U.S. patent application number 13/945441 was filed with the patent office on 2014-02-20 for method for cement evaluation with acoustic and nuclear density logs.
The applicant listed for this patent is Pingjun Guo, Richard J. Smith. Invention is credited to Pingjun Guo, Richard J. Smith.
Application Number | 20140052376 13/945441 |
Document ID | / |
Family ID | 50100640 |
Filed Date | 2014-02-20 |
United States Patent
Application |
20140052376 |
Kind Code |
A1 |
Guo; Pingjun ; et
al. |
February 20, 2014 |
Method for Cement Evaluation with Acoustic and Nuclear Density
Logs
Abstract
Method for evaluating cement quality in a cased well. A density
log of the well is obtained using, for example, a GammaRay sources
and detectors (51). The detector count rates are inverted to
provide initial estimates of cement density and thickness (53).
Acoustic waveform data are obtained from the well using an acoustic
logging tool (52). The acoustic data are inverted (54-56), using
the initial estimates of cement density and thickness obtained from
the density logs, and an updated density log is inferred. Cement
images are obtained from the updated density log, and cement bond
quality can be estimated (57).
Inventors: |
Guo; Pingjun; (Bellaire,
TX) ; Smith; Richard J.; (Calgary, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Guo; Pingjun
Smith; Richard J. |
Bellaire
Calgary |
TX |
US
CA |
|
|
Family ID: |
50100640 |
Appl. No.: |
13/945441 |
Filed: |
July 18, 2013 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
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61683528 |
Aug 15, 2012 |
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Current U.S.
Class: |
702/11 |
Current CPC
Class: |
E21B 47/005 20200501;
E21B 47/00 20130101 |
Class at
Publication: |
702/11 |
International
Class: |
E21B 47/00 20060101
E21B047/00 |
Claims
1. A method for evaluating cement quality in a cased well
environment using a density logging tool and an acoustic logging
tool, comprising: obtaining density logs from the cased well and
extracting from them initial estimates of cement density and
thickness; obtaining acoustic logs measured in the cased well;
using the initial estimates of cement density and thickness as an
initial medium model, inverting the acoustic logs to infer an
updated medium model, wherein the inverting is performed using a
computer; and using the updated medium model to evaluate cement
quality in the cased well.
2. The method of claim 1, wherein the inversion comprises using a
forward-modeling algorithm to predict the acoustic logs with the
initial medium model as input data, then comparing the predicted
logs to the measured logs and updating the initial medium model to
reduce misfit.
3. The method of claim 2, wherein the method is iterative,
continuing until the misfit is less than a preselected amount or
other stopping condition is reached.
4. The method of claim 1, wherein the cement quality evaluation is
based at least partly on cement density and thickness images
obtained from the updated medium model.
5. The method of claim 1, wherein the cement quality evaluation
includes evaluating cement bond.
6. The method of claim 1, wherein the density logging tool and the
acoustic logging tool are adapted to be lowered into a wellbore on
a wireline.
7. The method of claim 1, wherein the density tool comprises a
gamma ray source and at least one detector.
8. The method of claim 7, wherein the initial estimates of cement
density and thickness are obtained by inversion of the density
logs, said density logs comprising gamma ray count rates measured
by the at least one detector.
9. The method of claim 8, wherein the inversion of the density logs
is performed on a computer using an algorithm comprising
forward-modeling predicted gamma ray count rates using an assumed
density model, comparing the predicted count rates to the measured
count rates, and updating the assumed density model to reduce
misfit.
Description
CROSS-REFERENCE TO RELATED APPLICATION
[0001] This application claims the benefit of U.S. Provisional
Patent Application 61/683,528, filed Aug. 15, 2012, entitled METHOD
FOR CEMENT EVALUATION WITH ACOUSTIC AND NUCLEAR DENSITY LOGS, the
entirety of which is incorporated by reference herein.
FIELD OF THE INVENTION
[0002] This invention relates generally to the field of production
of oil or gas, and more particularly to well drilling.
Specifically, the invention is a method for evaluating cement
density in cementing around well casings.
BACKGROUND OF THE INVENTION
[0003] When a well is drilled and steel casing is placed, cement
slurry is pumped into the annular space between casing and
formations. The primary objectives of cementing are to provide
mechanical support for the steel casing string and zonal isolation
between earth strata or formations. Multiple-stage casing and
cementing operations are common procedures to establish pressure
barriers during drilling a well. It allows the use of heavier
drilling muds in drilling deeper sections without damaging or
fracturing the shallower formations due to hydrostatic pressure
gradient. An ideal cementing job would fill the casing and
formation annulus completely with cement. Less than ideal cementing
can result in fluid filled channels within the cement sheath and
fluid contaminated cement due to incomplete replacement or sweep of
drilling mud with cement slurry. Zonal isolation assessment is a
critical aspect of well integrity tests to ensure hydrocarbon
production in a safe manner. Cement evaluation measurements are
relied upon to demonstrate that fluid cross flow will not occur
from unwanted zones, i.e. zones other than the producing intervals.
This invention relates to in situ evaluation of cement quality
between steel casing and formations in a wellbore.
[0004] Shown in FIG. 1 is a traditional cement bond tool (CBL). It
has an acoustic transducer and two receivers located three and five
feet from the transmitter, respectively. The 3 ft receiver records
the amplitude of the acoustic signal and the 5 ft receiver records
a variable density log (VDL), which is the pressure waveform data
measured by the receivers. Typically, the 5 ft receiver may measure
both amplitude and phase as a function of time. The variable
density log, also called a microseismograph, is recorded and
plotted as a function of time as each logging depth. Interpreters
may qualitatively inspect these plots and look for certain
"chevron" patterns due to sonic wave attenuation as evidence of a
good bond. Parallel or "railroad" tracks often indicate the
presence of free pipe or fluid behind casing. The cement bond
logging tools generally do not have azimuthal sensitivity and
instead provide the averaged attenuation and impedance measurements
around the wellbore. The newer ultrasonic cement bond tools are
typically configured with either multiple arms or a rotating head
that have collocated transmitter and receivers. These types of
tools are operated in pulsed mode and are capable of recording
waveform data in multiple sectors around the wellbore. More
precisely, acoustic impedance, which is the product of density and
velocity, is inferred from the ultrasonic receiver data.
[0005] FIG. 2 illustrates ultrasonic waveform propagation paths and
the recorded waveform (pressure vs. time) data. The sector acoustic
impedance data is often interpreted and plotted in cement impedance
maps. Shown in FIG. 3 are examples of cement impedance maps for
drilling mud and two different grades of cement, as published in
Paper No. 145970 in the proceedings of the Society of Petroleum
Engineers Annual Technical Conference (Kessler, C., et al. 2011). A
natural gamma ray log and the averaged impedance of all azimuthal
sectors are shown on the first (left-most) track. A cement
impedance map obtained from the acoustic data is plotted in the
third track next to a depth track. The fourth track ("segmented
impedance curves") contains the sector impedance values, and the
fifth track illustrates the map of statistical variances of
impedances.
[0006] EP718641B1 describes a method for measurements of rock
properties while a well is being drilled. U.S. Pat. No. 7,398,837
discloses a method for a drill bit design with built-in logging
sensors, where real time logs are recorded during drilling
operations. Neither method is designed for cased wells.
[0007] U.S. patent application publication 2005/0234649 describes a
method for detecting presence of gas behind casing using nuclear
density and neutron porosity logs. The density log is first
corrected by removing casing and cement effects. Acquisition and
use of an acoustic log is not taught.
[0008] U.S. Pat. No. 3,815,677 discloses running an open hole sonic
log and a cased hole neutron log to detect fluid channels in
cement. Azimuthally oriented nuclear density logs are also run to
detect the channel orientations. The method does not involve data
inversion.
[0009] PCT International Patent Application WO 2011/127156
discloses a method for using sonic and neutron logs to evaluate
cement integrity. PCT International Patent Application WO
2012/036689 discloses a method for using sonic and pulsed neutron
logs simultaneously to evaluate rock properties and cement
integrity. Neither of these publications teaches acquisition or use
of a nuclear density log.
[0010] SONATA Software
(http://fxc-png.m/download/sonata_demo/sonata.zip) is a software
product developed by a Russian vendor which is capable of
interpreting acoustic and nuclear logs for cement evaluation. It
does not involve a data inversion engine with forward acoustic
modeling capabilities to integrate acoustic and nuclear data.
SUMMARY OF THE INVENTION
[0011] In one embodiment, the invention is a method for evaluating
cement quality in a cased well environment using a density logging
tool and an acoustic logging tool, comprising (a) obtaining density
logs from the cased well and extracting from them initial estimates
of cement density and thickness; (b) obtaining acoustic logs
measured in the cased well; (c) using the initial estimates of
cement density and thickness as an initial medium model, inverting
the acoustic logs to infer an updated medium model, wherein the
inverting is performed using a computer; and (d) using the updated
medium model to evaluate cement quality in the cased well.
BRIEF DESCRIPTION OF THE DRAWINGS
[0012] The present invention and its advantages will be better
understood by referring to the following detailed description and
the attached drawings in which:
[0013] FIG. 1 is a schematic diagram illustrating a Cement bond
logging tool (CBL);
[0014] FIG. 2 is a schematic diagram illustrating ultrasonic wave
propagation paths and waveforms;
[0015] FIG. 3 shows an example of acoustic impedance maps in
drilling mud, regular cement, and light weight cement;
[0016] FIG. 4 is a schematic diagram illustrating an integrated
cement evaluation system with acoustic and nuclear density
tools;
[0017] FIG. 5 is a flowchart showing basic steps in the present
inventive method for integrated cement evaluation with acoustic and
nuclear density logs;
[0018] FIG. 6 is a "spine-and-ribs" response plot of a two-detector
density tool as shown in the book by Ellis and Singer: Well Logging
for Earth Scientists, Elsevier, page 210 (1987);
[0019] FIG. 7 is a schematic diagram of a three-detector density
tool showing the different regions that each detector is
predominantly sensitive to;
[0020] FIG. 8 is a plot showing Density sensitivity as a function
of radial distance from a gamma ray source;
[0021] FIG. 9 is a schematic diagram illustrating ultrasonic wave
propagation within a wellbore;
[0022] FIG. 10 shows a comparison of modeled and measured
ultrasonic waveforms in a "good bond" interval;
[0023] FIG. 11 shows a comparison of modeled and measured
ultrasonic waveforms in a free pipe interval;
[0024] FIG. 12 shows a pair of examples of borehole density image
logs obtained with tools using a gamma ray source and
detectors;
[0025] FIGS. 13-16 illustrate wellbore models with casing but with
differing quality cement bonds;
[0026] FIG. 17 illustrates how acoustic impedance may be computed
from waveforms provided by an acoustic logging tool;
[0027] FIG. 18 shows the response of a short-spaced detector in the
form of measured density as function of formation density and
cement thickness;
[0028] FIG. 19 shows the response of a long-spaced detector in the
form of measured density as function of formation density and
cement thickness; and
[0029] FIG. 20 is a schematic diagram illustrating a density tool
with a gamma ray source, a short-spaced detector, and a long-spaced
detector.
[0030] Depending upon patent law restrictions, some of the drawings
may be black-and-white reproductions of colored originals. The
invention will next be described in connection with example
embodiments. However, to the extent that the following detailed
description is specific to a particular embodiment or a particular
use of the invention, this is intended to be illustrative only, and
is not to be construed as limiting the scope of the invention. On
the contrary, it is intended to cover all alternatives,
modifications and equivalents that may be included within the scope
of the invention, as defined by the appended claims.
DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS
[0031] The present invention is a method for evaluating cement
quality in a wellbore completed with casing and cement, where the
method uses acoustic logs and nuclear density logs. More
particularly, acoustic waveform data are acquired with sonic or
ultrasonic tools, and bulk density data are acquired with a nuclear
density tool, in a cased and cemented wellbore. One example of a
conceptual system design is illustrated in FIG. 4. Sonic and
density logs may be further analyzed with an integrated
multi-physics analysis method to quantify density and thickness of
casing and cement as well as to provide traditional cement bond
logs. An underlying principle of the new method is that a nuclear
density tool measures bulk density, .rho., and an acoustic tool
measures acoustic impedance, Z, which is defined as
Z=.rho..nu.
where .nu. is acoustic velocity. There is a fundamental coherence
among the density and acoustic impedance measurements. The method
may be implemented in a practical interpretation workflow
consisting of one or more forward acoustic (sonic or ultrasonic)
models and a nuclear density inversion algorithm.
[0032] In more detail, the acoustic velocities of steel, casing and
fluid are close to constant and do not vary much. The cement
acoustic velocity, for example, can be estimated using density,
.rho., from density log and acoustic impedance, Z, calculated from
field data as illustrated in FIG. 17. The velocity values can then
be adjusted in an iterative fashion within the workflow. A system
of wave propagation equations are then built with material
properties of borehole mud casing, cement, and formation under
proper boundary conditions at material interfaces. For example, an
equation describing propagation of incident and reflection wave
potentials, p, has the following form,
p=a.sub.1e.sup.ikx+a.sub.2e.sup.-ikx
where a.sub.1 and a.sub.2 are amplitudes. The wavenumber K equals
.omega./.nu..sub.p (angular frequency divided by acoustic
velocity), and x is distance.
[0033] The current invention provides a method to quantitatively
evaluate cement quality with acoustic and nuclear density tools.
The nuclear density log provides a volumetric density measurement
of surrounding media. In theory, a nuclear density tool may be
constructed with a gamma ray source and one gamma ray detector, and
its density response is calibrated to bulk density in a homogeneous
and infinite medium. However, density tools, in practice, have
multiple detectors in place to provide accurate density
measurements in layered media, such as mud cake, casing, and
cement. Shown in FIG. 6 is "spine-and-ribs" response plot of a
two-detector density tool behind mud cakes. The two axes represent
the short-spaced and long-spaced density values. The primary curve
61, also called the spine, represents the formation density
response line along which the short-spaced detector and long-spaced
detector read the identical formation density values, or bulk
density, e.g. .rho..sub.b=1.9 g/cc, 2.0 g/cc, etc., and the
departure "ribs" curves 62 indicate the characteristics of tool
response to mud cake properties. The short (SS) and long-spaced
(LS) detectors respond to mud cake in different manners due to the
difference in distance to the gamma ray source. Depending upon the
formation density values, the SS and LS densities follow distinct
rib lines. In general, SS and LS densities respond oppositely to
mud cakes with or without barite. As illustrated in FIG. 6, the rib
lines on the left side of the spine correspond to the SS and LS
densities reading lower density values in front of heavy barite mud
cakes, and the rib lines on the right side of the spine correspond
to the SS and LS densities reading higher density values in front
of non-barite mud cakes. The tables in FIG. 6 list the mud cake
density, .rho..sub.mc, and mud cake thickness, .rho..sub.th, values
that define the rib curves.
[0034] A density tool with three or more detectors as shown in FIG.
7 provides count rates from each detector. The drawing illustrates
in a qualitative way how distance from the source increases
penetration sensitivity. With known casing weight and dimension,
the density log is capable of simultaneously solving cement density
and thickness as well as formation density. A mathematical
inversion algorithm can be devised to accomplish these objectives.
One embodiment of such an algorithm would be to establish a set of
multi-parameter regression equations between detector spectral
count rates and cement density and thickness using a database built
by either Monte Carlo modeling or laboratory experiments for
specific well completions. With more than two measured parameters,
a graphical representation similar to FIG. 6 for the three detector
tool response is no longer feasible. FIG. 8 depicts a typical
nuclear density measurement sensitivity as a function of the
distance from gamma ray source. This follows the radiation
attenuation law in which gamma ray intensity decreases
exponentially as a function of distance from the source,
C=.lamda.S.sub.0e.sup.-.mu.L,
where C is the detector count rate, .lamda. is a factor to account
for gamma ray detection efficiency, S.sub.0 is source strength in
photons/sec, .mu. is gamma ray attenuation coefficient, and L is
distance from source to detector.
[0035] Shown in FIGS. 18 and 19 are examples of density tool
response to formation density and cement thickness for a density
tool with two detectors placed in a well with 7'' OD steel casing
and 16 lb/gal cement. FIG. 18 illustrates the short-spaced (SS)
detector response in the form of measured density in the Z-axis as
function of formation density and cement thickness. FIG. 19
illustrates the long-spaced (LS) detector response in the form of
measured density in the Z-axis as function of formation density and
cement thickness. The SS and LS detector density are used in these
plots to illustrate one form of detector response. Detector count
rates would be too complicated to display in a 3D surface plot.
Similar graphs can also be constructed to demonstrate the
sensitivity of density tool response to cement density. FIG. 20
illustrates schematically an example of a density tool with two
detectors.
[0036] FIG. 9 illustrates schematically how the acoustic tool
interacts with the cased well environment. The development of the
integrated acoustic and density interpretation method of the
present invention realizes the underlying principle wherein both
the acoustic and nuclear sensors measure a common earth model
consisting of the casing and cement as well as part of the
formation. A true wellbore model should satisfy both acoustic and
nuclear transport physics.
[0037] FIG. 5 shows basic steps in one embodiment of the present
inventive method. The two primary input logs to the workflow are
detector count rates recorded by a nuclear density tool 51 and
waveform (pressure) data recorded by an acoustic tool 52. The
acoustic tool includes a small seismic source and one or more
seismic receivers, and may, for example, be the Ultrasonic Imager
Tool (USIT) marketed by Schlumberger. The nuclear density tool
typically has a gamma ray source and at least one gamma ray
detector, and may, for example, be the Litho-Density Tool (LDT)
with two gamma ray detectors marketed by Schlumberger. However, the
standard density processing algorithms in these commercial devices
use raw detector count rates as input and produce formation density
logs that are corrected for borehole effects such as casing,
cement, and mud cake. Shown in FIG. 12 are examples of a
logging-while-drilling (LWD) density image log and a wireline
density log. The LWD density image log, shown on the left in FIG.
12, was acquired while the tool rotated in the wellbore and
describes the azimuthal formation density distribution around the
wellbore, and the density values are color-coded. The wireline
density log, shown on the right in FIG. 12, was acquired while the
tool was pulled with a cable in the well and it describes the
formation density along the tool path, but only at a particular
azimuth. Both the density tool 51 and the acoustic tool 52 may be
adapted to be lowered on a wireline into a wellbore, as shown in
FIG. 4.
[0038] At step 53, a computer is programmed with a nuclear density
algorithm that performs mathematical inversions on detector count
rates recorded with preferably three or more detectors and
calculates casing and cement density and thickness as well as
formation density. At step 54, this information is used to define a
wellbore model with casing and cement properties and thickness.
Schematic examples of wellbore models are shown in FIGS. 13 and 14.
FIG. 13 illustrates a wellbore model constructed with density and
geometric parameters calculated using the nuclear density algorithm
53. These parameters include formation density, cement density and
thickness, and steel casing size and weight. In many cases, casing
parameters are readily available in the well completion data, and
formation density is known from open hole density logs and/or core
data if open hole log data is acquired before the well is cased and
completed. With formation density and casing parameters known, the
number of unknown parameters is reduced to cement density and
thickness only, therefore, the nuclear density solution in step 53
is greatly simplified. FIG. 14 illustrates a wellbore model
constructed with model parameters calculated using nuclear density
in which annulus between casing and formation is partially filled
with fluid, i.e. a poor cement bond. Then, starting with the
predefined wellbore model, the sonic and/or ultrasonic modeling
codes perform (step 54) forward modeling and calculate waveforms.
Without an initial wellbore model 53 that is close to correct, it
would be extremely difficult to model the sonic or acoustic tool
response and update the wellbore model in an efficient manner.
Inversion of the nuclear density log provides the needed good
starting guess. To understand the value of this, assume that FIG.
14 represents close to the actual state of affairs, i.e. the cement
bond is poor. If the iteration cycle 56 were started assuming an
initial guess such as FIG. 13, the convergence process would be
greatly impeded.
[0039] The forward modeling code (54) solves a wave equation
(partial differential equation) governing propagation of acoustic
waves in a medium with boundary conditions (see FIG. 9), given as
input the numerical model from step 53 whose parameters are values
of the physical properties that governs such propagation, i.e. the
acoustic impedances of the borehole fluid, casing, cement, and
formation as well as thickness of casing and cement as described,
for example, in FIGS. 13 and 14. The modeled acoustic response is
further compared with the field waveform data using a predefined
cost function. An example of such acoustic modeling codes is
described in "Numerical simulation of sector bond log and improved
cement bond image," by Song et al. in Geophysics 77, pp. D95-D104,
(July-August 2012). The cost function is used as a convergence tool
to decide whether further iterations are required (step 55). During
each iteration 56, the wellbore model is modified within the
framework of the nuclear density solution range. A converged
solution may require several iterations, depending greatly on the
accuracy of the initial model from step 53. The converged solution
provides (56) cement density and thickness vs. depth in the
wellbore. Together (optionally) with an amplitude and cement
quality map inferred from conventional bond log interpretation and
a neutron log interpretation method disclosed in a related
application, "Method for cement evaluation with neutron logs" (U.S.
Provisional Patent Application No. 61/664,544) the present
inventive method provides an improved cement evaluation indicating
cement bond quality, zone isolation and presence of fluid
contaminations and/or fluid-filled channels. The output of the
embodiment of the present inventive method illustrated in FIG. 5
will be an updated wellbore model as a function of wellbore depth.
Examples of possible output wellbore models at certain depths are,
in addition to FIGS. 13 and 14, shown in FIGS. 15 and 16. As
described above, FIG. 13 illustrates a wellbore fully cemented
casing annulus that has a good bond and provides zone isolation.
FIG. 14 illustrates a wellbore with partially cemented casing
annulus and fluid behind the pipe, and therefore zone isolation is
not achievable. FIG. 15 illustrates a wellbore with a fluid filled
channel behind the casing. FIG. 16 illustrates a wellbore in which
a fraction of the casing annulus is filled with fluid contaminated
cement. Specifically, the output at step 57 will be a computation
of density as a function of radius for different depths in the
wellbore. From these density numbers, the interpreter can conclude
whether the annular region between the formation rock and the
casing is solid cement, fluid contaminated cement, or fluid, and
determine the corresponding volumes and locations of each. The
volumes and locations of channels such as shown in FIGS. 15 and 16
may be computed using the azimuthal geometry data in the final
wellbore model, because the nuclear density and acoustic tools have
directional sensitivity.
[0040] Thus, the present inventive method is a quantitative
analysis method and provides a much more accurate interpretation of
cement quality than traditional cement bond log interpretation
which is qualitative and is subject to individual interpreter's
observations. The final output 57 may also include the sonic
amplitudes/attenuation curves, cement impedance curves and maps.
Although not needed for a cement bond interpretation 60, the output
of the present inventive method may be supplemented by neutron log
results obtained by the method disclosed in the aforementioned
companion patent application entitled "Method for cement evaluation
with neutron logs" (U.S. provisional patent application No.
61/664,544). The neutron log 58 is especially useful when light
weight or foam cement is encountered where both density or acoustic
impedance contrasts between cement and fluid become very small. It
is standard industry practice to run a CBL tool 59; it would be
optional if the present inventive method is being utilized, but it
may add some value to the interpretation 60.
EXAMPLES
[0041] The present inventive method is tested using an ultrasonic
modeling software program that simulates the acoustic waveforms 52.
A well with fully cemented casing and free pipe was selected to
mock up for the modeling study. Actual acoustic data from the
subject well was also obtained. The modeling code essentially
predicts the field measurements that would result in an actual
experiment. The modeled and field ultrasonic waveform data are in
good agreement as shown in FIGS. 10 and 11, proving that the
modeling code works well. For FIG. 10, a fully cemented well with a
good cement bond was modeled and also measured experimentally. FIG.
13 shows a wellbore model with a good cement bond that is very
similar to the model in the calculation. For FIG. 11, free pipe (no
cement at all) was modeled and measured. FIG. 14 shows a wellbore
model with fluid behind casing that is not far off the model used
in the FIG. 11 calculation. The waveforms in FIGS. 10 and 11 depict
the cement bond quality accurately. Good cement bond is
characterized by a high rate of attenuation over time as the wave
amplitudes decay very fast. Free pipe is characterized by little
attenuation as the wave signal rings within the steel casing and
there is very little energy transfer from casing to the fluid
behind the casing.
[0042] The foregoing application is directed to particular
embodiments of the present invention for the purpose of
illustrating it. It will be apparent, however, to one skilled in
the art, that many modifications and variations to the embodiments
described herein are possible. All such modifications and
variations are intended to be within the scope of the present
invention, as defined in the appended claims. Persons skilled in
the art will readily recognize that in preferred embodiments of the
invention, at least some of the steps in the present inventive
method are performed on a computer, i.e. the invention may be
computer implemented. For example, the entire inversion module
(FIG. 5) will be programmed and run on a computer in all practical
applications of the present inventive method.
* * * * *
References