Exploration
A better way to extract fundamental rock properties with much less noise
Elastic inversion for Lamé parameters: An improved method to extract these parameters from seismic data
David Gray, Veritas DGC, Inc.
Recent successes in determining rock properties such as compressibility and rigidity can be improved upon by a new method. The method extracts the fundamental rock properties expressed by Lamé’s parameters, Lamé’s constant (l) and shear rigidity (m), from prestack seismic data in a way that is simple, direct, more stable and less ambiguous than the method currently used.
INTRODUCTION
It is easier to understand the connection of reservoir properties to fundamental rock properties such as compressibility and rigidity, than it is to understand their connection to traditional seismic attributes, like amplitude and velocity.^{1} Goodway et al.^{2} proposed a method to extract rock properties lr andmr. Lamé’s parameters are: Lamé’s constant (closely related to incompressibility), l, shear modulus, µ, and density, r and are often considered fundamental elastic constants. Goodway’s method has been shown to be generally applicable for exploration and development of reservoirs in various geological settings throughout the world,^{1,3} and for detailed reservoir characterization as well.^{4}
The author proposes an improvement to Goodway’s method, based on poststack inversion methods, and on the Gray, et al.,^{5} reexpression of Lamé’s parameters in Aki and Richards^{6} approximation to the Zoeppritz equation. This new method extracts l andm without the ambiguity introduced by the density parameter, r, inlr andmr. Even more significant, the new method should also be more stable statistically. Therefore, this new method is an improvement on Goodway’s method, which has already been used successfully in many reservoirs.
THE METHOD
Gray, et al., reexpressed Lamé’s parameters in Aki and Richards approximation to the Zoeppritz equation in terms of the parameters Dl/l, Dm/m and Dr/r; that is, the reflectivity of Lamé’s constant, the shear modulus reflectivity and density reflectivity, respectively. Amplitude vs. Offset (AVO) analysis using this equation allows the reflectivities Dl/l, Dm/m to be extracted from conventional, prestack, Pwave seismic data. These reflectivities can be inverted using poststack amplitude inversion to derive the individual, fundamental rock properties l andm from conventional seismic data.
It is possible to solve for the individual parameters using this new method. Their interpretation is less ambiguous than that of lr andmr because of elimination of the density term, r. Goodway’s method calculates lr from the squares of the Pimpedance (Ip) and the Simpedance (Is) using subtraction. These impedances are generated from seismic data and are therefore subject to measurement error. If it is assumed that the measurements of these impedances have a normal distribution, then it can be shown that squaring them introduces a bias into the results lr andmr that is approximately equal in magnitude to the variance of Is.
In addition, taking the square of these measurements approximately halves the signaltonoise (S/N) ratio. Since the squares of the impedances are positive, subtracting them to calculate lr increases its potential error. In fact, it can be shown (see sidebar) that the error associated with lr is greater than two times that associated with mr and, therefore, about four times greater than the error associated with Is.
The new approach derives l by inverting for it directly from the Dl/l, derived from Gray’s AVO equation. The same procedure is followed for the calculation of m from Dm/m. Since squaring is no longer required, then there is no bias in the result and the S/N ratio does not get worse. Since no subtraction is required to calculate lr, its potential error should be less than Goodway’s lr.
This presentation shows Gray’s result inverted directly for l and m for the synthetic data used in Gray, et al.^{5} Comparisons of the inversions to correct values of l andm derived from the logs are shown for these data. The new method is also tested on real seismic data containing both clastic and carbonate sequences from Erskine, Alberta, Canada. For these data, the new method is compared to Goodway’s to determine which method has a better S/N ratio, and the results are compared to l andm logs calculated for wells in this reservoir to test it for accuracy.
RESULTS
Fig. 1. The most striking observation is the comparison between Goodway’s lr andl calculated using the new method. Here, it is clear that lr is much noisier than l.

Fig. 1. On the left is lr, calculated using Goodway’s method. On the right is l, calculated using the new direct method. As expected, the lr plot is noisier than the new inversion for l. Inserted in the sections are lr and l logs at two well locations for comparison.


Fig. 2. This demonstrates elastic inversion derived by the new method. It is visual confirmation of the statistical result, derived in the sidebar, showing that the variance of lr should be about four times that of Is, while the variance of l should be close to that of Is. Additional benefits accrue from not having to deal with the density term, r, and from not having a bias in the answer for real seismic data.

Fig. 2. On the left is the new direct inversion for l from Dl/l, the l reflectivity derived from Gray‘s equation on synthetic data. Inserted is an exact l log calculated from the Pwave sonic, Swave sonic and density curves for this well. On the right is an example of one of the possibilities derived from using this method, an inversion for compressibility with the bulk modulus (1/compressibility) log inserted.


One of the benefits of removing the density term is that the results are isolated elastic constants. Therefore, other elastic constants can be calculated from them. In Fig. 2, synthetic prestack Pwave data are inverted using the new method. On the left is an inversion for l compared to l calculated directly from logs. On the right is a compressibility section derived from the reciprocal of the bulk modulus, derived using the new method, from its reflectivity as calculated from Gray et al’s Equation 1.^{5} The inversion for compressibility can be done because the direct inversion method inverts for individual elastic parameters. Information derived from the inversion for compressibility may be of interest to reservoir engineers.




Mathematical estimation of signal to noise ration
Using Goodway, et al.’s (1997) notation:
Assuming that measurements of the P and Swave impedances, Îp and Îs, follow normal distributions,
and with means I_{p} and I_{s} and variances s_{p}^{2} and s_{s}^{2} respectively, then their distributions can be represented as follows:
Therefore their expectations are:
If s_{p} s_{s,} then:
Using the Moment Generating Function for a random variable, x, distributed by a Normal Distribution,^{7} then the expectations of powers of x are:
Since e_{s}~N(0,s_{s}^{2}) and e_{p}~N(0,s_{p}^{2}), then the variances of mr andlr can be calculated:
Similarly:
Assuming that I_{p}^{2} and I_{s}^{2} are independent random variables:
If s_{p} s_{s}, then:






CONCLUSIONS
A new method of extracting the fundamental rock properties, Lamé’s parameters, l andm, by poststack inversion of their reflectivities derived from conventional, prestack, Pwave seismic data using Gray’s AVO equation is demonstrated. This new method successfully predicts l and m, producing results that are similar to l and m logs. It produces more stable results than can presently be achieved from the method of Goodway, improving the S/N by a factor of two for µ and four for l. It also avoids ambiguity caused by the density component, r, in lr and mr. The absence of the density component allows other elastic parameters, such as compressibility, to be calculated from the results of the new method. As a result, this new method should be considered as an important extension of Goodway’s method.
LITERATURE CITED
^{1} Gray, F. D. and E. A. Andersen, “The application of AVO and inversion to formation properties," World Oil, Vol. 221, No. 7., 2000.
^{2} Goodway, W., Chen, T. and J. Downton, “Improved AVO fluid detection and lithology discrimination using Lamé petrophysical parameters; ‘LambdaRho’, ‘MuRho’, & ‘Lambda/Mu fluid stack’ from P and S inversions,” 67th Annual Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, pp. 183 – 186, 1997.
^{3} Soldo, J., et al., “3D AVO and seismic inversion in María Inés Oeste field, Santa Cruz, Argentina, a case study,” 71st Annual Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 2001.
^{4} Chen, T., et al., “Integrating geophysics, geology and petrophysics: A 3D seismic AVO and borehole/logging case study,” 68th Annual Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, pp. 615 – 618, 1998.
^{5} Gray, F. D., Goodway, W. N. and T. Chen, “Bridging the gap: Using AVO to detect changes in fundamental elastic constants,” 69th Annual Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, pp. 852 – 855, 1999.
^{6} Aki, K. and P. G. Richards, “Quantitative seismology,” W.H. Freeman and Co., 1980.
^{7} Hogg, R. V. and A. T. Craig, “Introduction to mathematical statistics,” 4th Edition, MacMillan Publishing, Inc., 1978.
THE AUTHOR


David Gray is a research geophysicist and AVO specialist for Veritas in Calgary, Canada. He received a BS in honours geophysics from the University of Western Ontario in 1984 and a masters of mathematics in statistics from the University of Waterloo in 1989. He joined Veritas in 1988 where he has held positions as processor, AVO specialist and research programmer. He is presently developing applications for Veritas Exploration Services, to provide advanced interpretation and reservoir services. His areas of interest are in extracting more information from Pwave seismic by using AVO for derivation of elastic rock properties and fracture detection. He has authored/coauthored more than 50 papers on AVO techniques and methodology, which have been published in The Leading Edge, World Oil, the CSEG Recorder and the AAPG Explorer, and presented at numerous meetings. He has served on technical committees for the EAGE, SEG and CSEG. Mr. Gray is a professional geophysicist certified by APEGGA, and he is a member of the CSEG, SEG, EAGE and SPE.



