Dipole shear anisotropy logging

Parametric inversion is used to advantage in a technique and apparatus for borehole logging to determine properties of anisotropic formations, and a dispersion function that varies with frequency is used in the modeling of the formations. An embodiment of the method includes the following steps: providing a logging device that is moveable through the borehole; exciting a sonic source at a transmitter location on the logging device to establish flexural waves in the surrounding formations; measuring at each of a plurality of receiver locations on the logging device, which are spaced at a respective plurality of distances from the transmitter location, orthogonal wave components of split flexural waves that have travelled through the formations; computing, for each of the plurality of distances and for multiple frequencies, model orthogonal wave components which would result from the superposition of model split-flexural waves having selected wave parameters including respective fast and slow model slownesses which are variable functions of frequency and model polarizations; determining an error value which depends on the differences, at each of the plurality of receiver locations, between measured wave components and the model composite waves; modifying the model parameters; iteratively repeating the computing, determining, and modifying steps to reduce the error; and storing the ultimately modified model parameters as being indicative of properties of the formations.

FIELD OF THE INVENTION 
This invention relates to investigation of earth formation and, more 
particularly, to a method and apparatus for determining properties of 
anisotropic earth formations. 
BACKGROUND OF THE INVENTION 
It is well known that mechanical disturbances can be used to establish 
acoustic waves in earth formations and the properties of these waves can 
be measured to obtain important information about the formations through 
which the waves have propagated. Parameters of compressional and shear 
waves, such as their velocity and polarization directions, can be 
indicators of formation characteristics that help in evaluation of the 
location and/or producibility of hydrocarbon resources. Reference can be 
made, for example, to my U.S. Pat. Nos. 4,684,039 and 4,809,239. 
Detection of fractured zones and estimation of their properties, 
identification of principal stress directions and stress magnitudes, and 
measuring intrinsic anisotropy of formations such as shales, are of great 
interest in exploration and production geophysics. Shear wave splitting 
occurs when a shear wave separates into two phases with different 
velocities and different polarizations. Split-shear waves are one of the 
main acoustic indicators of anisotropy caused by stress, oriented 
inclusions or intrinsic properties of rocks. Anisotropy measurements using 
split-shear waves have been widely used in conjunction with surface 
seismic and borehole seismic techniques. Reference can be made, for 
example, to S. Crampin, Evaluation of Anisotropy by Shear-Wave Splitting: 
Geophysics, 50, 142-152, 1985; and to my U.S. Pat. No. 5,214,613. In 
borehole seismics, a source of acoustic energy is located on the surface 
of the earth, near a borehole. The source may be, for example, an 
explosive device or vibrating device. A logging tool within the borehole 
is equipped with sensors, such as geophones that receive acoustic energy 
from the source that has propagated through the earth formations. 
Shear anisotropy measurements in boreholes using dipole-shear tools (which 
have dipole sources as well as sensors, on the logging device) is a 
relatively new development. Reference can be made, for example, to C. 
Esmersoy, Dipole Shear Anisotropy Logging, 10th Petroleum Congress of 
Turkey, Expanded Abstracts, 1004; C. Esmersoy, K. Koster, M. Williams, A. 
Boyd, and M. Kane, Dipole Shear Anisotropy Logging: 64th Ann. Internat. 
Mtg., Soc. Explor. Geophys., Expanded Abstracts, 1994; K. Koster, M. 
Williams, C. Esmersoy, and J. Walsh, Applied Production Geophysics Using 
Shear-Wave Anisotropy: Production Applications For The Dipole Shear Imager 
and the Multi-component VSP, 64th Ann. Internat. Mtg., Soc. Explor. 
Geophys., Expanded Abstracts, 1994; M. Muller, A. Boyd, and C. Esmersoy, 
Case Studies of The Dipole Shear Anisotropy Log, 64th Ann. Internat. Mtg., 
Soc. Explor. Geophys., Expanded Abstracts, 1994; and C. Esmersoy, A. Boyd. 
M. Kane, and S. Denoo, Fracture and Stress Evaluation Using Dipole-Shear 
Anisotropy Logs, SPWLA, 36th Ann. Logging Symposium, paper J, 1995. 
Dipole-shear measurement resembles a small-scale shear borehole seismic 
survey inside the borehole where dipole transducers replace the surface 
shear source and downhole geophones. One of the major differences between 
the shear seismic and dipole shear data is that the former involves 
non-dispersive shear body waves, and the latter is typically dominated by 
dispersive borehole flexural waves. Flexural waves can be visualized as 
waves resulting from a part of the borehole wall flexing sideways as a 
consequence of the directional force exerted by a dipole source which acts 
through the borehole fluid.! However, in anisotropic formations the 
flexural waves split very much like the shear waves as illustrated in FIG. 
1. See K. Ellefsen, C. Cheng, and N. Toksoz, Applications of Perturbation 
Theory To Acoustic Logging: J. Geophys. Res., 96, 537-549, 1991; D. Leslie 
and C. Randall, Multipole Sources In Deviated Boreholes Penetrating 
Anisotropic Formations, J. Acoust. Soc. Am., 91, 12-27, 1992; and B. 
Sinha, A. Norris, and S. Chang, Borehole Flexural Modes in Anisotropic 
Formations, Geophysics, 59, 1037-1052, 1994.! In the Figure, 105 is an 
anisotropic formation traversed by a borehole with an axis 110, with a 
dipole source oriented at an unknown angle .theta. from the fast shear 
polarization direction u, and with g.sub.f and g.sub.s being the Green's 
functions representing propagation in the fast and slow shear polarization 
planes. The unit vectors u.sub.i and u.sub.c are the in-line and 
cross-component receiver signals. 
A borehole seismic technique for split-shear processing is the so-called 
four-component rotation or "4C" technique (see R. Alford, Shear Data In 
The Presence Of Azimuthal Anisotropy, Dilley, Tex.: 56th Ann. Internat. 
Mtg., Soc. Expl. Geophys., Expanded Abstracts, 476-479, 1986). The 4C 
technique employs data acquired using a two-component source and an array 
of two-component receivers. Two-component source/receiver refers to two 
elements with orthogonal directions placed at the same physical location. 
The 4C technique has been adapted to dipole shear applications. (See C. 
Esmersoy, K. Koster, M. Williams, A. Boyd, and M. Kane, Dipole Shear 
Anisotropy Logging: 64th Ann. Internat. Mtg., Soc. Explor. Geophys., 
Expanded Abstracts, 1994.) 
Another borehole seismic technique is parametric inversion (see U.S. Pat. 
No. 5,214,613 and C. Esmersoy, Split-shear Wave Inversion for Fracture 
Evaluation, SEG Ann. Internat. Meeting Abstracts, pp. 1400-1403, 1990), 
which requires one or more one-component sources and an array of 
two-component receivers. As described further hereinbelow (see e.g. 
equation (1)), the technique uses model values of fast shear and slow 
shear slowness that apply for all frequencies, and this is suitable for 
the borehole seismic application in which it is used. However, for 
wideband processing of split flexural waves in shear anisotropy borehole 
logging, the fast shear and slow shear slownesses are dispersive. 
It is among the objects of the present invention to address limitations of 
the prior art regarding shear anisotropy borehole logging. 
SUMMARY OF THE INVENTION 
In the present invention, parametric inversion is used to advantage in a 
technique and apparatus for borehole logging to determine properties of 
anisotropic formations, and a dispersion function that varies with 
frequency is used in the modeling of the formations. 
In accordance with an embodiment of the method of the invention, the 
following steps are performed: providing a logging device that is moveable 
through the borehole; exciting a sonic source at a transmitter location on 
the logging device to establish flexural waves in the surrounding 
formations; measuring at each of a plurality of receiver locations on the 
logging device, which are spaced at a respective plurality of distances 
from the transmitter location, orthogonal wave components of split 
flexural waves that have travelled through the formations; computing, for 
each of the plurality of distances and for multiple frequencies, model 
orthogonal wave components which would result from the superposition of 
model split-flexural waves having selected wave parameters including 
respective fast and slow model slownesses which are variable functions of 
frequency and model polarizations; determining an error value which 
depends on the differences, at each of the plurality of receiver 
locations, between measured wave components and the model composite waves; 
modifying the model parameters; iteratively repeating the computing, 
determining, and modifying steps to reduce the error; and storing the 
ultimately modified model parameters as being indicative of properties of 
the formations. 
Further features and advantages of the invention will become more readily 
apparent from the following detailed description when taken in conjunction 
with the accompanying drawings.

DETAILED DESCRIPTION 
Referring to FIG. 2, there is shown an apparatus which can be used in 
practicing an embodiment of the invention. Subsurface formations 231 are 
traversed by a borehole 232 which is typically, although not necessarily, 
filled with drilling fluid or mud. A logging tool 210 is suspended on an 
armored cable 212 and may have optional centralizers (not shown). The 
cable 212 extends up the borehole, over a sheave wheel 220 on a derrick 
221 to a winch forming part of surface equipment 250. Known depth gauging 
apparatus (not shown) is provided to measure cable displacement over the 
sheave wheel 220 and accordingly the depth of the logging tool 210 in the 
borehole 232. A device of a type well known in the art is included in the 
tool 210 to produce a signal indicative of orientation of the body of the 
tool 210. Processing and interface circuitry within the tool 210 
amplifies, samples and digitizes the tool's information signals for 
transmission and communicates them to the surface equipment 250 via the 
cable 212. Electrical power and control signals for coordinating operation 
of the tool 210 are generated by the surface equipment 250 and 
communicated via the cable 212 to circuitry provided within the tool 210. 
The surface equipment includes processor 270, standard peripheral 
equipment (not shown), and recorder 226. 
The logging device 210 may be, for example, a Dipole Shear Sonic Imager 
("DSI"--trademark of Schlumberger) of the type described in Harrison et 
al., "Acquisition and Analysis of Sonic Waveforms From a Borehole Monopole 
and Dipole Source for the Determination of Compressional and Shear Speeds 
and Their Relation to Rock Mechanical Properties and Surface Seismic 
Data", Society of Petroleum Engineers, SPE 20557, 1990. Pertinent portions 
of the logging device are shown in greater detail in FIG. 3. The logging 
device 210 includes crossed dipole transmitters 315 and 320 (only one end 
of dipole 320 being visible) and a monopole transmitter 325. Eight spaced 
apart receiver stations, designated 331 through 338 each comprise four 
receiver hydrophones mounted azimuthally at ninety degree intervals in the 
surface of the cylindrical logging device. FIG. 4 shows the hydrophones, 
designated A, B, C, and D. In the example, of FIG. 4, the X component is 
obtained by subtracting the signals received at A and C (i.e., A-C), and 
the Y component is obtained by subtracting the signals received at B and D 
(i.e., B-D). With four receiver elements at each receiver station, there 
are a total of thirty two receiver elements. The receiver stations are 
also configurable for monopole reception. 
The transmitter electronics contain a power amplifier and switching 
circuitry capable of driving the two crossed-dipole transmitter elements 
and the monopole element from a programmable waveform. Separate waveforms 
with appropriate shape and frequency content are used for dipole, Stoneley 
and compressional measurements. The receiver electronics processes the 
signals from the 32 individual receiver elements located at the eight 
receiver stations which are spaced six inches apart. At each station, four 
receivers are mounted as shown in FIG. 4 which allows measurement of the 
dipole and crossed-dipole waveforms by differencing the outputs from 
opposite receivers, as previously described. Summing the outputs of the 
receivers can be used to produce a monopole equivalent signal. As further 
described in Harrison et al., supra, the receiver electronics multiplexes, 
filters, amplifies and channels the signals from the 32 receiver elements 
to 8 parallel signal paths. These eight parallel analog signals are passed 
to an acquisition electronics cartridge where eight 12-bit 
analog-to-digital converters digitize the signals from the receiver 
electronics. The telemetry circuitry passes the digitized information to 
the earth's surface. 
FIG. 5 shows the acquisition signal path in block diagram form for one of 
the eight receiver stations, as described in Harrison et al., supra. Each 
receiver has its own charge preamplifier (represented at 505). The output 
of the receivers, odd numbered pairs being in-line with the upper dipole 
transmitter and even numbered pairs with the lower dipole transmitter, 
passes into both a summing circuit (for monopole measurements) and a 
differencing circuit (for dipole measurements), as represented at 510. 
Under software control the sum or difference is selected by a multiplexer 
stage (block 520) and the signal passed to one of eight programmable gain 
amplifier stages (540) and filters (545). The other similar channels are 
represented by block 550. The eight parallel analog signals are passed to 
eight parallel 12-bit A/D converters (represented at 560) where 
simultaneous waveform digitization is performed. After digitization, the 
eight waveforms are passes to the memory section associated with downhole 
processor 580. The processor also provides control signals and waveforms 
to transmitter and receiver electronics. An alternate path directs the 8 
analog receiver signals into threshold crossing detection circuitry or 
digital first motion detection, as represented at block 565. This 
circuitry detects the time of all up or down going threshold crossings. 
The digitized waveform data and the threshold crossing time data are 
passed to the surface using telemetry circuitry 590. 
In the FIG. 2 embodiment, the processing of signals recorded uphole can be 
implemented using a processor 170, such as a suitably programmed general 
purpose digital processor with memory and peripherals conventionally 
provided. It will be understood, however, that the processing need not be 
performed at the wellsite, and that signals derived at the wellsite can be 
processed at a remote location. 
Before describing the routine for controlling the processor for operation 
in accordance with an embodiment of the invention, underlying theory will 
be set forth. The parametric inversion technique has been previously used 
for shear seismic and borehole seismic applications to invert the 
anisotropic parameters (i.e., the characteristic direction and slow/fast 
slownesses) of subsurface formations. See e.g. my U.S. Pat. No. 
5,214,613.! In this technique, data recorded at an array of M 
two-component receivers, at some angular frequency .omega., are 
represented in the frequency domain by a parametric model 
##EQU1## 
where s.sub.f, s.sub.s, and a.sub.f (.omega.), and a.sub.s (.omega.) are, 
respectively, the fast- and slow-shear slownesses and Fourier amplitudes 
(at frequency .omega.) of the waveforms respectively, u.sub.m =(x.sub.m, 
y.sub.m).sup.T is the two-component modeled data vector at receiver 
station m, and h.sub.f =(cos.theta., -sin .theta.).sup.T and h.sub.s 
=(sin.theta., cos.theta.).sup.T are the fast- and slow-shear polarization 
vectors. The next step is to find the best set of unknown parameters, 
.theta., s.sub.f, s.sub.s, a.sub.f (.omega.), a.sub.s (.omega.) , such 
that this parametric model fits observed data. By choosing a least-squares 
criterion for fitness, the parameters are estimated by minimizing the 
squared error 
##EQU2## 
between the model u and the observed data d. The apparent bottleneck in 
this approach is the number of unknown parameters to be estimated. In 
particular, the number of Fourier components a.sub.f (.omega.) and a.sub.s 
(.omega.) can quickly reach to hundreds, since they are different for each 
frequency. However, it was shown C. Esmersoy, Split-Shear Wave Inversion 
for Fracture Evaluation, SEG Ann. Internat. Meeting Abstracts, pp. 
1400-1403, 1990! that the Fourier components can be eliminated from the 
problem. In other words, the angle .theta., and apparent slowness s.sub.f, 
and s.sub.s, for each wave (all of which are independent of frequency) can 
be estimated without knowing the Fourier components. Reference can again 
be made to my U.S. Pat. No. 5,214,613. 
FIG. 6 shows the recording geometry for observation of split-flexural waves 
by an array of dipole receivers (represented by the dots on the vertical 
axis). The reference numerals 105 and 110 correspond to those of FIG. 1. 
The x and y components of receivers make an angle .theta. with the fast- 
and slow- shear polarization planes in the azimuthal plane. The flexural 
waves are generated by a dipole source with an arbitrary and unknown 
azimuthal direction with respect to the polarization planes. For the 
flexural waves excited using the described type of logging device, it is 
known that there is wave dispersion; that is, variation of the slowness 
parameters with frequency. In the present invention, the parametric 
inversion considers wave dispersion, and the technique is not limited to 
narrow band operation. Signals recorded at an array of M dipole receivers, 
at some angular frequency .omega., are represented in the frequency domain 
by the parametric model 
##EQU3## 
where h.sub.f =(cos .theta., -sin .theta.).sup.T and h.sub.s =(sin 
.theta., cos .theta.).sup.T are the fast- and slow-shear polarization 
vectors where the two elements in the vectors represent the x and y 
directions indicated in FIG. 6. Comparing equations (1) and (3), the 
frequency independent, fast/slow shear slowness s.sub.f, s.sub.s in (1) 
are replaced by the phase slowness dispersion functions p.sub.f (.omega.) 
and p.sub.s (.omega.) which are dependent on frequency. 
The best set of unknowns, .theta., p.sub.f (.omega.) and p.sub.s (.omega.) 
is sought such that this parametric model fits observed data using some 
measure of fitness. The functional dependency of the slowness function p() 
on its parameters is assumed known. This may be given in analytical form, 
exact or with approximations, or it can be obtained numerically and stored 
in tables. For example, one simple analytical form can be obtained by 
approximating the dispersion curve in a frequency band with a line. A 
similar approach to estimate the dispersion curves of flexural waves is 
shown in K. Hsu, and C. Esmersoy, Parametric Estimation of Phase and Group 
Slowness From Sonic Logging Waveforms, Geophysics, 57, No. 8, 978-985, 
1992. Using the Taylor series expansion, the phase slownesses around 
frequency .omega..sub.0 can be approximated by the two parameter model, 
EQU p(.omega.).apprxeq.p.sub.0 +(.omega.-.omega..sub.0)q.sub.0,(4) 
where p.sub.o =p(.omega..sub.0) is the phase slowness at .omega..sub.0, and 
##EQU4## 
is the difference between the group and phase slownesses (i.e., amount of 
dispersion), at .omega..sub.0. 
By choosing the least-squares criterion for fitness, the parameters are 
estimated by minimizing the squared error 
##EQU5## 
between the model u and the observed data d. As described before, the 
Fourier components a.sub.f (.omega.) and a.sub.s (.omega.) can be 
eliminated from the minimization problem by substitution, and the above 
squared error is minimized over the remaining five parameters. 
In its simplest form, the model for dipole-shear waveforms in an 
anisotropic medium consists of one fast and one slow flexural wave. It has 
been shown B. Sinha, A. Norris, and S. Chang, Borehole Flexural Modes in 
Anisotropic Formations, Geophysics, 59, 1037-1052, 1994! that for flexural 
wave propagation, the anisotropic medium can be adequately represented by 
two effective isotropic media; one for fast and another for the slow 
waves. Therefore, we can use the dispersion function for an isotropic 
medium, which has fewer parameters, to represent p(). 
The phase slowness functions of these waves, in general, depends on many 
formation and borehole parameters and it can be complex valued due to 
attenuation. However, using some assumptions and approximations the number 
of parameters can be reduced. The flexural dispersion curve for an 
elastic, isotropic medium and circular borehole depends on five parameters 
A. Kurkjian, and S. Chang, Acoustic Multipole Sources In Fluid-Filled 
Boreholes, Geophysics, 51,148-163, 1986; U.S. Pat. No. 5,278,805!; the 
shear and compressional slownesses of the formation, the slowness of the 
mud, the formation to mud density ratio, and the borehole diameter. In the 
simplest case it can be assumed that all the other parameters are known 
and the flexural dispersion curves (both for slow and fast waves) can be 
parameterized with only the shear slowness. For the simplest case, the 
model is given by 
##EQU6## 
where the flexural phase slowness dispersion p(s,.omega.) is a known 
function (either analytical or numerically computed and stored) 
parameterized by the slowness s. After the elimination of the waveform 
Fourier components C. Esmersoy, Split-Shear Wave Inversion for Fracture 
Evaluation, SEG Ann. Internat. Meeting Abstracts, pp. 1400-1403, 1990!, 
the inversion problem becomes the estimation of three parameters; the 
fast-flexural polarization angle .theta., and the fast- and slow shear- 
slowness s.sub.f and s.sub.s by solving 
##EQU7## 
An advantage of the parametric inversion technique is that it minimally 
requires only a one-component source and two-component receivers. Inline 
and cross component data from a single dipole source is sufficient for 
inversion. Of course, further source and/or receiver data permits 
determination of more unknowns and/or increase in the degree of over 
determination. Further advantages are that additional sources need not be 
at the same position as the first source and need not have identical 
characteristics. 
Data recorded at an array of M dipole receivers, at some angular frequency 
.omega., given in equation (3), can be written by using more comprehensive 
parametric models for flexural wave propagation. 
##EQU8## 
where c(s.sub.f,.omega.,.alpha.) and c(s.sub.s,.omega.,.alpha.) are the 
complex dispersion curves for the fast- and slow-flexural waves for an 
anisotropic medium characterized by: s.sub.f, s.sub.s the fast- and 
slow-shear slownesses, and .alpha. the vector representing all other 
parameters of the anisotropic medium and borehole such as all elastic 
constants (Sinha et al., supra), unelastic properties such as attenuation, 
borehole size and shape, and properties of the borehole fluid. h.sub.f 
=(cos .theta., -sin .theta.).sup.T and h.sub.s =(sin .theta., cos 
.theta.).sup.T are the fast- and slow-shear polarization vectors where the 
two elements in the vectors represent the x and y directions indicated in 
FIG. 6. Comparing equations (6) and (8), the real-valued fast/slow phase 
slowness functions p(s.sub.f,.omega.) , p(s.sub.f,.omega.) in (6) are 
replaced by the complex slowness functions c(s.sub.f,.omega.,.alpha.) and 
c(s.sub.s,.omega.,.alpha.) which are dependent on the parameter vector 
.alpha. in addition to the fast/slow shear slownesses themselves. The 
functional dependency of the complex slowness function c() on its 
parameters is assumed known. This may be given in analytical form, exact 
or with approximations, or it can be obtained numerically and stored in 
tables. It is also possible to represent the functional form of c() by a 
parameterized function and include the unknown parameters in the parameter 
vector .alpha.. 
As before, the next step is to find the best set of unknown parameters, 
.theta., s.sub.f, s.sub.s, .alpha., a.sub.f (.omega.) , a.sub.s (.omega.), 
such that this parametric model fits observed data using some measure of 
fitness. By choosing a least-squares criterion for fitness, we estimate 
the parameters by minimizing the squared error 
##EQU9## 
between the model u and the observed data d. As described before, the 
Fourier components a.sub.f (.omega.) and a.sub.s (.omega.) can be 
eliminated from the minimization problem by substitution, and the above 
squared error is minimized over the remaining parameters. 
Equations (1), (3), (6) and (8) represent a superposition of two waves. As 
described in C. Esmersoy, Split-Shear Wave Inversion for Fracture 
Evaluation, SEG Ann. Internat. Meeting Abstracts, pp. 1400-1403, 1990, 
this model can be generalized to include more waves, such as fast and slow 
direct shear waves, compressional waves, Stoneley waves or any other wave 
or mode. The model may also include waves that propagate in opposite 
direction across the receiver array, e.g. due to reflections. These waves 
would be represented by the same polarization vectors, such as h.sub.f, 
and by the sign-reversed, complex conjugate, dispersion curves, such as 
-c*(s.sub.f,.omega.,.alpha.) 
FIG. 6, previously referenced, shows the recording geometry for observation 
of split-flexural waves by an array of dipole receivers. As noted, the 
flexural waves are generated by a dipole source with an arbitrary and 
unknown azimuthal direction. If there is more than one dipole source 
available, this data can still be utilized by the parametric inversion 
technique in an optimal way without the constraints of other techniques. 
By dropping .omega. and by using bold letters for the data and model 
vectors, equation (3) or (5) can be represented as 
EQU u=e.sub.f a.sub.f +e.sub.s a.sub.s. (10) 
This is the model for one dipole source. The model for multiple dipole 
sources (two or more), is obtained by augmenting the model vector. For 
example, the augmented model for two sources is given by 
##EQU10## 
Note that the model vectors e.sub.f and e.sub.s containing the unknowns 
.theta., s.sub.f, and s.sub.s are the same for each source, because the 
unknowns represent the measured medium independent from the source 
position or orientation. Therefore, addition of more sources does not add 
new unknowns but adds new measured data points. As is known in inversion 
techniques, adding new measurements makes the inversion problem more over 
determined, resulting in more reliable results. Equation (8) has the same 
mathematical form as the equations (3) or (5), and the augmented 
minimization problem is given by 
##EQU11## 
where L is the number of sources, and this is solved as described above. 
A dipole receiver is usually constructed by differencing the pressures 
measured at two closely spaced hydrophones, as first shown in FIG. 4. 
Ideally these hydrophones would be perfectly matched, but in practice the 
quality of the match can vary between tools and can change in time as the 
materials age. The parametric inversion technique can be extended to 
represent such possible mismatches by either representing the model 
explicitly for the recorded pressures rather than their differences. Data 
recorded at the mth 2-component receiver station is given by (dropping 
.omega. for brevity) 
EQU u.sub.m =h.sub.f exp.phi..sub.f.sup.m !a.sub.f +h.sub.s 
exp.phi..sub.s.sup.m !a.sub.s, (13) 
where .phi..sup.m.sub.f and .phi..sup.m.sub.s represent the complex 
exponents in equations (3) or (5) for the fast and slow waves 
respectively. From FIG. 4, the x and y components of the vector can be 
written in terms of the pressure differences by 
##EQU12## 
where A.sub.m, B.sub.m, C.sub.m, and D.sub.m are the measured pressures, 
and G.sub.Am, G.sub.Bm, G.sub.Cm, and G.sub.Dm are the gain factors for 
individual hydrophones. If the gain responses, as a function of frequency, 
are given or measured by some means with respect to a reference, then 
these correction factors (inverses of the gains) are used as known 
coefficients in the model. If the hydrophone gains are not known, then 
they can be included explicitly in the model for the individual pressures 
as 
##EQU13## 
These gains are included in the parameter vector .alpha. in equation (8) 
and estimated together with the other unknown quantities by minimizing the 
fit error. 
Referring to FIG. 7, there is shown a flow diagram of a routine that can be 
used for programming the processor 170, or other suitable general or 
special purpose processor, in accordance with an embodiment of the 
invention. The block 702 represents the deriving of measurement data taken 
with the tool (e.g. FIG. 3) over a range of depth levels to be processed. 
In the present example, the data comprises the previously described two 
component data taken at the receiver stations (eight of them in this 
case). Data can be processed in real time, or can be processed after 
collection and storage. The block 705 represents rotation of the data 
waveforms, in a manner known in the art, to correct for the tool 
rotational orientation. Gyro data can be used to implement the necessary 
corrections. The rotation at this stage is optional. An alternative is to 
not rotate the waveforms beforehand, but to later correct the angle 
.theta. determined from the inversion by an angle based on the measured 
tool rotation orientation. If equation (15) is used, for example, when 
hydrophone gains are matched, the data waveform rotation cannot be used, 
and the alternative technique of post-inversion adjustment of .theta. can 
be implemented.! Next, the block 708 represents the transformation of the 
data from the time domain to the frequency domain. In the present 
embodiment this is implemented, in the known fashion, using a Fourier 
transformation technique. A time window of several milliseconds, for 
example, can be used for the transformation. 
The block 710 is then entered, this block representing the initializing to 
a first depth level to be processed. The block 713 is next entered, this 
block representing the selection of model values, for the current depth 
level. In the present example, the angle .theta. (which determines the 
polarization vectors h.sub.f and h.sub.s) and the flexural phase slowness 
dispersion functions p.sub.f (.omega.) and p.sub.s (.omega.), which are 
variable functions of frequency. As also described above, these functions 
can be complex, with real and imaginary parts. As previously described, 
the flexural dispersion curve for an isotropic medium and a circular 
borehole depends on several parameters. In one example of the present 
embodiment, it is assumed that these parameters are known, except the 
shear slowness for the fast and slow waves (that is, s.sub.f and s.sub.s, 
respectively), so, as in equation (6) above, the flexural dispersion 
functions are p(s.sub.f,.omega.) and p(s.sub.s,.omega.), respectively. 
Accordingly, the selection of model values s.sub.f and s.sub.s determines, 
at each frequency, .omega., the model values p(s.sub.f,.omega.) and 
p(s.sub.s,.omega.). 
The block 716 is next entered, this block representing the initializing of 
an error accumulator to zero. As will be described, this accumulator is 
used in computing the error value E. The frequency, .omega., is then 
initialized at the first Fourier frequency component to be considered 
(block 719). The number of Fourier frequency components to be used can be 
selected as a trade-off between accuracy of representation and computation 
time. In an example hereof, 20 frequencies were employed, although any 
suitable number can be used. The index m is then initialized to the first 
receiver position, m=1 (block 722). The values h.sub.f 
exp.tau..omega.p.sub.f (.omega.)z.sub.m !a.sub.f (.omega.) and h.sub.s 
exp.tau..omega.p.sub.s (.omega.)z.sub.m !a.sub.s (.omega.) are then 
computed and summed, to obtain the model value u.sub.m (.omega.) that is, 
u.sub.1 (.omega.) for the first pass!. This is represented by the block 
725. For each set of model parameters (for a given frequency), the Fourier 
components can be obtained, in known manner, by equating the frequency 
components obtained from the measured waveforms in the depth window with 
the equations for the model, and solving simultaneously for unknown values 
of a.sub.f (.omega.) and a.sub.s (.omega.) . Reference can also be made to 
Esmersoy, P and SV Inversion From Multicomponent Offset VSPs, Geophysics 
(1990). The error component for the current frequency component, .omega., 
is then computed (block 728), e.g. in accordance with the relationship 
(5), and this error component is added to the error accumulator (block 
730). The quantity u.sub.m (.omega.) in the general case will have two 
orthogonal components (e.g. FIG. 1) and, as is well known, a vector 
difference (for the frequency component currently being considered) will 
be the difference between the two-dimensional vector for the measured 
values d.sub.m (.omega.)!, and the two-dimensional vector for the model 
composite wave u.sub.m (.omega.)!, which was computed above, as 
represented by block 725. Inquiry is then made (decision diamond 731) as 
to whether the last receiver position (M) has been reached. If not, m is 
incremented (block 734), block 725 is re-entered, and the loop 740 is 
continued as each receiver station is considered and the computed error 
components are added to the error accumulator. When all receiver positions 
have been processed, inquiry is made (diamond 745) as to whether all 
frequency components have been considered. If not, the frequency is 
incremented (block 748), block 722 is re-entered, and the loop 750 is 
continued as all frequencies are considered and error components are 
appropriately accumulated. When the loop 750 is complete, the error, E, in 
the error accumulator is in accordance with relationship (2). 
Inquiry is then made (diamond 755) as to whether E is greater than 
E.sub.min. In this embodiment, E.sub.min is the error threshold of 
acceptability, it being understood that other criteria can be utilized. 
For example, as is well known in parametric inversion techniques, the 
number of iterations can be kept track of and acceptability can be a 
function of the number of iterations or the extent of improvement that is 
obtained. Among other alternatives would be to permit a certain maximum 
number of iterations. In the present embodiment, if the computed error is 
above the predetermined threshold, the block 758 is entered, this block 
representing the computing of the direction and step size of the model 
vector increment; in other words, the manner in which the model parameters 
should be incremented or decremented. This type of determination is well 
known in the art and not, of itself, a novel feature hereof. Reference can 
be made, for example, to J. E. Dennis et al., Numerical Methods For 
Unconstrained Optimization And Non-Linear Equations, Prentice Hall (1983), 
and to suitable known techniques for solution of non-linear least squares 
problems. The model parameter values (functions) are then appropriately 
modified (block 761), the block 716 is re-entered, the error accumulator 
is reset to zero, and the loop 765 continues until the computed error is 
below the predetermined threshold (or other conditions are indicated, as 
discussed above). The block 770 is then entered, this block representing 
the storing of the current model parameter values, that is, at least, 
.theta., p(s.sub.f, .omega.) and p(s.sub.s, .omega.). Inquiry is then made 
(diamond 773) as to whether the routine has reached the last depth level 
to be processed. If not, the position of the depth index is incremented 
(block 776), block 713 is re-entered, and the loop 780 is continued until 
all desired depth levels have been processed. Regarding the re-entry to 
block 713 for selection of initial model values (functions) for the next 
depth level, it will be understood that the final functions for the 
previous depth level can be advantageously used for this purpose. The 
initial estimates, for the first depth level, can be from indications in 
the data itself, other data, or arbitrary guesses.! If desired, and as 
indicated by the optional block 785, formation properties such anisotropy 
and its direction can be computed in known fashion from the wave model 
parameter values. 
The invention has been described with reference to particular preferred 
embodiments, but variations within the spirit and scope of the invention 
will occur to those skilled in the art. For example, although the 
described embodiment is set forth in terms of wireline logging, it will be 
understood that the invention can also be applied in 
logging-while-drilling.