This invention generally relates to seismic data processing. More specifically, the present invention relates to estimation of azimuthal amplitude gradient using correlation of seismic attributes within a sliding volume of seismic data.
In reflection seismology, azimuthal amplitude gradient represents the change in reflected seismic amplitude with respect to source-receiver azimuth. It is often desirable to use the azimuthal amplitude gradient, derived from seismic data, to determine orientation and magnitude of reservoir fractures and of geologic stress fields. Conventionally, azimuthal amplitude gradient is typically estimated at each point separately in a seismic survey. However, this point-wise estimation approach can be limited by various factors such as noise, data misalignments at different azimuths, and ambiguity between gradient polarity and symmetry azimuth.
One method of determining or estimating azimuthal amplitude gradient is known as amplitude variations with azimuth (AVAZ). AVAZ analysis takes advantage of the fact that rocks often exhibit different properties along paths of maximum and minimum stress and/or along paths parallel and perpendicular to fractures. A fundamental ambiguity in AVAZ analysis has been recognized for some time and can be seen in
This invention computes the azimuthally-varying component of seismic amplitudes in a more stable and robust way as compared to conventional point-wise method generally in use today.
One example a method for processing 3-D seismic data includes: a) forming, using a computing processor, common image location gathers of seismic traces as a function of time or depth, offset, and azimuth; b) performing a linear regression on a gather at each time or depth to generate a model that includes an intercept term A, an average gradient term Bavg, an anisotropic gradient term Bcos, and an anisotropic gradient term Bsin; c) calculating joint correlations of Bcos and Bsin seismic traces over a plurality of rectangular prisms in lateral position and time or depth, each prism centered about its own analysis point; d) forming a vector whose rectangular coordinates are (M, R) for each analysis point, wherein M represent half of difference between mean-squared amplitudes of Bcos and Bsin within the prism, and R represent the average product of Bcos and Bsin within the prism; e) calculating angles 2θup or 2θdown the vector makes with the M axis; f) taking one-half the rotation angles θup or θdown as symmetry azimuths of vertical reservoir fractures; and g) determining azimuthal amplitude gradient based on the determined symmetry azimuth.
Another example of a method for processing 3-D seismic data includes: a) forming common image location gathers of seismic traces as a function of time or depth, offset, and azimuth; b) performing a linear regression, using a computing processor, on a gather at each time or depth to generate a model that includes an intercept term A, an average gradient term Bavg, an anisotropic gradient term Bcos, and an anisotropic gradient term Bsin; c) calculating joint correlations of Bcos and Bsin seismic traces over a plurality of rectangular prisms in lateral position and time or depth, each prism centered about its own analysis point; d) forming a vector whose rectangular coordinates are (M, R) for each analysis point, wherein M represent half of difference between mean-squared amplitudes of Bcos and Bsin within the prism, and R represent the average product of Bcos and Bsin within the prism; e) calculating angles 2θup or 2θdown the vector makes with the M axis; f) determining one-half the rotation angles θup or θdown as symmetry azimuths of vertical reservoir fractures; g) determining azimuthal amplitude gradient based on the determined symmetry azimuth; and h) detecting one or more subsurface heterogeneity utilizing the determined azimuthal amplitude gradient.
A more complete understanding of the present invention and benefits thereof may be acquired by referring to the follow description taken in conjunction with the accompanying drawings in which:
Turning now to the detailed description of the preferred arrangement or arrangements of the present invention, it should be understood that the inventive features and concepts may be manifested in other arrangements and that the scope of the invention is not limited to the embodiments described or illustrated.
The following examples of certain embodiments of the invention are given. Each example is provided by way of explanation of the invention, one of many embodiments of the invention, and the following examples should not be read to limit the scope of the invention.
The present invention provides tools and methods for estimating azimuthal amplitude gradient using correlation of seismic attributes within a sliding volume of seismic data (“windowed statistical method”). As used herein, “azimuthal amplitude gradient” represents a change in seismic amplitude with respect to source-receiver azimuth or orientation of source-receiver line. In one embodiment, a new method of performing AVAZ analysis is provided. One of the goals of this method is to obtain estimates of symmetry azimuth αsym that is more stable in the presence of noise and small misalignments of data from one azimuth to the next and to resolve the ambiguity between gradient polarity and symmetry azimuth. These estimations can be used to detect subsurface heterogeneities such as, but not limited to, faults, fractures, stratigraphic discontinuities, and the like. Other advantages will be apparent from the disclosure herein.
In combination with well information or prior geologic constraints, the estimation azimuthal amplitude gradient can infer the orientations and magnitudes of fractures away from the well. Such information is valuable for developing oil and gas resources, particularly where artificial fracturing of reservoir rock is required. Moreover, the present invention can utilize well control and/or prior geologic information to resolve fundamental ambiguity problem generally encountered in AVAZ analysis.
Ambiguity of AVAZ Analysis
Rp(α,ψ)=A+B(α)sin2ψ (1)
Constant term A is normal-incidence reflection coefficient, which equals half of the fractional change of acoustic impedance across the interface. The term A depends neither on the azimuth nor the incidence angle. Coefficient B(α) is the amplitude gradient, and it specifies how rapidly the reflected amplitude changes with incidence angle ψ at any azimuth α. The amplitude gradient has the following two components:
B(α)=Biso+Bani cos2(α−αsym) (2)
In equation (2), αsym represents symmetry azimuth of the fracture sets, as shown in
Fracture Detection by AVAZ Analysis Using the Standard Point-Wise Method
In order to use a linear regression to extract the desired parameters from equation (2), namely Biso, Bani and αsym, the terms can be linearized as a sum coefficients to be determined, each multiplied by a fixed basis function that is independent of the data:
B(α)=Bavg+Bcos cos 2α+Bsin sin 2α (5)
where Bavg is an average over all azimuths and where
Bcos=½Bani cos 2αsym; Bsin=½Bani sin 2αsym (6)
Equation (5) is convenient in that three basis functions (1, cos α, sin α) are mutually orthogonal over full range of azimuths. A vector of parameters
{right arrow over (X)}=[A,Bavg,Bcos,Bsin]T (7)
is estimated from linear regression using following equation:
{right arrow over (X)}≈(Σi{right arrow over (g)}i{right arrow over (g)}iT+ξl)−1(Σidi{right arrow over (g)}i) (8)
where I is the identity matrix, {right arrow over (g)}i is the vector of basis functions of the i-th seismic sample di evaluated at its imaged point,
{right arrow over (g)}i=[1,sin2 ψi,cos 2αi sin2 ψi,sin 2αi sin2 ψi]T (9)
and ξ is a stability factory sometimes needed to constrain L-2 norm of {right arrow over (X)}.
It can be seen from equation (5) that the azimuthal gradient is a vector quantity that takes two numbers to describe it at every reflection time and position. It can be expressed in Cartesian coordinates aligned with the seismic survey coordinates as it {right arrow over (u)}=(a, b), where a and b to be the Bcos and Bsin elements of the regression vector {right arrow over (X)}. It can also be expressed in polar coordinates as pointwise magnitude Bmag=2√{square root over (a2+b2)} and azimuth of the maximum signed amplitude gradient, αmax. This azimuth can then be found as half the counter-clockwise angle the {right arrow over (u)} vector makes with the α-axis, as shown in
Biso=Bavg−½Bani (10)
Seismic data will generally have no zero-frequency component. Whatever reflections present in the earth are convolved by a zero-mean wavelet, which has equal positive and negative areas. For example, an interface having a positive Bani coefficient with a zero-phase wavelet would yield value correct of αsym in central wavelet lobe, while an azimuth that is ±90° opposite αsym would appear in the negative side lobes of the wavelet. Since this literal interpretation of azimuth is unrealistic from a geologic perspective, it becomes critical to ensure that the polarity of Bcos remains fixed through horizon extractions of αmax or else striping can occur.
a. αup=αmax if |αmax|<45°
b. αup=αmax−90° if αmax≥45°
c. αup=αmax+90° if αmax≤45°
A corresponding spin-up gradient (Bup) can be defined as |{right arrow over (u)}| when |αmax|<45° and −|{right arrow over (u)}| otherwise.
Referring to
a. αdown=αup−90° if αup>0
b. αdown=αup+90° if αup<0
Synthetic Data Example
To obtain a more realistic example, synthetic data for a 1-D model was generated. 141 synthetic traces with random source-receiver azimuths and offsets ranging from 0 to 3.1 km were selected. These data underwent azimuthal velocity analysis using trim statistics to align events to be as flat as possible at each time sample. The data were top muted to an angle of about 45° and a 20 db/s gain was applied. Every other one of the resulting traces are shown in
Comparing these results with theoretical ones computed from equation 4 and listed in Table 1, there is a qualitative (but not quantitative) agreement of the average and spin-up gradient estimates. The lack of quantitative agreement may be due to computation of spherical divergence and incidence angle, which assumes isotropic media. Estimates of spin-up and maximum-amplitude azimuths were very unstable, taking on virtually every possible value, depending precisely on where they were sampled. This instability may be due to slight time misalignments of the seismic data at different source-receiver azimuths and/or slightly amount of noise that is present in the synthetic data.
Windowed Statistical Method
The windowed statistical method of performing AVAZ analysis can obtain estimates of symmetry azimuth αsym that is more stable in the presence of noise and small misalignments of the data from one azimuth to the next as well as resolve the ambiguity problem. For a single interface with ideal data, a crossplot of these Bsin versus Bcos estimates would form a straight line, as the band-limited Bani varies over positive and negative values. The slope of this line is affected by the symmetry azimuth, as shown in
More realistically, a crossplot of Bsin versus Bcos over an arbitrary window of data would produce an approximate elliptical distribution of points, similar to ellipse shown in
An input vector can be formed from Bcos and Bsin input traces:
The joint correlation matrix of {right arrow over (s)} is the expected value of {right arrow over (s)}{right arrow over (s)}T averaged over a sliding time-space window. This correlation is the 2×2 matrix Φs, which is computed as
whose elements are slowly-varying functions of time and space. They are computed by convolving the data in a super-gather by the weighting functions described.
A unitary matrix, U†, can be found which transforms input signal vector {right arrow over (s)} into a new predicted anisotropy signal
{right arrow over (s)}′=U†{right arrow over (s)}=[a′b′]T (13)
whose components a′ and b′ are uncorrelated with each other. The stronger prediction component, a′, referred to as the background while the weaker component, b′, is referred to as the anomaly. The correlation matrix Φ{right arrow over (s′)} of the predicted signal is given by
It is required that the unitary transform U† make σa′≥σb′ and R′=0.
Real Solutions
If a(t) and b(t) are real seismic traces, with no quadrature (imaginary) component, joint statistics can be computed as follows:
where the sums are over a sliding window. The statistics P is defined to be the average power in a and b, and M to be half the difference in powers between a and b:
P=½(σa2+σb2) (18)
M=½(σa2−σb2) (19)
A coordinate rotation by an angle of θ can be obtained by pre-multiplying the signal vector {right arrow over (s)} by the unitary matrix
where 2θ is the angle (M, R) vector makes with the Maxis. As shown in
2θ=a tan 2[R,M]−2nπ (21)
where a tan 2 [y, x] is the clockwise angle from the x-axis to the vector (y, x) and where n is any integer. If n is an even number, θ is said to be a spin-up solution. If it is odd, it is said to be a spin-down solution. Since all solutions having the same parity are equivalent, n=0 can be used for the spin-up solution and n=sign (θup) for the spin-down solution. These choices ensure that both up and down rotation angles will always be between ±180° as shown in
Complex Solutions
The real solution can suffer from several limitations. Slight misalignments in the Bcos and Bsin traces can lead to large variations in the computed azimuths. Due to the narrow-band nature of seismic data, these time differences can manifest themselves as approximate phase difference between these traces. To overcome this difficulty, equation (11) can be generalized to:
Where Hi{ } denotes the 90° phase-rotated signal (also known as the Hilbert transform). The complex correlation matrix
and where aj and bj are samples of the continuous functions a(t) and b(t). In the complex case, the matrix U†, which diagonalizes Φ{right arrow over (S)} and makes R′=0 in equation (14), is not unique. However, a preferred embodiment chooses a unitary matrix of the form
Note that it is a function of two angles, θ and ϕ. The latter represents half the phase difference between Bcos and Bsin. To diagonalize Φ{right arrow over (S)}
2ϕ=a tan 2[Risign(Rr),|Rr|]−2mπ (26)
2θ=a tan 2[|R|sign(Rr),M]−2nπ (27)
The result is an exact complex solution. If the exact anisotropic power Δe is denoted to be the length of the (M, R) vector, then eigenvalues {σa′, σb′} of the predicted correlation matrix Φ{right arrow over (S′)} (which will match those of the original correlation matrix Φ{right arrow over (S)}) are P+Δ and P−Δ, respectively. The Cauchy-Schwarz inequality establishes that |R|2≤σ2aσ2b, so
Δe=√{square root over (|R|2+M2)}≤√{square root over (σa2σb2+M2)}=P (28)
If the coefficient of anisotropy Ca is denoted to be Δe/P, then value of Ca lies between 0 and 1. The magnitude of anisotropy Bmag is √{square root over (2Δe)}.
For m=0, value ϕ is restricted to lie within range ±45°. This means that actual phase difference that is tolerated is ±90°. If actual phase difference gradually exceeds this, it will result in a sudden discontinuity in the estimate of θ. This is a branch line in the ϕ direction. Since such discontinuities are generally undesirable, an alternative solution can be expressed as
2ϕ=sin−1(Ri/P) (29)
2θ=a tan 2[Rr,M cos 2ϕ]−2nπ (30)
These solutions yield identical results as the exact solution given noise-free data, but gradually drive the estimate of the phase difference ϕ to zero as the noise level increases. Phase differences between the a and b traces approaching 45° masquerade as isotropic zones. Without determining spin boundaries in the phase direction, it is difficult or impossible to eliminate discontinuities in the a′ and b′ gradients. For this reason, a modified definition of anisotropic power may be preferable:
Δr=√{square root over (Rr2+M2 cos2 2ϕ)}=√{square root over (Rr2+Mr2)} (31)
where Mr=M cos 2ϕ is the robust power difference in a and b, taking into account possible phase differences between them. It becomes clear that Δr≤Δe≤P. The spin-up azimuth αup is found to be ½θup and the spin-down azimuth αdown is ½θdown=αup−½πsign(αup).
Referring to
Regions Analysis
The rotation angle θ and the symmetry azimuth a are obtained from the windowed statistic method by computing the angle of the {right arrow over (S)}=(Mr, Rr) statistics vector shown in the upper right portion
PS=½(σa2,σb2)+[other term] (32)
The 3-D seismic volume can then be subdivided into a set of contiguous interconnected regions, each of which resides within a single zone in the (Mr, Rr) plane (
Valid boundaries between regions are those that do not skip across Zone 0. These would include, for example, a boundary between Zones 1 and −1 but not between Zones 1 and −2. If too many boundaries are not valid, the (Mr, Rr) plane may have to be subdivided again using a larger radius for the incoherent zone. Chains of coherent regions that are connected through valid boundaries are then assembled and recursively inspected for parity violations. A parity violation occurs if any circular pathway through the 3-D volume passes through an odd number of valid coherent boundaries. For each parity violation, one of the boundaries can be designated as weakest. The weakest boundary could be one having smallest surface area or smallest average coherence. Conflicts may be resolved or avoided by marking the smaller of the two regions across the weakest boundary as being incoherent. If this eliminates too much of the 3-D volume, an alternative is to eliminate the weakest boundary although this can cause that boundary to retain its abrupt change in polarity of Bback and Banon and cause the estimation of symmetry azimuth to suddenly shift by 90°.
Starting from an arbitrary region in the chain, initial spins are consistently assigned to the coherent regions contained in the chain so that a reversal in spin occurs only when a valid spin-flip boundary is crossed from either direction. Spin-flip boundaries are shown as dashed lines in
In this case, the preferred azimuth θp=0° and the width of the indicator was X=1°. This indicator is used to determine when branch line opposite the preferred azimuth (180° in this case) is crossed. It will suddenly switch polarity when this happens (i.e., at spin-reversal points 544 and 594 ms). The trace labeled “CA” represents the Coefficient of Anisotropy. This is always a positive number between 0 and 1, which denotes the normalized anisotropy magnitude, CA=Δr/Ps. Additional term included in equation (32) was 5% of the average AVO gradient Bavg smoothed over a 1000 ms window. This was done so as to cause the CA to fall to zero in the no-data zones between events. Minimum threshold of anisotropy (k) was set to 0.1 which caused each reflector to be placed into a separate coherent region. The trace labeled αsym is the estimate of the symmetry azimuth. Once the azimuth is estimated, it can be used as a component of the azimuthal gradient in its direction (Bani) and in its perpendicular direction (Banom).
In comparing the right-most traces of
In closing, it should be noted that the discussion of any reference is not an admission that it is prior art to the present invention, especially any reference that may have a publication date after the priority date of this application. At the same time, each and every claim below is hereby incorporated into this detailed description or specification as additional embodiments of the present invention.
Although the systems and processes described herein have been described in detail, it should be understood that various changes, substitutions, and alterations can be made without departing from the spirit and scope of the invention as defined by the following claims. Those skilled in the art may be able to study the preferred embodiments and identify other ways to practice the invention that are not exactly as described herein. It is the intent of the inventors that variations and equivalents of the invention are within the scope of the claims, while the description, abstract and drawings are not to be used to limit the scope of the invention. The invention is specifically intended to be as broad as the claims below and their equivalents.
All of the references cited herein are expressly incorporated by reference. The discussion of any reference is not an admission that it is prior art to the present invention, especially any reference that may have a publication data after the priority date of this application. Incorporated references are listed again here for convenience:
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2. Davidson, M., Swan, H., Sil, S., Howell, J., Olson, R., and Zhou, C., 2011, A robust workflow for detecting azimuthal anisotropy: SEG, Expanded Abstracts, 30, no. 1, 259-263.
3. Davison, C., Ratcliffe, A., Grion, S., Johnston, R., Duque, C., and Maharramov, M., 2011, Azimuthal AVO analysis: Stabilizing themodel parameters: SEG, Expanded Abstracts, 30, no. 1, 330-334.
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5. Downton, J., Roure, B., and Hunt, L., 2011, Azimuthal Fourier coefficients: CSEG Recorder, 36, no. 10, 22-36.
6. Gray, D., and Head, K., 2000a, Fracture detection in Manderson field: A 3-D AVAZ case history: The Leading Edge, 19, no. 11, 1214-1221.
7. Gray, D., Roberts, G., and Head, K., 2002, Recent advances in determination of fracture strike and crack density from P-wave seismic data: The Leading Edge, 21, no. 3, 280-285.
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12. Mallick, S., Chambers, R. E., and Gonzalez, A., 1996, Method for determining the principal axes of azimuthal anisotropy from P-wave data: U.S. Pat. No. 5,508,973.
13. Mallick, S., Craft, K. L., Meister, L. J., and Chambers, R. E., 1998, Determination of the principal directions of azimuthal anisotropy from P-wave seismic data: Geophysics, 63, no. 2, 692-706.
14. Rüger, A., and Tsvankin, I., 1997, Using AVO for fracture detection: Analytic basis and practical solutions: The Leading Edge, 16, no. 10, 1429-1434.
15. Rüger, A., 1995, P-wave reflection coefficients for transversely isotropic media with vertical and horizontal axis of symmetry: SEG, Expanded Abstracts, 14, 278-281.
16. Rüger, A., 1997, Plane wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry: Geophysics, 62, no. 3, 713-722.
17. Rüger, A., 2000, Variation of P-wave reflectivity with offset and azimuth in anisotropic media: SEG Books, 20, 277-289.
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19. Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, no. 10, 1954-1966.
20. Tsvankin, I., 1996, Pwave signatures and notation for transversely isotropic media: An overview: Geophysics, 61, no. 2, 467-483.
21. Wang, J., Zheng, Y., and Perz, M., 2007, VVAZ vs. AVAZ: Practical implementation and comparison of two fracture-detection methods: SEG, Expanded Abstracts, 26, no. 1, 189-193.
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This application is a non-provisional application which claims benefit under 35 USC § 119(e) to U.S. Provisional Application Ser. No. 62/181,009 filed Jun. 17, 2015, entitled “SEISMIC AZIMUTHAL GRADIENT ESTIMATION,” which is incorporated herein in its entirety.
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Al-Shuhail, A. A., 2007, Fracture-porosity inversion from P-wave AVOA data along 2D seismic lines: An example from the Austin chalk of southeast Texas: Geophysics, 72, No. 1, B1-B7. |
Davidson, M., Swan, H., Sil, S., Howell, J., Olson, R., and Zhou, C., 2011, A robust workflow for detecting azimuthal anisotropy: SEG, Expanded Abstracts, 30, No. 1, 259-263. |
Davison, C., Ratcliffe, A., Grion, S., Johnston, R., Duque, C., and Maharramov, M., 2011, Azimuthal AVO analysis: Stabilizing themodel parameters: SEG, Expanded Abstracts, 30, No. 1, 330-334. |
Downton, J., and Gray, D., 2006, AVAZ parameter uncertainty estimation: SEG, Expanded Abstracts, 25, No. 1, 234-238. |
Downton, J., Roure, B., and Hunt, L., 2011, Azimuthal Fourier coefficients: CSEG Recorder, 36, No. 10, 22-36. |
Gray, D., and Head, K., 2000a, Fracture detection in Manderson field: A 3-D AVAZ case history: The Leading Edge, 19, No. 11, 1214-1221. |
Gray, D., Roberts, G., and Head K., 2002, Recent advances in determination of fracture strike and crack density from P-wave seismic data: The Leading Edge, 21, No. 3, 280-285. |
Jenner, E., 2002, Azimuthal AVO: Methodology and data examples: The Leading Edge, 21, No. 8, 782-786. |
Mallick, S., and Frazer, L. N., 1987, Practical aspects of reflectivity modeling: Geophysics, 52, No. 10, 1355-1364. |
Mallick, S., Craft, K. L., Meister, L. J., and Chambers, R. E., 1998, Determination of the principal directions of azimuthal anisotropy from P-wave seismic data: Geophysics, 63, No. 2, 692-706. |
Ruger, A., and Tsvankin, I., 1997, Using AVO for fracture detection: Analytic basis and practical solutions: The Leading Edge, 16, No. 10, 1429-1434. |
Ruger, A., 1995, P-wave reflection coefficients for transversely isotropic media with vertical and horizontal axis of symmetry: SEG, Expanded Abstracts, 14, 278-281. |
Ruger, A., 1997, Plane wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry: Geophysics, 62, No. 3, 713-722. |
Ruger, A., 2000, Variation of P-wave reflectivity with offset and azimuth in anisotropic media: Geophysics, vol. 63, No. 3, p. 935-947 (1998). |
Swan, H. W., 1990, Amplitude versus offset measurement errors in a finely layered medium: Geophysics, 56, No. 1, 41-49. |
Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, No. 10, 1954-1966. |
Tsvankin, I., 1996, P-wave signatures and notation for transversely isotropic media: An overview: Geophysics, 61, No. 2, 467-483. |
Wang, J., Zheng, Y., and Perz, M., 2007, VVAZ vs. AVAZ: Practical implementation and comparison of two fracture-detection methods: SEG, Expanded Abstracts, 26, No. 1, 189-193. |
Whitcombe, D. N., Dyce, M., McKenzie, C. J. S., and Hoeber, H., 2004, Stabilizing the AVO gradient: SEG, Expanded Abstracts, 23, No. 1, 232-235. |
Xu, X., and Tsvankin, I., 2006, Azimuthal AVO analysis with anisotropic spreading correction: A synthetic study: The Leading Edge, 25, No. 11, 1336-1342. |
Zheng, Y., and Larson, G., 2004, Seismic fracture detection: Ambiguity and practical solution: SEG, Expanded Abstracts, 23, No. 1, 1575-1578. |
Veeken, P.C.H., et al., “AVO attribute analsyis and seismic reservoir characterization.” In: First Break. Feb. 2006 (Feb. 2006) Retrieved from <http://kms.geo.net/fileadmin/docs/AVO_attribute_analysis_and seismic reservoir characterization.pdf> entire document. |
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20170031042 A1 | Feb 2017 | US |
Number | Date | Country | |
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62181009 | Jun 2015 | US |