The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

Reduced amplitude and distorted dispersion of seismic waves caused by attenuation, especially strong attenuation, always degrades the resolution of migrated images. To improve image resolution, we evaluated a methodology of compensating for attenuation (∼1∕Q) effects in reverse-time migration (Q-RTM). The Q-RTM approach worked by mitigating the amplitude attenuation and phase dispersion effects in source and receiver wavefields. Source and receiver wavefields were extrapolated using a previously published time-domain viscoacoustic wave equation that offered separated amplitude attenuation and phase dispersion operators. In our Q-RTM implementation, therefore, attenuation- and dispersion-compensated operators were constructed by reversing the sign of attenuation operator and leaving the sign of dispersion operator unchanged, respectively. Further, we designed a low-pass filter for attenuation and dispersion operators to stabilize the compensating procedure. Finally, we tested the Q-RTM approach on a simple layer model and the more realistic BP gas chimney model. Numerical results demonstrated that the Q-RTM approach produced higher resolution images with improved amplitude and phase compared to the noncompensated RTM, particularly beneath high-attenuation zones.

Q-compensated reverse-time migration

We evaluated a time-domain wave equation for modeling acoustic wave propagation in attenuating media. The wave equation was derived from Kjartansson’s constant-Q constitutive stress-strain relation in combination with the mass and momentum conservation equations. Our wave equation, expressed by a second-order temporal derivative and two fractional Laplacian operators, described very nearly constant-Q attenuation and dispersion effects. The advantage of using our formulation of two fractional Laplacians over the traditional fractional time derivative approach was the avoidance of time history memory variables and thus it offered more economic computations. In numerical simulations, we formulated the first-order constitutive equations with the perfectly matched layer absorbing boundaries. The temporal derivative was calculated with a staggered-grid finite-difference approach. The fractional Laplacians are calculated in the spatial frequency domain using a Fourier pseudospectral implementation. We validated our numerical results through comparisons with theoretical constant-Q attenuation and dispersion solutions, field measurements from the Pierre Shale, and results from 2D viscoacoustic analytical modeling for the homogeneous Pierre Shale. We also evaluated different formulations to show separated amplitude loss and dispersion effects on wavefields. Furthermore, we generalized our rigorous formulation for homogeneous media to an approximate equation for viscoacoustic waves in heterogeneous media. We then investigated the accuracy of numerical modeling in attenuating media with different Q-values and its stability in largecontrast heterogeneous media. Finally, we tested the applicability of our time-domain formulation in a heterogeneous medium with high attenuation.

Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians

Gas hydrates grown at gas−ice interfaces were examined by electron microscopy and found to have a submicron porous structure. In situ observations of the formation of porous CH4- and CO2-gas hydrates from deuterated ice Ih powders were made, using time-resolved neutron diffraction on the high-flux diffractometer D20 (ILL, Grenoble) at different pressures and temperatures. For the first time neutron diffraction experiments were also performed with methane in hydrogenated samples. The isotopic differences between H2O and D2O are found insignificant concerning the clathrate formation kinetics. At similar excess fugacities, the reaction of CO2 was distinctly faster than that of CH4. The transient formation of the CO2-hydrate crystal structure II was also observed in coexistence with the usual type-I hydrate reaching a maximum of 5% after 5 h of the reaction at 272 K. A phenomenological model for the kinetics of the gas hydrate formation from ice powders is developed with special account of sample consolidatio...

Formation of porous gas hydrates from ice powders: Diffraction experiments and multistage model

A key step in sparsifying signals is the choice of a sparsity-promoting dictionary. There are two basic approaches to design such a dictionary: the analytic approach and the learning-based approach. Although the analytic approach enjoys the advantage of high efficiency, it lacks adaptivity to various data patterns. On the other hand, the learning-based approach can adaptively sparsify different data sets but has a heavier computational complexity and involves no prior-constraint pattern information for particular data. We have developed a double-sparsity dictionary (DSD) for seismic data to combine the benefits of both approaches. We have evaluated two models to learn the DSD: the synthesis model and the analysis model. The synthesis model learns DSD in the data domain, and the analysis model learns DSD in the model domain. We tested the analysis model and proposed to use the seislet transform and data-driven tight frame (DDTF) as the base transform and adaptive dictionary, respectively, in the DS...

Double Sparsity Dictionary for Seismic Noise Attenuation

A constant-Q wave equation involving fractional Laplacians was recently introduced for viscoacoustic modeling and imaging. This fractional wave equation has a convenient mixed-domain space-wavenumber formulation, which involves the fractional-Laplacian operators with a spatially varying power. We have applied the low-rank approximation to the mixed-domain symbol, which enables a space-variable attenuation specified by the variable fractional power of the Laplacians. Using the proposed approximation scheme, we formulated the framework of the Q-compensated reverse time migration (Q-RTM) for attenuation compensation. Numerical examples using synthetic data demonstrated the improved accuracy of using low-rank wave extrapolation with a constant-Q fractional-Laplacian wave equation for seismic modeling and migration in attenuating media. Low-rank Q-RTM applied to viscoacoustic data is capable of producing images comparable in quality with those produced by conventional RTM from acoustic data.

https://www.jsg.utexas.edu/tyzhu/files/SEG2014_lowrank_CQ.pdf

Viscoacoustic modeling and imaging using low-rank approximation

I have developed and solved the constant-Q model for the attenuation of P- and S-waves in the time domain using a new modeling algorithm based on fractional derivatives. The model requires time derivatives of order m 2 applied to the strain components, where m 0,1,... and 1/tan 1 1/Q, with Q the P-wave or S-wave quality factor. The derivatives are computed with the Grunwald-Letnikov and central-difference fractional approximations, which are extensions of the standard finite-difference operators for derivatives of integer order. The modeling uses the Fourier method to compute the spatial derivatives, and therefore can handle complex geometries and general materialproperty variability. I verified the results by comparison with the 2D analytical solution obtained for wave propagation in homogeneous Pierre Shale. Moreover, the modeling algorithm was used to compute synthetic seismograms in heterogeneous media corresponding to a crosswell seismic experi ment.

/pdf/theory-and-modelling-of-constant-q-p-and-s-waves-using-um5nhegbk0.pdf

Theory and modelling of constant-Q P- and S-waves using fractional spatial derivatives

In this study, we investigate the accuracy of approximating constant‐Q wave propagation by series of Zener or standard linear solid (SLS) mechanisms. Modelling in viscoacoustic and viscoelastic media is implemented in the time domain using the finite‐difference (FD) method. The accuracy of numerical solutions is evaluated by comparison with the analytical solution in homogeneous media. We found that the FD solutions using three SLS relaxation mechanisms as well as a single SLS mechanism, with properly chosen relaxation times, are quite accurate for both weak and strong attenuation. Although the RMS errors of FD simulations using a single relaxation mechanism increase with increasing offset, especially for strong attenuation (Q = 20), the results are still acceptable for practical applications. The synthetic data of the Marmousi‐II model further illustrate that the single SLS mechanism, to model constant Q, is efficient and sufficiently accurate. Moreover, it benefits from less computational costs in computer time and memory.

Tieyuan Zhu

Papers

Q-compensated reverse-time migration

Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians

Viscoacoustic modeling and imaging using low-rank approximation

Theory and modelling of constant-Q P- and S-waves using fractional spatial derivatives

Approximating constant-Q seismic propagation in the time domain