                     j 
                     XZ 
                   
                 
                 ) 
               
             
             = 
             
               
                 GT 
                 i 
                 XZ 
               
               · 
               
                 GR 
                 j 
                 XZ 
               
               · 
               
                 ( 
                 
                   
                     
                       ZZ 
                       ij 
                     
                     + 
                     
                       
                         
                           XX 
                           ij 
                         
                         + 
                         
                           YY 
                           ij 
                         
                       
                       2 
                     
                     + 
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         ϕ 
                         · 
                         
                           ( 
                           
                             
                               ZX 
                               ij 
                             
                             + 
                             
                               XZ 
                               ij 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         ϕ 
                         · 
                         
                           ( 
                           
                             
                               ZY 
                               ij 
                             
                             + 
                             
                               YZ 
                               ij 
                             
                           
                           ) 
                         
                       
                     
                     + 
                   
                    
                   
                     
                       
                         
                           
                             
                               XX 
                               ij 
                             
                             + 
                             
                               YY 
                               ij 
                             
                           
                           2 
                         
                         · 
                         cos 
                       
                        
                       
                           
                       
                        
                       2 
                        
                       
                           
                       
                        
                       ϕ 
                     
                     + 
                     
                       
                         
                           
                             XY 
                             ij 
                           
                           + 
                           
                             YX 
                             ij 
                           
                         
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       2 
                        
                       
                           
                       
                        
                       ϕ 
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               V 
                
               
                 ( 
                 
                   
                     T 
                     i 
                     XZ 
                   
                   , 
                   
                     R 
                     j 
                     
                       XZ 
                       down 
                     
                   
                 
                 ) 
               
             
             = 
             
               
                 GT 
                 i 
                 XZ 
               
               · 
               
                 GR 
                 j 
                 XZ 
               
               · 
               
                 ( 
                 
                   
                     
                       ZZ 
                       ij 
                     
                     - 
                     
                       
                         
                           XX 
                           ij 
                         
                         + 
                         
                           YY 
                           ij 
                         
                       
                       2 
                     
                     + 
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         ϕ 
                         · 
                         
                           ( 
                           
                             
                               ZX 
                               ij 
                             
                             - 
                             
                               XZ 
                               ij 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         ϕ 
                         · 
                         
                           ( 
                           
                             
                               ZY 
                               
                                 ij 
                                 - 
                               
                             
                             + 
                             
                               YZ 
                               ij 
                             
                           
                           ) 
                         
                       
                     
                     - 
                   
                    
                   
                     
                       
                         
                           
                             
                               XX 
                               ij 
                             
                             - 
                             
                               YY 
                               ij 
                             
                           
                           2 
                         
                         · 
                         cos 
                       
                        
                       
                           
                       
                        
                       2 
                        
                       
                           
                       
                        
                       ϕ 
                     
                     - 
                     
                       
                         
                           
                             XY 
                             ij 
                           
                           + 
                           
                             YX 
                             ij 
                           
                         
                         2 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       2 
                        
                       
                           
                       
                        
                       ϕ 
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             where 
              
             
                 
             
              
             
               GT 
               i 
               XZ 
             
              
             
                 
             
              
             is 
              
             
                 
             
              
             the 
              
             
                 
             
              
             channel 
              
             
                 
             
              
             complex 
              
             
                 
             
              
             gain 
              
             
                 
             
              
             and 
              
             
                 
             
              
             ϕ 
              
             
                 
             
              
             is 
              
             
                 
             
              
             the 
              
             
                 
             
              
             tool 
              
             
                 
             
              
             orientation 
           
         
       
     
         [0061]    With this antenna pair combination, all couplings are available (rotation is required) and will not require any calibrations. The formulas for the measurement generation are shown in the Table 2. As a note, the channel gains for TX and RCV do not include any possible variation related to the direction of the antenna magnetic moment due to temperature and pressure. A characterization is needed to validate these effect, but these effects are expected to be negligible. 
         [0000]                            TABLE 2               Measurement               type   Definition   Generation                   Symmetrized Directional               ZZ   -   ZX       ZZ   +   ZX       ·       ZZ   +   XZ       ZZ   -   XZ                                           V        (       T   i   XZ     ,     R   j   XZ       )       0     +     V        (       T   i   XZ     ,     R   j     XZ   down         )       -     (         V        (       T   i   XZ     ,     R   j   XZ       )       1     +       V        (       T   i   XZ     ,     R   j     XZ   down         )       1       )             V        (       T   i   XZ     ,     R   j   XZ       )       0     +     V        (       T   i   XZ     ,     R   j     XZ   down         )       +     (         V        (       T   i   XZ     ,     R   j   XZ       )       1     +       V        (       T   i   XZ     ,     R   j     XZ   down         )       1       )         ·                     V        (       T   i   XZ     ,     R   j   XZ       )       0     +     V        (       T   i   XZ     ,     R   j     XZ   down         )       +     (         V        (       T   i   XZ     ,     R   j   XZ       )       1     -       V        (       T   i   XZ     ,     R   j     XZ   down         )       1       )             V        (       T   i   XZ     ,     R   j   XZ       )       0     +     V        (       T   i   XZ     ,     R   j     XZ   down         )       -     (         V        (       T   i   XZ     ,     R   j   XZ       )       1     -       V        (       T   i   XZ     ,     R   j     XZ   down         )       1       )                                   Anti- Symmetrized Directional               ZZ   -   ZX       ZZ   +   ZX       ·       ZZ   -   XZ       ZZ   +   XZ                                           V        (       T   i   XZ     ,     R   j   XZ       )       0     +     V        (       T   i   XZ     ,     R   j     XZ   down         )       -     (         V        (       T   i   XZ     ,     R   j   XZ       )       1     +       V        (       T   i   XZ     ,     R   j     XZ   down         )       1       )             V        (       T   i   XZ     ,     R   j   XZ       )       0     +     V        (       T   i   XZ     ,     R   j     XZ   down         )       +     (         V        (       T   i   XZ     ,     R   j   XZ       )       1     +       V        (       T   i   XZ     ,     R   j     XZ   down         )       1       )         ·                     V        (       T   i   XZ     ,     R   j   XZ       )       0     +     V        (       T   i   XZ     ,     R   j     XZ   down         )       -     (         V        (       T   i   XZ     ,     R   j   XZ       )       1     -       V        (       T   i   XZ     ,     R   j     XZ   down         )       1       )             V        (       T   i   XZ     ,     R   j   XZ       )       0     +     V        (       T   i   XZ     ,     R   j     XZ   down         )       +     (         V        (       T   i   XZ     ,     R   j   XZ       )       1     -       V        (       T   i   XZ     ,     R   j     XZ   down         )       1       )                                   Resistivity Harmonic             2                 ZZ       XX   +   YY                           V        (       T   i   XZ     ,     R   j   XZ       )       0     +       V        (       T   i   XZ     ,     R   j     XZ   down         )       0             V        (       T   i   XZ     ,     R   j   XZ       )       0     -       V        (       T   i   XZ     ,     R   j     XZ   down         )       0                           Resistivity Anisotropy           XX   YY                         V        (       T   i   XZ     ,     R   j   XZ       )       0     -       V        (       T   i   XZ     ,     R   j     XZ   down         )       0     +       V        (       T   i   XZ     ,     R   j   XZ       )       2             V        (       T   i   XZ     ,     R   j   XZ       )       0     -       V        (       T   i   XZ     ,     R   j     XZ   down         )       0     -       V        (       T   i   XZ     ,     R   j   XZ       )       2                                
While having a combination of the coils in series implemented in hardware is desirable because of the simplicity of the electronics (fewer channels needed and absence of calibration), the coil pair measurements can also be acquired individually and a combination performed as part of an algorithm or software procedure.
 
         [0062]    Similarly, in the case of an XYZ configuration and with different combinations of XZ and YZ in series and an alignment angle α, the following formulas involving combinations of R 1 XZ up  and R 1 XZ down  can be used to get the ZZ coupling by addition of both measurements (DC term) and to get the (XX+YY)/2 and (XY−YX)/2 terms weighted by cos α and sin α by subtraction of both measurements. Note that the alignment angle can be determined from the measurements using the DC term if two other measurements are used (for example R 1 XY left  and R 1 XY right ), allowing solving for angle α, ZZ, (XX+YY)/2 and (XY−YX)/2 from four equations. 
         [0000]    
       
         
           
             
               V 
               
                 R 
                  
                 
                     
                 
                  
                 1 
                  
                 
                     
                 
                  
                 XZup 
               
             
             = 
             
               ZZ 
               + 
               
                 
                   
                     XX 
                     + 
                     YY 
                   
                   2 
                 
                  
                 
                   cos 
                    
                   
                     ( 
                     α 
                     ) 
                   
                 
               
               + 
               
                 
                   
                     XY 
                     - 
                     YX 
                   
                   2 
                 
                  
                 
                   sin 
                    
                   
                     ( 
                     α 
                     ) 
                   
                 
               
               + 
               
                 ZX 
                  
                 
                     
                 
                  
                 
                   cos 
                    
                   
                     ( 
                     φ 
                     ) 
                   
                 
               
               + 
               
                 ZY 
                  
                 
                     
                 
                  
                 
                   sin 
                    
                   
                     ( 
                     φ 
                     ) 
                   
                 
               
               + 
               
                 XZ 
                  
                 
                     
                 
                  
                 
                   cos 
                    
                   
                     ( 
                     
                       α 
                       + 
                       φ 
                     
                     ) 
                   
                 
               
               + 
               
                 YZ 
                  
                 
                     
                 
                  
                 
                   sin 
                    
                   
                     ( 
                     
                       α 
                       + 
                       φ 
                     
                     ) 
                   
                 
               
               + 
               
                 
                   
                     XX 
                     + 
                     YY 
                   
                   2 
                 
                  
                 
                   cos 
                    
                   
                     ( 
                     
                       α 
                       + 
                       
                         2 
                          
                         
                             
                         
                          
                         φ 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   
                     XY 
                     + 
                     YX 
                   
                   2 
                 
                  
                 
                   sin 
                    
                   
                     ( 
                     
                       α 
                       + 
                       
                         2 
                          
                         
                             
                         
                          
                         φ 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               V 
               
                 R 
                  
                 
                     
                 
                  
                 1 
                  
                 
                     
                 
                  
                 XZdown 
               
             
             = 
             
               ZZ 
               - 
               
                 
                   
                     XX 
                     + 
                     YY 
                   
                   2 
                 
                  
                 
                   cos 
                    
                   
                     ( 
                     α 
                     ) 
                   
                 
               
               - 
               
                 
                   
                     XY 
                     - 
                     YX 
                   
                   2 
                 
                  
                 
                   sin 
                    
                   
                     ( 
                     α 
                     ) 
                   
                 
               
               - 
               
                 ZX 
                  
                 
                     
                 
                  
                 
                   cos 
                    
                   
                     ( 
                     φ 
                     ) 
                   
                 
               
               - 
               
                 ZY 
                  
                 
                     
                 
                  
                 
                   sin 
                    
                   
                     ( 
                     φ 
                     ) 
                   
                 
               
               + 
               
                 XZ 
                  
                 
                     
                 
                  
                 
                   cos 
                    
                   
                     ( 
                     
                       α 
                       + 
                       φ 
                     
                     ) 
                   
                 
               
               + 
               
                 YZ 
                  
                 
                     
                 
                  
                 
                   sin 
                    
                   
                     ( 
                     
                       α 
                       + 
                       φ 
                     
                     ) 
                   
                 
               
               - 
               
                 
                   
                     XX 
                     - 
                     YY 
                   
                   2 
                 
                  
                 
                   cos 
                    
                   
                     ( 
                     
                       α 
                       + 
                       
                         2 
                          
                         
                             
                         
                          
                         φ 
                       
                     
                     ) 
                   
                 
               
               - 
               
                 
                   
                     XY 
                     + 
                     YX 
                   
                   2 
                 
                  
                 
                   sin 
                    
                   
                     ( 
                     
                       α 
                       + 
                       
                         2 
                          
                         
                             
                         
                          
                         φ 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0063]    A three antenna group configuration is shown in  FIG. 11  with T 1 X, R 1 X, and R 2 X as saddle transverse coils and T 1 Z, R 1 Z, and R 2 Z as standard axial coils. Each of the transmitter antenna group or receiver antenna group is an XZ. However, it is clear that the group can alternatively be a YZ or XY. Keeping the same angle between antennas, the axial coils will provide the standard ZZ propagation measurement while the transverse coils will provide the transverse (XX) propagation measurement. With this configuration, the axial and transverse attenuation and phase shift measurements can be generated according to the formulas below, where the transverse formula uses the DC terms (  V ) but without the borehole compensated feature (discussed below). A similar formula may also be generated based on (XX−YY)/2. Note that in this case, the receiver channel gains need to be known. 
         [0000]    
       
         
           
             
               
                 
                   ATT 
                   ZZ 
                 
                  
                 
                   andPS 
                   ZZ 
                 
               
               = 
               
                 
                   
                     GR 
                     1 
                     Z 
                   
                   · 
                   
                     V 
                      
                     
                       ( 
                       
                         
                           T 
                           1 
                           Z 
                         
                         , 
                         
                           R 
                           1 
                           Z 
                         
                       
                       ) 
                     
                   
                 
                 
                   
                     GR 
                     2 
                     Z 
                   
                   · 
                   
                     V 
                      
                     
                       ( 
                       
                         
                           T 
                           1 
                           Z 
                         
                         , 
                         
                           R 
                           2 
                           Z 
                         
                       
                       ) 
                     
                   
                 
               
             
             , 
             and 
           
         
       
       
         
           
             
               
                 ATT 
                 XX 
               
                
               
                 andPS 
                 XX 
               
             
             = 
             
               
                 
                   
                     
                       GR 
                       1 
                       X 
                     
                     · 
                     
                       
                         V 
                          
                         
                           ( 
                           
                             
                               T 
                               1 
                               X 
                             
                             , 
                             
                               R 
                               1 
                               X 
                             
                           
                           ) 
                         
                       
                       0 
                     
                   
                   
                     
                       GR 
                       2 
                       X 
                     
                     · 
                     
                       
                         V 
                          
                         
                           ( 
                           
                             
                               T 
                               1 
                               X 
                             
                             , 
                             
                               R 
                               2 
                               X 
                             
                           
                           ) 
                         
                       
                       0 
                     
                   
                 
                 . 
                 
                   
                     where 
                      
                     
                         
                     
                   
                   0 
                 
               
                
               
                   
               
                
               stands 
                
               
                   
               
                
               for 
                
               
                   
               
                
               the 
                
               
                   
               
                
               DC 
                
               
                   
               
                
               
                 term 
                 . 
               
             
           
         
       
     
         [0064]    A three antenna group configuration is shown in  FIG. 12  with T 1 X, T 1 Y, R 1 X, R 1 Y, and R 2 X, R 2 Y as transverse saddle coils and T 1 Z, R 1 Z, and R 2 Z as standard axial coils. Keeping the same angle between antennas, the axial coils will provide the standard ZZ coupling measurements while the transverse coils will provide the transverse (XX) and (YY) measurements. In this configuration, assuming calibrated measurements, the complete elementary coupling Z T1R1  and Z T1R2  can be measured and a generalized ratio Z T1R1 /Z T1R2  can be constructed. This generalized ratio can also be computed with the XZ antenna system (explained above) when rotation is also used. Different combinations of coils in series (ZX, YZ, XY) can be used in a manner similar to the that described above to yield a similar measurement with the advantage of being calibrated. 
         [0065]    An exemplary dual antenna configuration is shown in  FIG. 13  with T 1 X, R 1 X, R 2 X, and T 2 X as transverse saddle coils and T 1 Z, R 1 Z, R 2 Z, and T 2 Z as standard axial coils. Keeping the same angle between antennas, the axial coils will provide the standard ZZ coupling propagation measurement while the transverse coils will provide the transverse propagation (XX) measurement. The co-located receiver antennas can also be wired in series to become a single tilted antenna (R 1 XZ and R 2 XZ). The corresponding series combination for the transmitter is possible but would require a third tuning circuitry to be added. 
         [0066]    The axial and transverse borehole compensated attenuation and phase shift measurements can be generated according to the formulas where the transverse formula uses the DC terms (  V ). A similar formula may also be generated based on (XX−YY)/2. 
         [0000]    
       
         
           
             
               
                 
                   ATT 
                   ZZ 
                 
                  
                 
                   andPS 
                   ZZ 
                 
               
               = 
               
                 
                   
                     V 
                      
                     
                       ( 
                       
                         
                           T 
                           1 
                           Z 
                         
                         , 
                         
                           R 
                           1 
                           Z 
                         
                       
                       ) 
                     
                   
                   
                     V 
                      
                     
                       ( 
                       
                         
                           T 
                           1 
                           Z 
                         
                         , 
                         
                           R 
                           2 
                           Z 
                         
                       
                       ) 
                     
                   
                 
                 · 
                 
                   
                     V 
                      
                     
                       ( 
                       
                         
                           T 
                           2 
                           Z 
                         
                         , 
                         
                           R 
                           2 
                           Z 
                         
                       
                       ) 
                     
                   
                   
                     V 
                      
                     
                       ( 
                       
                         
                           T 
                           2 
                           Z 
                         
                         , 
                         
                           R 
                           1 
                           Z 
                         
                       
                       ) 
                     
                   
                 
               
             
             , 
             and 
           
         
       
       
         
           
             
               
                 ATT 
                 XX 
               
                
               
                 andPS 
                 XX 
               
             
             = 
             
               
                 
                   
                     
                       V 
                        
                       
                         ( 
                         
                           
                             T 
                             1 
                             X 
                           
                           , 
                           
                             R 
                             1 
                             X 
                           
                         
                         ) 
                       
                     
                     0 
                   
                   
                     
                       V 
                        
                       
                         ( 
                         
                           
                             T 
                             1 
                             X 
                           
                           , 
                           
                             R 
                             2 
                             X 
                           
                         
                         ) 
                       
                     
                     0 
                   
                 
                 · 
                 
                   
                     
                       V 
                        
                       
                         ( 
                         
                           
                             T 
                             2 
                             Z 
                           
                           , 
                           
                             R 
                             2 
                             Z 
                           
                         
                         ) 
                       
                     
                     0 
                   
                   
                     
                       V 
                        
                       
                         ( 
                         
                           
                             T 
                             2 
                             X 
                           
                           , 
                           
                             R 
                             1 
                             X 
                           
                         
                         ) 
                       
                     
                     0 
                   
                 
               
                
               
                 
                   where 
                    
                   
                       
                   
                 
                 0 
               
                
               
                   
               
                
               stands 
                
               
                   
               
                
               for 
                
               
                   
               
                
               the 
                
               
                   
               
                
               DC 
                
               
                   
               
                
               
                 term 
                 . 
               
             
           
         
       
     
         [0000]    These measurements provide standard calibrated formation evaluation anisotropy measurements in both vertical and horizontal boreholes, but do not have any directionality. The axial borehole compensated measurement allows determination of the resistivity anisotropy when in high angle wells, but not in nearly vertical well (where high angle or nearly vertical well refers to the relative orientation of the anisotropy and the tool axis). 
         [0067]    The four antenna group configuration can also be wired in a series in a similar way as described above. In such an implementation, the same axial and transverse borehole compensation measurement can be constructed, but with only half of the channels, as shown in Table 3. 
         [0000]    
       
         
               
               
               
             
           
               
                 TABLE 3 
               
               
                   
               
               
                 Measurement 
                   
                   
               
               
                 type 
                 Definition 
                 Generation 
               
               
                   
               
             
             
               
                 Standard propagation axial 
                 ATTzz,PSzz 
                 BHC using ZZ ij  = V(T i   XZ , R j   XZ ) 0  + V(T i   XZ , R j   XZ     down   ) 0   
               
               
                   
               
               
                 Standard propagation transverse 
                 ATTxx,PSxx 
                 
                   
                     
                       
                           
                         
                           
                             
                               
                                 
                                   BHC 
                                    
                                   
                                       
                                   
                                    
                                   using 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       
                                         XX 
                                         ij 
                                       
                                       + 
                                       
                                         YY 
                                         ij 
                                       
                                     
                                     2 
                                   
                                 
                                 = 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     V 
                                      
                                     
                                       ( 
                                       
                                         
                                           T 
                                           i 
                                           XZ 
                                         
                                         , 
                                         
                                           R 
                                           j 
                                           XZ 
                                         
                                       
                                       ) 
                                     
                                   
                                   0 
                                 
                                 - 
                                 
                                   
                                     V 
                                      
                                     
                                       ( 
                                       
                                         
                                           T 
                                           i 
                                           XZ 
                                         
                                         , 
                                         
                                           R 
                                           j 
                                           
                                             XZ 
                                             down 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   0 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
         [0068]    The electronics impact for a transverse and axial implementation is shown in  FIG. 15 . The additional pockets and TX board would not change the TX sub chassis length significantly. Note also that implementing the transverse axial series configuration reduces by half the number of pockets. 
         [0069]    The bottomhole assembly (BHA) shown, for example, in  FIG. 16  is an embodiment of the antenna configuration discussed above. In this BHA, tilted antennas provide a measurement with a deep depth of investigation to sense the resistivity and boundary using the resistivity contrast of the formation around and ahead of the bit. The antenna system allows a look-around and look-ahead with the location of the TX coil as close to the bit as possible. The saddle/axial co-located antennas close to the bit provide a shallow directional resistivity measurement.  FIG. 17  shows corresponding tilted antenna magnetic moment orientations. The tilted receivers can be co-located, but are not required to be co-located. The shallow saddle/axial coil system is shown immediately above the tilted transmitter.  FIG. 18  shows an example of an implementation in a collar close to the bit with a four-antenna group (for borehole compensation). Note that the last antenna is a standard tilted antenna. 
         [0070]    The BHA shown in the  FIG. 19  is an alternate embodiment of the antenna configuration discussed above. In this embodiment, saddle/axial co-located antennas provide a measurement with a deep depth of investigation to sense the resistivity and boundary using the resistivity contrast of the formation around and ahead of the bit. As before, the saddle/axial co-located antennas close to the bit provide an FE shallow directional resistivity. The corresponding saddle/axial co-located antenna magnetic moment orientation are shown in  FIG. 20 . The transmitter may be a tilted coil or a saddle/axial co-located coil system (driven in series or individually driven). The shallow saddle/axial coil system is shown immediately above the tilted transmitter in this system.  FIG. 21  shows an example of an implementation in a collar close to the bit with a four-antenna group (for borehole compensation). Note that the last antenna close to the bit is also a saddle/axial co-located antenna system instead of a standard tilted antenna (used for deep measurements).  FIG. 22  shows another possible implementation of a shallow saddle/axial coil system with a combination of propagation formation evaluation measurements and multiple two antenna tri-axial systems with different depths of investigation for deeper dip and formation evaluation anisotropy measurements. 
         [0071]    A shallow resistivity measurement close to the bit significantly enhances the results of a look-ahead inversion. In this inversion, by using the look-around capability (i.e., by using shallower measurements), the look-ahead is improved since the measurement sensitivity is mostly look-around with some look-ahead. The look-ahead allows for resistivity profile (ramp) estimation and detection of an oil-water contact (OWC). A longer spacing (e.g., 100 ft) combined with shallow resistivity close to the bit offers the best sensitivity to ramp and OWC ahead of the bit. The look-ahead also allows for detection of a thin layer ahead of the bit. 
         [0072]    A study was conducted using the parametric inversion looking at the effect of knowing the anisotropy of the formation (shallow resistivity sensor measuring anisotropy) versus not knowing the anisotropy. The general consensus is that the answer product with an anisotropic shallow resistivity measurement is qualitatively better, but not significantly better. Typically, the anisotropy of the upper layers is solved as an overall anisotropy. It is expected that instead of using an overall anisotropy, the inversion will be improved when including the anisotropy information found below the resistivity sensor measure point at a previous measurement depth. 
         [0073]    For simulation purposes, the antenna configuration shown in  FIG. 23  was used. All measurements are shown relative to an isotropic formation. The sensitivity to resolution for the transverse propagations and the harmonic anisotropy resistivity channel HRA, P=Z/(XX+YY)/2 are shown in  FIG. 24  and  FIG. 25 , respectively, for a zero dip situation. The HRA and HRP plots have a better resolution as function of anisotropy, but the accuracy of that measurement is expected to be lower than the standard propagation measurement. In addition, the dependence with respect to anisotropy and dip may not necessarily be a single value, unlike the HRA and HRP measurements. In both measurements, the overall spread of the different dip curves as function of anisotropy allows one to discriminate and isolate the anisotropy and dip effect. At higher resistivity, the anisotropy dependence is rather weak. 
         [0074]    A dual antenna induction configuration is shown in  FIG. 26  with T 1 X, R 1 X, R 2 X, and T 2 X as transverse saddle coils and T 1 Z, R 1 Z, R 2 Z, and T 2 Z as standard axial coils. In this implementation, the receiver coils are connected in series so as to eliminate the direct coupling from the transmitter and yield a dependence nearly linear with respect to conductivity. In one embodiment, the electronics for a triaxial induction implementation will need a current drive transmitter board, a receiver board, and a calibration board, as shown in  FIG. 27 . 
         [0075]    We can measure the apparent conductivity tensor σ app   k  for the k-th spacing and invert a 1D-dipping layered earth model for the Rh_k, Rv_k, dip_k, azi_k, or bed thickness. This is symbolically shown by: 
         [0000]    
       
         
           
             
               σ 
               app 
               k 
             
             = 
             
               
                 ( 
                 
                   
                     
                       
                         σ 
                         xx 
                       
                     
                     
                       
                         σ 
                         xy 
                       
                     
                     
                       
                         σ 
                         xz 
                       
                     
                   
                   
                     
                       
                         σ 
                         yx 
                       
                     
                     
                       
                         σ 
                         yy 
                       
                     
                     
                       
                         σ 
                         yz 
                       
                     
                   
                   
                     
                       
                         σ 
                         zx 
                       
                     
                     
                       
                         σ 
                         zy 
                       
                     
                     
                       
                         σ 
                         zz 
                       
                     
                   
                 
                 ) 
               
               → 
               
                 
                   Inversion 
                    
                   
                     ( 
                     
                       
                         R 
                         h 
                       
                       , 
                       
                         R 
                         v 
                       
                       , 
                       dip 
                       , 
                       azi 
                       , 
                       h 
                     
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
         [0000]    The calibration of the triaxial or tensor resistivity tool can be shown for the i-th transmitter, the j-th receiver, and the k-th spacing functionally as: 
         [0000]      σ ijk   app =           elec ( T   e )           ijk   TTL (σ ijk   meas −σ ijk   SEC ( T   a )).
 
         [0000]    where σ ijk   app  is the calibrated-complex apparent-conductivity and            ijk   TTL  is the gain correction defined for a modeled reference tilted test loop and given by: 
         [0000]    
       
         
           
             
               g 
               ij 
               TTL 
             
             = 
             
               
                 
                    
                   
                     σ 
                     ijk 
                     
                       TTL 
                       Ref 
                     
                   
                    
                 
                 
                    
                   
                     σ 
                     ijk 
                     
                       TTL 
                       Meas 
                     
                   
                    
                 
               
               . 
             
           
         
       
     
         [0000]    The background correction σ ijk   SEC (T) is given by: 
         [0000]    
       
         
           
             
               
                 σ 
                 ijk 
                 SEC 
               
                
               
                 ( 
                 T 
                 ) 
               
             
             = 
             
               
                 
                   η 
                   jk 
                 
                 
                   K 
                   ijk 
                 
               
                
               
                 
                   
                     V 
                     jk 
                     SEC 
                   
                   
                     I 
                     i 
                   
                 
                 . 
               
             
           
         
       
     
         [0000]    An embodiment of an LWD tenser resistivity tool is shown in  FIG. 28 . A typical tool response to a zero azimuth formation versus dip and anisotropy is shown in  FIG. 29 . 
         [0076]    For completeness the elementary coupling formula as function of rotation is shown below. 
         [0000]    
       
         
           
             
               
                 
                   RotationofelementarycouplingmeatrixM 
                    
                   
                     ( 
                     φ 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     
                       
                         
                           M 
                            
                           
                             ( 
                             φ 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                 
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       sin 
                                     
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                 
                                 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                             
                             ] 
                           
                           · 
                           
                             [ 
                             
                               
                                 
                                   XX 
                                 
                                 
                                   XY 
                                 
                                 
                                   XZ 
                                 
                               
                               
                                 
                                   YX 
                                 
                                 
                                   YY 
                                 
                                 
                                   YZ 
                                 
                               
                               
                                 
                                   ZX 
                                 
                                 
                                   ZY 
                                 
                                 
                                   ZZ 
                                 
                               
                             
                             ] 
                           
                           · 
                           
                             [ 
                             
                               
                                 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                 
                                 
                                   
                                     
                                       - 
                                       sin 
                                     
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                 
                                 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                 
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       sin 
                                     
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                 
                                 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                             
                             ] 
                           
                           · 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         · 
                                         XX 
                                       
                                     
                                     + 
                                     
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         · 
                                         XY 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         · 
                                         YX 
                                       
                                     
                                     + 
                                     
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         · 
                                         YY 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         · 
                                         ZX 
                                       
                                     
                                     + 
                                     
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         · 
                                         ZY 
                                       
                                     
                                   
                                 
                               
                             
                              
                             
                               
                                 
                                   
                                     
                                       
                                         - 
                                         sin 
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         · 
                                         XX 
                                       
                                     
                                     + 
                                     
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         · 
                                         XY 
                                       
                                     
                                   
                                 
                                 
                                   XZ 
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         - 
                                         sin 
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         · 
                                         YX 
                                       
                                     
                                     + 
                                     
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         · 
                                         YY 
                                       
                                     
                                   
                                 
                                 
                                   YZ 
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         - 
                                         sin 
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         · 
                                         ZX 
                                       
                                     
                                     + 
                                     
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         φ 
                                         · 
                                         ZY 
                                       
                                     
                                   
                                 
                                 
                                   ZZ 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             cos 
                                             2 
                                           
                                            
                                           
                                             φ 
                                             · 
                                             XX 
                                           
                                         
                                         + 
                                         
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             · 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           cos 
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             · 
                                             XY 
                                           
                                         
                                         + 
                                         
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             · 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           cos 
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             · 
                                             YX 
                                           
                                         
                                         + 
                                         
                                           
                                             sin 
                                             2 
                                           
                                            
                                           
                                             φ 
                                             · 
                                             YY 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   - 
                                                   sin 
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   φ 
                                                   · 
                                                   cos 
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   φ 
                                                   · 
                                                   XX 
                                                 
                                               
                                               + 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 
                                                   cos 
                                                   2 
                                                 
                                                  
                                                 
                                                   φ 
                                                   · 
                                                   XY 
                                                 
                                               
                                               - 
                                               
                                                 
                                                   sin 
                                                   2 
                                                 
                                                  
                                                 
                                                   φ 
                                                   · 
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         YX 
                                         + 
                                         
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             · 
                                             cos 
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             · 
                                             YY 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           cos 
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             · 
                                             XZ 
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                             
                                         
                                          
                                         
                                           φ 
                                           · 
                                           YZ 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   - 
                                                   sin 
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   φ 
                                                   · 
                                                   cos 
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   φ 
                                                   · 
                                                   XX 
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 
                                                   sin 
                                                   2 
                                                 
                                                  
                                                 
                                                   φ 
                                                   · 
                                                   XY 
                                                 
                                               
                                               + 
                                               
                                                 
                                                   cos 
                                                   2 
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   φ 
                                                   · 
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         YX 
                                         + 
                                         
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             · 
                                             cos 
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             · 
                                             YY 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   sin 
                                                   2 
                                                 
                                                  
                                                 
                                                   φ 
                                                   · 
                                                   XX 
                                                 
                                               
                                               - 
                                               
                                                 sin 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   φ 
                                                   · 
                                                 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 cos 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   φ 
                                                   · 
                                                   XY 
                                                 
                                               
                                               - 
                                               
                                                 sin 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   φ 
                                                   · 
                                                   cos 
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   φ 
                                                   · 
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         YX 
                                         + 
                                         
                                           
                                             cos 
                                             2 
                                           
                                            
                                           
                                             φ 
                                             · 
                                             YY 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             - 
                                             sin 
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             φ 
                                             · 
                                             XZ 
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         cos 
                                          
                                         
                                             
                                         
                                          
                                         
                                           φ 
                                           · 
                                           YZ 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       φ 
                                       · 
                                       ZX 
                                     
                                   
                                   + 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       φ 
                                       · 
                                       ZY 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       sin 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       φ 
                                       · 
                                       ZX 
                                     
                                   
                                   + 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       φ 
                                       · 
                                       ZY 
                                     
                                   
                                 
                               
                               
                                 ZZ 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
             
             
               
                 with 
               
             
             
               
                 
                   
                     
                       
                         cos 
                         2 
                       
                        
                       φ 
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       · 
                       
                         ( 
                         
                           1 
                           + 
                           
                             cos 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             φ 
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     
                       
                         sin 
                         2 
                       
                        
                       φ 
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       · 
                       
                         ( 
                         
                           1 
                           - 
                           
                             cos 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                                 
                             
                              
                             ϕ 
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     sin 
                      
                     
                         
                     
                      
                     
                       φ 
                       · 
                       cos 
                     
                      
                     
                         
                     
                      
                     φ 
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                       · 
                       sin 
                     
                      
                     
                         
                     
                      
                     2 
                      
                     
                         
                     
                      
                     φ 
                   
                 
               
             
           
         
       
     
         [0000]    and the receiver voltage is given by: 
         [0000]        V (φ)= m   T   t   M (φ) m   R ,
 
         [0000]    where m T   t  and m R  are the transmitter and receiver vector moments, respectively. 
         [0077]    It should be appreciated that while the invention has been described with respect to a limited number of embodiments, those skilled in the art, having benefit of this disclosure, will appreciate that other embodiments can be devised which do not depart from the scope of the invention as disclosed herein. Accordingly, the scope of the invention should be limited only by the attached claims.