Frequency-Domain Modeling Techniques for the Scalar Wave Equation : An Introduction

Frequency-domain finite-difference (FDFD) modeling offers several advantages over traditional timedomain methods when simulating seismic wave propagation, including a convenient formulation within the context of wavefield inversion and a straight-forward extension for adding complex attenuation mechanisms. In this short paper we introduce the FDFD method, develop a simple solver for the scalar Helmholtz problem, and explore some possible approaches for solving large scale seismic modeling problems in the frequency domain.Massachusetts Institute of Technology. Earth Resources Laborator

Temporal Integration of Seismic Traveltime Tomography

Time-lapse geophysical measurements and seismic imaging methods in particular are powerful techniques for monitoring changes in reservoir properties. Traditional time-lapse processing methods treat each dataset as an independent unit and estimate changes in reservoir state through differencing these separate inversions. We present a general least-squares approach to jointly inverting time-varying property models through use of spatio-temporal coupling operators. Originally developed within the medical imaging community, this extension of traditional Tikhonov regularization allows us to constrain the way in which models vary in time, thereby reducing artifacts observed in traditional time-lapse imaging formulations. The same methodology can also accommodate changes in experiment geometry as a function of time thus allowing inversion of incremental or incomplete surveys. In this case, temporal resolution is traded for improved spatial coverage at individual timesteps. We use seismic traveltime tomography as a model problem although almost any geophysical inversion task can be posed within this formalism. We apply the developed time-lapse inversion algorithm to a synthetic crosswell dataset designed to replicate a CO2 sequestration monitoring experiment

Non-Linear Constraints with Application to Self-Potential Source Inversion

We investigate the use of non-linear constraints for geophysical inverse problems, with specific examples applied to source inversion of self-potential data. Typical regularization methods often produce smooth solutions by introducing a quadratic term in the objective function that minimizes the L2 norm of a low-order differential operator applied to the model. In some cases, however, the properties of interest may not vary smoothly. Two alternative constraints are examined that provide inversion stability while allowing for solutions with non-smooth properties. One method, often referred to as ‘compactness’ or ‘minimum support’, seeks to minimize the area (in 2D) or volume (in 3D) occupied by non-zero model parameters. The second method, ‘total variation’, minimizes an approximation of the L1 norm of the gradient of the model. Both approaches involve a non-linear regularization functional, and must therefore be solved iteratively. We discuss the practical aspects of implementing these regularization methods and compare several examples using self-potential source inversion on a synthetic model. We also apply the compactness constraint for self-potential source inversion using a field data example.Kuwait-MIT Center for Natural Resources and the EnvironmentMassachusetts Institute of Technology. Earth Resources Laborator

Applying Compactness Constraints to Differential Traveltime Tomography

Tomographic imaging problems are typically ill-posed and often require the use of regularization techniques to guarantee a stable solution. Minimization of a weighted norm of model length is one commonly used secondary constraint. Tikhonov methods exploit low-order differential operators to select for solutions that are small, flat, or smooth in one or more dimensions. This class of regularizing functionals may not always be appropriate, particularly in cases where the anomaly being imaged is generated by a non-smooth spatial process. Timelapse imaging of flow-induced velocity anomalies is one such case; flow features are often characterized by spatial compactness or connectivity. By performing inversions on differenced arrival time data, the properties of the timelapse feature can be directly constrained. We develop a differential traveltime tomography algorithm which selects for compact solutions i.e. models with a minimum area of support, through application of model-space iteratively reweighted least squares. Our technique is an adaptation of minimum support regularization methods previously explored within the potential theory community. We compare our inversion algorithm to the results obtained by traditional Tikhonov regularization for two simple synthetic models; one including several sharp localized anomalies and a second with smoother features. We use a more complicated synthetic test case based on multiphase flow results to illustrate the efficacy of compactness constraints for contaminant infiltration imaging. We conclude by applying the algorithm to a CO[subscript 2] sequestration monitoring dataset acquired at the Frio pilot site. We observe that in cases where the assumption of a localized anomaly is correct, the addition of compactness constraints improves image quality by reducing tomographic artifacts and spatial smearing of target features.Massachusetts Institute of Technology. Earth Resources Laborator

Applying Compactness Constraints to Seismic Traveltime Tomography

Tomographic imaging problems are typically ill-posed and often require the use of regularization techniques to guarantee a stable solution. Minimization of a weighted norm of model length is one commonly used secondary constraint. Tikhonov methods exploit low-order differential operators to select for solutions that are small, flat, or smooth in one or more dimensions. This class of regularizing functionals may not always be appropriate, particularly in cases where the anomaly being imaged is generated by a non-smooth spatial process. Timelapse imaging of flow-induced seismic velocity anomalies is one such case; flow features are often characterized by spatial compactness or connectivity. We develop a traveltime tomography algorithm which selects for compact solutions through application of model-space iteratively reweighted least squares. Our technique is an adaptation of minimum support regularization methods previously developed within the potential theory community. We emphasize the application of compactness constraints to timelapse datasets differenced in the data domain, a process which allows recovery of compact perturbations in model properties. We test our inversion algorithm on a simple synthetic dataset generated using a velocity model with several localized velocity anomalies. We then demonstrate the efficacy of the algorithm on a CO2 sequestration monitoring dataset acquired at the Frio pilot site. In both cases, the addition of compactness constraints improves image quality by reducing spatial smearing due to limited angular aperture in the acquisition geometry.Toksoz, M. NafiMassachusetts Institute of Technology. Earth Resources Laborator

Computation of 3D Frequency-Domain Waveform Kernals for c(x,y,z) Media

Seismic tomography, as typically practiced on both the exploration, crustal, and global scales, considers only the arrival times of selected sets of phases and relies primarily on WKBJ theory during inversion. Since the mid 1980’s, researchers have explored, largely on a theoretical level, the possibility of inverting the entire seismic record. Due to the ongoing advances in CPU performance, full waveform inversion is finally becoming feasible on select problems with promising results emerging from frequency-domain methods. However, frequency-domain techniques using sparse direct solvers are currently constrained by memory limitations in 3D where they exhibit a O(n4) worst-case bound on memory usage. We sidestep this limitation by using a hybrid approach, calculating frequency domain Green’s functions for the scalar wave equation by driving a high-order, time-domain, finite-difference (FDTD) code to steady state using a periodic source. The frequency-domain response is extracted using the phase sensitive detection (PSD) method recently developed by Nihei and Li (2006). The resulting algorithm has an O(n3) memory footprint and is amenable to parallelization in the space, shot, or frequency domains. We demonstrate this approach by generating waveform inversion kernels for fully c(x,y,z) models. Our test examples include a realistic VSP experiment using the geometry and velocity models obtained from a site in Western Wyoming, and a deep crustal reflection/refraction profile based on the LARSE II geometry and the SCEC community velocity model. We believe that our 3D solutions to the scalar Helmholtz equation, for models with upwards of 100 million degrees of freedom, are the largest examples documented in the open geophysical literature. Such results suggest that iterative 3D waveform inversion is an achievable goal in the near future.Shell GameChangerMassachusetts Institute of Technology. Earth Resources Laborator

The potential of distributed acoustic sensing (DAS) in teleseismic studies: insights from the Goldstone experiment

Distributed acoustic sensing (DAS) is a recently developed technique that has demonstrated its utility in the oil and gas industry. Here we demonstrate the potential of DAS in teleseismic studies using the Goldstone OpticaL Fiber Seismic experiment in Goldstone, California. By analyzing teleseismic waveforms from the 10 January 2018 M7.5 Honduras earthquake recorded on ~5,000 DAS channels and the nearby broadband station GSC, we first compute receiver functions for DAS channels using the vertical‐component GSC velocity as an approximation for the incident source wavelet. The Moho P‐to‐s conversions are clearly visible on DAS receiver functions. We then derive meter‐scale arrival time measurements along the entire 20‐km‐long array. We are also able to measure path‐averaged Rayleigh wave group velocity and local Rayleigh wave phase velocity. The latter, however, has large uncertainties. Our study suggests that DAS will likely play an important role in many fields of passive seismology in the near future

Continuous active-source seismic monitoring of CO2 injection in abrine aquifer

Continuous crosswell seismic monitoring of a small-scale CO2injection was accomplished with the development of a noveltubing-deployed piezoelectric borehole source. This piezotube source wasdeployed on the CO2 injection tubing, near the top of the saline aquiferreservoir at 1657-m depth, and allowed acquisition of crosswellrecordings at 15-minute intervals during the multiday injection. Thechange in traveltime recorded at various depths in a nearby observationwell allowed hour-by-hour monitoring of the growing CO2 plume via theinduced seismic velocity change. Traveltime changes of 0.2 to 1.0 ms ( upto 8 percent ) were observed, with no change seen at control sensorsplaced above the reservoir. The traveltime measurements indicate that theCO2 plume reached the top of the reservoir sand before reaching theobservation well, where regular fluid sampling was occuring during theinjection, thus providing information about the in situ buoyancy ofCO2

Reactive transport model of sulfur cycling as impacted by perchlorate and nitrate treatments

Microbial souring in oil reservoirs produces toxic, corrosive hydrogen sulfide through microbial sulfate reduction, often accompanying (sea)­water flooding during secondary oil recovery. With data from column experiments as constraints, we developed the first reactive-transport model of a new candidate inhibitor, perchlorate, and compared it with the commonly used inhibitor, nitrate. Our model provided a good fit to the data, which suggest that perchlorate is more effective than nitrate on a per mole of inhibitor basis. Critically, we used our model to gain insight into the underlying competing mechanisms controlling the action of each inhibitor. This analysis suggested that competition by heterotrophic perchlorate reducers and direct inhibition by nitrite produced from heterotrophic nitrate reduction were the most important mechanisms for the perchlorate and nitrate treatments, respectively, in the modeled column experiments. This work demonstrates modeling to be a powerful tool for increasing and testing our understanding of reservoir-souring generation, prevention, and remediation processes, allowing us to incorporate insights derived from laboratory experiments into a framework that can potentially be used to assess risk and design optimal treatment schemes