High Resolution AVO NMO bc High Resolution AVO NMO Jon Downton Larry Lines

Theme This paper demonstrates a methodology to produce high-resolution AVO reflectivity attribute estimates similar to sparse spike deconvolution. The AVO estimate is performed prior to NMO thus avoiding NMO stretch distortions and incorporating differential tuning into the forward model. Approach similar to Downton and Lines (2002) but better able to handle tuning

Message High-resolution AVO NMO is better able to estimate reflectors undergoing NMO stretch and differential tuning than traditional AVO performed on a sample-by-sample basis on NMO corrected gathers Provides framework to do elastic impedance inversion similar to poststack sparse spike impedance inversion

Methodology Theory for new algorithm developed using Bayesian formulism Instead of typical assumption of Gaussian priors will assume long tailed distribution Linear operator based on AVO NMO model presented by Downton and Lines (2002) assuming some input wavelet Constraints analogous to the high-resolution Radon transform (Sacchi and Ulrych, 1995) Similar to the poststack sparse spike impedance inversion presented by Sacchi (1999) and Trad (2002) Inversion is nonlinear solved by iterative CG

Outline High-Resolution AVO Prestack Inversion Theory Example 1: model with NMO stretch and differential tuning Example 2: well log model Example 3: Data example Prestack Inversion Example 4

Theory: Convolutional model Use convolutional model as basis of AVO NMO inversion scheme Assumes locally, earth is composed of a series of flat, homogenous, isotropic layers Approximation of Zoeppritz equation used to model how reflectivity changes as a function of offset Ray tracing is done to map relationship between angle of incidence and offset Transmission losses, converted waves and multiples are not incorporated in this model and so must be addressed through pre-processing Gain corrections such as spherical divergence, absorption, directivity and array corrections can be incorporated in this model but are not considered for brevity and simplicity

Theory: Convolutional model AVO NMO formulation (Downton and Lines, 2002) where W is the wavelet, Nk is the NMO operator, F and G are diagonal matrices containing AVO weights, rp is the P-impedance reflectivity series, rf is the fluid stack reflectivity series, and dk is the data at offset k This model may be generalized for more offsets and interfaces as the linear equation Choose this particular parameterization since uncorrelated

Theory: constraints Need to regularize problem Problem is typically underdetermined or ill-conditioned due to the Band-limited nature of the data Differential tuning in the NMO operator Need to regularize problem Choose a weighting function that will treat certain reflection coefficients as being more reliable than others Choosing a long tailed a priori distribution will lead to such a weighting function Good results were obtained with Huber, L1, and Cauchy norms For illustration purposes show weighting function that arises from L1 norm

Theory: constraints Sacchi (1999) shows that for the case when p=1 this leads to a prior of the form where J is the objective function and Q is a diagonal matrix defined as where sm is the L1 standard deviation of the reflectivity and e is some user chosen threshold. Matrix Q must be calculated iteratively in a bootstrap fashion

Theory: Nonlinear inversion The Likelihood function may be combined with the a priori probability function using Bayes’ theorem. The most likely solution leads to the nonlinear equation where Q is regularization parameter, L is the linear operator and =Lm-d two sources of nonlinearity the estimate of the regularization parameter m the calculation of Q Solve in iterative fashion Q improves the condition number significantly

Theory: Numerical Solution Inverse problem is quite large Most efficiently solved using iterative techniques such as conjugate gradient Solving inverse problem requires two nested loops Inner loop conjugate gradient algorithm is used to estimate reflectivity using the previously calculated values of m and Q maximum number of conjugate gradient iterations is used as a trade-off parameter between resolution and reliablity (Skewchuk, 1994 ) Outer loop After solving for the reflectivity the estimate of m and the matrix Q is updated

Outline High-Resolution AVO Prestack Inversion Theory Example 1: model with NMO stretch and differential tuning Example 2: well log model Example 3: Data example Prestack Inversion Example 4

NMO Stretch and Differential tuning Dong (1999) quantified this error for the intercept A and the Gradient B where f is a simple function Conclusions Most of error is in the gradient term Size of error is dependent on B/A ratio Small for class II (A=0) Small for class I (A<<B) Potentially large for class III and IV (A>B) Reflectors on mudrock trend error is small (A<<B)

NMO stretch AVO distortions Offset (m) Generated simple model Generated Class III and IV anomalies Most reflections follow mudrock trend Density follows Gardner relationship Convolution model Reflectivity generated by Aki and Richards equation Ricker wavelet 32.5 Hz central frequency S/N ratio = 4 Constant velocity so simple angle to offset relationships Maximum offset is 4X target depth to give angles out to 65 degrees Use large angles to highlight distortions Time (s)

Traditional Flow: Error due to NMO Offset (m) Time (s) Theoretical errors introduced by NMO correction Systematic error biases S-reflectivity estimate Input (generated with no moveout) NMO corrected data Difference

Traditional Flow (45 degree ) Time (s) Extraction to 45 degrees estimate actual

High Resolution AVO NMO Offset (m) Offset (m) Time (s) Input Reconstruction 5X difference

High Resolution AVO NMO Time (s) 45 degree results S-reflectivity is more noisy than P-reflectivity due to noise amplification resulting in fatter cross-plot Summarize AVO NMO results estimate actual Filtered

High Resolution AVO NMO Time (s) estimate actual filtered unfiltered

Outline High-Resolution AVO Prestack Inversion Theory Example 1: model with NMO stretch and differential tuning Example 2: well log model Example 3: Data example Prestack Inversion Example 4

Example 2: Well log synthetic Offset (m) Offset (m) Generated synthetic based on well log Most reflections follow mudrock trend but significant deviations at Charlie Lake and Halfway Convolution model Reflectivity generated by Zoeppritz equation Wavelet 10/14 – 90/110 Hz S/N ratio = 4 AVO inversion done using angles 0 – 35 degrees

Example 2 Time (s) estimate actual filtered unfiltered

Example 2 Time (s) estimate actual filtered unfiltered

Outline High-Resolution AVO Prestack Inversion Theory Example 1: model with NMO stretch and differential tuning Example 2: well log model Example 3: Colony data example Prestack Inversion Example 4

Constraints even improve zero offset P-impedance section Gas Gas Describe wells Constraints even improve zero offset P-impedance section

Xscl tscl noband Describe wells Gas Gas Describe wells Constraints even improve zero offset P-impedance section Xscl tscl noband

Data example Offset (m) Time (s) Input Reconstruction Difference

Outline High-Resolution AVO Prestack Inversion Theory Example 1: model with NMO stretch and differential tuning Example 2: well log model Example 3: Data example Prestack Inversion Example 4

Theory: well log constraints Relationship between reflectivity r and impedance x Write a series of constraints that specify that the average estimated impedance from the seismic over some time interval must be equal the average impedance of the well log over that same interval with Sw variance Pm=w where P is the linear operator for integration and w is the well log data This leads to an equation of the form which can be solved in an iterative fashion similar to before

Example 4 Time (s) Generated simple model Processing sequence Offset (m) Generated simple model Geologic model sparse reflectivity most reflections follow mudrock trend class III anomaly Density follows Gardner relationship Convolution model Reflectivity generated by the Zoeppritz equation Wavelet 5/10-40/50 Hz S/N ratio = 4 Processing sequence High-resolution AVO NMO inversion Convert sparse spike reflectivity to impedance Time (s)

Example 4 Offset (m) Time (s) Input Reconstruction Difference

Example 4 Time (s) estimate actual filtered unfiltered

Conclusions High-resolution AVO NMO is able to resolve reflectors undergoing NMO stretch and differential tuning is more reliable than the estimates provided by the traditional AVO analysis that is performed on a sample-by-sample basis on NMO corrected gathers greater reliability is due to the classic trade-off between resolution and reliability a few sparse reflectivity values are estimated with greater certainty than the dense reflectivity at every time sample as in the traditional AVO analysis. Provides framework to do elastic impedance inversion similar to poststack sparse spike impedance inversion Algorithm is fast only a few iterations of Conjugate Gradient are needed to get satisfactory results

Acknowledgements The authors would like to thank Core Lab, N.S.E.R.C. and the C.R.E.W.E.S. sponsors for funding this research.