A gradient‐based Markov chain Monte Carlo algorithm for elastic pre‐stack inversion with data and model space reduction

Well logs are geophysical measurements of rock properties acquired continuously along a borehole. Because of the physics underlying their operation, borehole logging instruments perform local spatial averages of rock properties in the vicinity of the borehole, yielding well logs that are often affected by tool design, layer boundaries, borehole environmental conditions and mud‐filtrate invasion. Such environmental effects are ubiquitous and can lead to errors in petrophysical–property estimations from well logs if not accounted for in the interpretation. Separate well‐log inversion mitigates environmental effects by matching well logs with their numerical simulations. The latter simulations rely on specific assumptions about the relative geometry of the borehole and the rocks penetrated by the well. For vertical wells penetrating horizontal layers, it is common to assume piecewise‐constant layer properties that we collectively refer to as the earth model; some of these properties have a direct relationship to well logs (e.g. resistivity, gamma ray, density and acoustic slowness). Well‐log inversion is traditionally approached with deterministic methods such as the Levenberg–Marquardt algorithm to minimize the differences between well logs and their numerical simulations, often without accounting for the relationship between measurement noise and model uncertainty. Bayesian inversion, on the other hand, yields the posterior probability distribution of estimated properties and intrinsically quantifies their uncertainty. However, Bayesian Well‐log inversion usually involves the implementation of Markov chain Monte Carlo sampling, which requires a prohibitive number of forward simulations and is therefore not suitable for rapid petrophysical and/or elastic and mechanical evaluations of rocks. We introduce an efficient Bayesian Well‐log inversion method using a gradient‐based Markov chain Monte Carlo method. Gradient‐based Markov chain Monte Carlo is a group of relatively new Markov chain Monte Carlo algorithms that combine gradient updates with Hessian‐based sampling. gradient‐based Markov chain Monte Carlo draws samples efficiently from the posterior probability distribution using the gradient information to guide samples towards high‐probability regions and the Hessian to approximate the posterior probability distribution locally. We verify the gradient‐based Markov chain Monte Carlo inversion method with the inversion of synthetic well logs, including gamma ray, resistivity, density, neutron porosity, photoelectric factor and compressional/shear‐wave slowness. Results show that gradient‐based Markov chain Monte Carlo decreases the computational cost by more than 90% compared to conventional Markov chain Monte Carlo sampling. Next, gradient‐based Markov chain Monte Carlo inversion is applied to a field example from the Central North Sea, where conventional interpretation methods yield shale concentration, sandstone porosity and water saturation with errors up to 19.2%, 14.9% and 48.8%, respectively. Gradie...

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Efficient Bayesian inversion of borehole geophysical measurements with a gradient‐based Markov chain Monte Carlo method

Deng

Ambía

Torres‐Verdín

2023

“…The choice of the number of DCT coefficients to compress the model and data space should always constitute a compromise between the desired data and model resolution and the resulting computational cost of the inversion procedure. For the estimation of the optimal number of DCT coefficients needed to approximate the model and data spaceswe quantify how the variability of the true model and the observed data changes as the number of DCT coefficients varies, where the variability is defined as the ratio between the variance of the compressed model and data and the corresponding variances before the compression (Aleardi, 2021).…”

Section: Methodsmentioning

confidence: 99%

A computationally efficient Bayesian approach to full‐waveform inversion

Berti,

Aleardi,

Stucchi

2023

Conventional methods solve the full‐waveform inversion making use of gradient‐based algorithms to minimize an error function, which commonly measure the Euclidean distance between observed and predicted waveforms. This deterministic approach only provides a ‘best‐fitting’ model and cannot account for the uncertainties affecting the predicted solution. Local methods are also usually prone to get trapped into local minima of the error function. On the other hand, casting this inverse problem into a probabilistic framework has to deal with the formidable computational effort of the Bayesian approach when applied to non‐linear problems with expensive forward evaluations and large model spaces. We present a gradient‐based Markov Chain Monte Carlo full‐waveform inversion in which the posterior sampling is accelerated by compressing the data and model spaces through the discrete cosine transform, and by also defining a proposal that is a local, Gaussian approximation of the target posterior probability density. This proposal is constructed using the local Hessian and gradient informations of the log posterior, which are made computationally manageable thanks to the compression of the data and model spaces. We demonstrate the applicability of the approach by performing two synthetic inversion tests on portions of the Marmousi and BP acoustic model. In these examples, the forward modelling is performed using Devito, a finite difference domain‐specific language that solves the discretized wave equation on a Cartesian grid. For both examples, the results obtained by the implemented method are also validated against those obtained using a classic deterministic approach. Our tests illustrate the efficiency of the proposed probabilistic method, which seems quite robust against cycle‐skipping issues and also characterized by a computational cost comparable to that of the local inversion. The outcomes of the proposed probabilistic inversion can also play the role of starting models for a subsequent local inversion step aimed at improving the spatial resolution of the probabilistic result, which was limited by the model compression.

“…It is common in seismic data inversion to assume that the data uncertainty is uncorrelated and stationary (Aleardi, 2021;Aleardi & Salusti, 2020;Aleardi et al, 2018;Grana, 2020;Grana et al, 2022;Gunning & Sams, 2018;Li et al, 2022;Sengupta et al, 2021). Cordua et al (2009) assessed the impact of correlated noise on cross-borehole groundpenetrating radar data.…”

Section: Introductionmentioning

confidence: 99%

Assessment of the impact of noise magnitude and bandwidth variations on a probabilistic inversion of reflection seismic data

Heidari

Madsen

Amini

et al. 2023

Accounting for an accurate noise model is essential when dealing with real data, which are noisy due to the effect of environmental noise, failures and limitations in data acquisition and processing. Quantifying the noise model is a challenge for practitioners in formulating an inverse problem, and usually, a simple Gaussian noise model is assumed as a white noise model. Here we propose a pragmatic approach to use an estimated seismic wavelet to capture the correlated noise model (coloured noise) for the processed reflection seismic data. We assess the proposed method through a direct inversion of post‐stack seismic data associated with a carbonate reservoir of an oil field in southwest Iran to porosity, using a probabilistic sampling‐based inversion algorithm. In the probabilistic formulation of the inverse problem, we assume eight different noise models with varying bandwidth and magnitude and investigate the corresponding posterior statistics. The results indicate that if the correlated nature of the noise samples is ignored in the noise covariance matrix, some unrealistic features are generated in porosity realizations. In addition, if the noise magnitude is underestimated, the inversion algorithm overfits the data and generates a biased model with low uncertainty. Furthermore, by considering an imperfect bandwidth for the noise model, the error is propagated to the posterior realizations. Assuming the correlated noise in a probabilistic inversion resolves these issues significantly. Therefore, for inverting real seismic data where the estimation of the magnitude and correlations of the noise is not straightforward, the wavelet, which is estimated from the real seismic data, provides a good proxy for describing the correlation of the noise samples or equivalently the bandwidth of the noise model. In addition, it might be better to overestimate the noise magnitude than to underestimate it. This is true especially for an uncorrelated noise model and to a lesser degree also for the correlated noise model.