Modern visualization can be formulated as inversion problems that aim to obtain structural information about a complex medium through wave excitations. However, without numerically efficient forward calculations, even state-of-the-art inversion procedures are too computationally intensive to implement. We adapt a method previously used to treat transport in electronic waveguides to describe acoustic wave motion in complex media with high gains in computational time. The method consists of describing the system as if it was made of disconnected parts that are patched together. By expressing the system in this manner, wave-propagation calculations that otherwise would involve a very large matrix can be done with considerably smaller matrices instead. In particular, by treating one of such patches as a target whose parameters are changeable, we are able to implement target-oriented optimization in which the model parameters can be continuously refined until the ideal result is reproduced. The so-called Patched Green's function (PGF) approach is mathematically exact and involves no approximations, thus improving the computational cost without compromising accuracy. Given the generality of our method, it can be applied to a wide variety of inversion problems. Here we apply it to the case of seismic modeling where acoustic waves are used to map the earth subsurface in order to identify and explore mineral resources. The technique is tested with realistic seismic models and compared to standard calculation methods. The reduction in computational complexity is remarkable and paves the way to treating larger systems with increasing accuracy levels.
We have designed a workflow to apply full-waveform inversion (FWI) to a new type of acquisition recorded on ocean-bottom nodes (OBNs) in the Brazilian presalt area. The data set consists of three large-radius concentric circles of seismic source points recorded on OBN placed within the area of the circles. This geometry provides full azimuth distribution and benefits from long offsets in which the effect of diving waves is strong. These diving waves carry information about the presalt targets of interest. In the FWI workflow, we retrieve the P-wave velocity using the density as a propagation parameter. We test three objective functions: ℓ2-norm, ℓ1-norm, and nonparametric residuals objective function (NPR-FWI), which have different abilities to handle low signal-to-noise ratio data. In our inversion tests, the ℓ1-norm and NPR-FWI retrieve more information about the presalt reservoir than the ℓ1-norm. We find through several inversion experiments that we can recover structures related to the presalt reservoir, demonstrating the potential for this new type of OBN circular source geometry. We also discuss implications for low-cost data acquisition that take advantage of the sparse nature of sources and receivers needed in this survey geometry, including its extension to time-lapse target-oriented reservoir monitoring.
A shot-encoding technique can be used in seismic waveform inversion to significantly reduce the computational cost by reducing the number of seismic simulations in the inversion procedure. Here we developed two alternative shot-encoding schemes to perform simultaneous-sources waveform inversion. The first scheme (I) encodes shot gathers with random-phase rotations applied to seismic traces. The second scheme (II) encodes shot gathers with random static time shifts. The well-known polarity encoding scheme (III) is just a special case of the random-phase rotation scheme. The second scheme is a variation of the conventional static shift encoding (IV), but the static time shifts in the second scheme are limited to one period of the dominant frequency. All encoded shot gathers are added up into a single super-shot gather for seismic waveform inversion. We perform the time-domain waveform inversion, using these shot-encoding schemes in conjunction with a restarted L-BFGS algorithm in the iterative inversion. The effectiveness and efficiency analyses demonstrate that the two shot-encoding schemes (I and II) proposed in this paper may improve the convergence of the iterative inversion, reduce the crosstalk effect among shots and consequently produce a subsurface velocity model with a high resolution.
We have designed a target-oriented methodology to perform Full Waveform Inversion using a frequency-domain wave propagator based on the so-called Patched Greens Function (PGF) technique. Originally developed in condensed matter physics to describe electronic waves in materials, the PGF technique is easily adaptable to the case of wave propagation in a spatially variable media in general. By dividing the entire computational domain into two sections, namely the target area and the outside target area, we calculate the Green Functions related to each section separately. The calculations related to the section outside the target are performed only once at the beginning of inversion, whereas the calculations in the target area are performed repeatedly for each iteration of the inversion process. With the Green Functions of the separate areas, we calculate the Green Functions of the two systems patched together through the application of a Recursive Dyson equation. By performing 2D and time-lapse experiments on the Marmousi model and a Brazilian Pre-salt velocity model, we demonstrate that the target-oriented PGF reduces the computational time of the inversion without compromising accuracy. In fact, when compared with conventional FWI results, the PGF-based calculations are identical but done in a fraction of the time.
Summary In an attempt to overcome the difficulties of the full waveform inversion (FWI), several alternative objective functions have been proposed over the last few years. Many of them are based on the assumption that the residuals (differences between modelled and observed seismic data) follow specific probability distributions when, in fact, the true probability distribution is unknown. This leads FWI to converge to an incorrect probability distribution if the assumed probability distribution is different from the real one and, consequently it may lead the FWI to achieve biased models of the subsurface. In this work, we propose an objective function which does not force the residuals to follow a specific probability distribution. Instead, we propose to use the non-parametric kernel density estimation technique (KDE) (which imposes the least possible assumptions about the residuals) to explore the probability distribution that may be more suitable. As evidenced by the results obtained in a synthetic model and in a typical P-wave velocity model of the Brazilian pre-salt fields, the proposed FWI reveals a greater potential to overcome more adverse situations (such as cycle-skipping) and also a lower sensitivity to noise in the observed data than conventional L2 and L1-norm objective functions and thus making it possible to obtain more accurate models of the subsurface. This greater potential is also illustrated by the smoother and less sinuous shape of the proposed objective function with fewer local minima compared with the conventional objective functions.
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