Mattia Aleardi

1D elastic full-waveform inversion and uncertainty estimation by means of a hybrid genetic algorithm–Gibbs sampler approach

ABSTRACT Stochastic optimization methods, such as genetic algorithms, search for the global minimum of the misfit function within a given parameter range and do not require any calculation of the gradients of the misfit surfaces. More importantly, these methods collect a series of models and associated likelihoods that can be used to estimate the posterior probability distribution. However, because genetic algorithms are not a Markov chain Monte Carlo method, the direct use of the genetic‐algorithm‐sampled models and their associated likelihoods produce a biased estimation of the posterior probability distribution. In contrast, Markov chain Monte Carlo methods, such as the Metropolis–Hastings and Gibbs sampler, provide accurate posterior probability distributions but at considerable computational cost. In this paper, we use a hybrid method that combines the speed of a genetic algorithm to find an optimal solution and the accuracy of a Gibbs sampler to obtain a reliable estimation of the posterior probability distributions. First, we test this method on an analytical function and show that the genetic algorithm method cannot recover the true probability distributions and that it tends to underestimate the true uncertainties. Conversely, combining the genetic algorithm optimization with a Gibbs sampler step enables us to recover the true posterior probability distributions. Then, we demonstrate the applicability of this hybrid method by performing one‐dimensional elastic full‐waveform inversions on synthetic and field data. We also discuss how an appropriate genetic algorithm implementation is essential to attenuate the “genetic drift” effect and to maximize the exploration of the model space. In fact, a wide and efficient exploration of the model space is important not only to avoid entrapment in local minima during the genetic algorithm optimization but also to ensure a reliable estimation of the posterior probability distributions in the subsequent Gibbs sampler step.

A feasibility study on the expected seismic AVA signatures of deep fractured geothermal reservoirs in an intrusive basement

The deep geothermal reservoirs in the Larderello-Travale field (southern Tuscany) are found in intensively fractured portions of intrusive/metamorphic rocks. Therefore, the geothermal exploration has been in search of possible fracture signatures that could be retrieved from the analysis of geophysical data. In the present work we assess the feasibility of finding seismic markers in the pre-stack domain which may pinpoint fractured levels. Thanks to the availability of data from boreholes that ENEL GreenPower drilled in the deep intrusive basement of this geothermal field, we derived the expected amplitude versus angle (AVA) responses of the vapour reservoirs found in some intensely, but very localized, fractured volumes within the massive rocks. The information we have available limit us to build 1D elastic and isotropic models only and thus anisotropy effects related to the presence of fractures cannot be properly modelled.We analysed the velocities and the density logs pertaining to three wells which reached five deep fractured zones in the basement. The AVA response of the fractured intervals is modelled downscaling the log data to seismic scale and comparing the analytical AVA response (computed with the Aki and Richards approximation) and the AVA extracted from a synthetic common mid point (calculated making use of a reflectivity algorithm). The results show that the amplitude of the reflections from the fractured level is characterized by negative values at vertical incidence and by decreasing absolute amplitudes with the increase of the source to receiver offset. This contrasts with many observations from hydrocarbon exploration in clastic reservoirs where gas-sand reflections often exhibit negative amplitudes at short offsets but increasing absolute amplitudes for increasing source to receiver offsets. Thereby, some common AVA attributes considered in silicoclastic lithologies would lead to erroneous fracture localization. For this reason we propose a modified AVA indicator which may highlight fracture locations in this peculiar rock type.

Comparison of Stochastic Optimization Methods on Two Analytic Objective Functions and on a 1D Elastic FWI

We compare the performance of three different stochastic optimization methods on two analytic objective functions varying the number of parameters, and on a 1D elastic full waveform inversion (FWI) problem. The three methods that we consider are the Adaptive Simulated Annealing (ASA), the Genetic Algorithm (GA), and the Neighbourhood Algorithm (NA) which are frequently used in seismic inversion. The application of these algorithms on the two analytic functions is aimed at evaluating the rate of convergence for different model space dimensions. The first function consists in a convex surface, and the second one is a multi-minima objective function which also permits to verify the ability of each method to escape from entrapment in local minima. Our study shows that among the three optimization methods GA displays the better scaling with the number of parameters. The ASA method is often the most efficient in case of low dimensional model spaces, whereas NA seems to perform less efficiently than the other two and to be more prone to get trapped in local minima. Tests of 1D elastic FWI on synthetic data, inverting for density, P and S-wave velocity for a total of 21 unknowns confirm the conclusions drawn from the previous examples.

Comparing the performances of four stochastic optimisation methods using analytic objective functions, 1D elastic full-waveform inversion, and residual static computation

We compare the performances of four different stochastic optimisation methods using four analytic objective functions and two highly non-linear geophysical optimisation problems: 1D elastic full-waveform inversion (FWI) and residual static computation. The four methods we consider, namely, adaptive simulated annealing (ASA), genetic algorithm (GA), neighbourhood algorithm (NA), and particle swarm optimisation (PSO), are frequently employed for solving geophysical inverse problems. Because geophysical optimisations typically involve many unknown model parameters, we are particularly interested in comparing the performances of these stochastic methods as the number of unknown parameters increases. The four analytic functions we choose simulate common types of objective functions encountered in solving geophysical optimisations: a convex function, two multi-minima functions that differ in the distribution of minima, and a nearly flat function. Similar to the analytic tests, the two seismic optimisation problems we analyse are characterized by very different objective functions. The first problem is a 1D elastic FWI, which is strongly ill-conditioned and exhibits a nearly flat objective function, with a valley of minima extended along the density direction. The second problem is the residual static computation, which is characterized by a multi-minima objective function produced by the so-called cycle-skipping phenomenon. According to the tests on the analytic functions and on the seismic data, GA generally displays the best scaling with the number of parameters. It encounters problems only in the case of irregular distribution of minima, that is, when the global minimum is at the border of the search space and a number of important local minima are distant from the global minimum. The ASA method is often the best-performing method for low-dimensional model spaces, but its performance worsens as the number of unknowns increases. The PSO is effective in finding the global minimum in the case of low-dimensional model spaces with few local minima or in the case of a narrow flat valley. Finally, the NA method is competitive with the other methods only for low-dimensional model spaces; its performance stability sensibly worsens in the case of multi-minima objective functions. This article is protected by copyright. All rights reserved

Two-grid stochastic full waveform inversion of 2D marine seismic data

We apply stochastic Full Waveform Inversion (FWI) to 2D marine seismic data to estimate the macro-model velocity field which can be a suitable input for subsequent local (gradient based) FWI. Genetic Algorithms are used as the global optimization method. Our two-grid representation of the subsurface, made of a coarse grid for the inversion and of a fine grid for the modeling, allows us to reduce the number of unknowns to an acceptable number for the given computer resources and to perform a stable and reliable finite difference modeling. Thus, notwithstanding the known high computational costs that characterize global inversion methods, we are able to reconstruct a smooth, low wavenumber, acoustic velocity model of the subsurface. The reliability of the estimated velocity macro-model is checked through the inspection of prestack depth migrated gathers and through the superposition of observed and modeled seismograms. The method we propose is less affected by the risk of being trapped in local minima of the misfit functional than gradient based FWI methods, and can be a viable alternative to estimate proper starting models for gradient based full waveform inversions.

Estimation of a High Resolution P-wave Velocity Model of the Seabed Layers by Means of Global and a Gradient-based FWI

Summary We present a two-step procedure to full-waveform inversion (FWI) that combines a stochastic, genetic algorithm optimization and a subsequent gradient-based inversion with the aim to estimate a high-resolution P-wave velocity (Vp) model of the shallow seabed layers. In particular, we take advantage of the broad band frequency content of the seismic well-site (WSS) data to extend the frequency range up to 70 Hz. The first step is a genetic algorithm optimization aimed at deriving a reliable starting model for the subsequent gradient-based FWI. The lack of low frequencies and the limited maximum offset of the WSS acquisition, make the GA inversion particularly crucial as it provides a Vp field that contains the low-medium wavelengths of the subsurface compressional velocity field. These wavelengths are essential to attenuate the risk for the following gradient-based FWI of being trapped into local minima. The gradient-based FWI is performed in the acoustic approximation thus inverting for the Vp only. This two-step procedure yields a final Vp model characterized by an improved resolution with respect to the outcomes of GA-FWI and many fine details of the layering. The fair match between the reflections kinematics in the actual and predicted data supports the reliability of the final model.

Reservoir Characterization Through Target-Oriented AVA-Petrophysical Inversions with Spatial Constraints

We apply three methods that use different regularization strategies to insert spatial constraints into the seismic-petrophysical inversion. The first method is what we call the structurally constrained inversion (SCI), which directly uses the structural information brought by the seismic stack image to insert geological (structural) constraints into the inversion. The second method is based on anisotropic Markov random field (AMRF) and uses the Huber energy function to reasonably model the spatial heterogeneity of petrophysical reservoir properties. Finally, the last method includes both geostatistical information (describing the lateral variability of reservoir properties) and hard data (i.e. well log data) constraints into the inversion kernel (GHDC inversion). For computationally feasibility, we apply a target-oriented inversion that uses the amplitude versus angle (AVA) responses extracted along the top reservoir reflections to infer the petrophysical properties of interest (i.e. porosity, water saturation and shaliness) for the target layer. In particular, an empirical, linear rock-physics model, properly calibrated for the investigated area, is used to rewrite the P-wave reflectivity equation as a function of the petrophysical contrasts instead of the elastic constants. This reformulation allows for a direct and a simultaneous estimation of petrophysical properties from AVA data. The implemented approaches are tested both on synthetic and field seismic data and compared against the standard method in which each AVA response is inverted independently (laterally unconstrained Bayesian inversion; LUBI). In the case of poor signal-to-noise ratio it turns out that the three considered spatially constrained methods achieve better delineations of reservoir bodies and provide more reliable results than the standard LUBI approach. More in detail, the AMRF recovers sharper geological boundaries than the SCI and GHDC algorithms. The SCI algorithms is more sensitive to data noise, whereas the key advantage of the GHDC consists in analytically providing posterior uncertainties for the model parameters. Finally, for what concerns the computational cost the GHDC and the SCI methods result the most and the least computationally demanding, respectively.

Estimation of reservoir properties from seismic data through a Markov Chain Monte Carlo-AVA inversion algorithm

We formulate the amplitude versus angle (AVA) inversion in terms of a Markov Chain Monte Carlo (MCMC) algorithm and apply it for reservoir characterisation and litho-fluid facies prediction in offshore Nile Delta. A linear empirical rock physics model is used to link the petrophysical properties (porosity, water saturation and shaliness) to the elastic attributes (P-wave velocity, S-wave velocity and density), whereas the exact Zoeppritz equations are used to convert the elastic properties into AVA responses. The exact Zoeppritz equations allow us to take advantage of the long offset seismic acquisition and thus to consider a wide range of incidence angles in the inversion. The proposed algorithm reliably estimates the non-uniqueness of the solution that is the uncertainties affecting the estimated subsurface characteristics (both in terms of litho-fluid facies and petrophysical properties), taking into consideration the uncertainties in the prior information, the uncertainties in the estimated rock-physics model and the errors affecting the observed AVA responses. A blind test, based on available well log information, demonstrates the applicability of the proposed method and the reliability of the results. In addition, comparisons between the results provided by the implemented MCMC algorithm with those yielded by a linear AVA inversion and an analytical approach to facies prediction, show the benefits introduced by wide-angle reflections in better constraining the inverted parameters and in attenuating the noise in the predicted subsurface models. We formulate the amplitude versus angle (AVA) inversion in terms of a Markov Chain Monte Carlo algorithm and apply it for reservoir characterisation and litho-fluid facies prediction in offshore Nile Delta. A blind test, based on available well log information, demonstrates the applicability of the proposed method and the reliability of the results.

A two-step elastic full-waveform inversion applied to reflection seismic data for shallow hazard identification

We apply a two-step elastic full-waveform inversion (FWI) to well-site survey (WSS) marine seismic data to estimate high-resolution P-wave (Vp) and S-wave (Vs) velocity models. Our approach combines a first global, genetic-algorithm optimization and a subsequent gradient-based inversion. The broad-band frequency content of the available seismic data makes it possible to extend the frequency range considered in the inversion up to 70 Hz and thus to derive a high-resolution elastic characterization of the shallowest part of the subsurface. The lack of low frequencies and the limited maximum source-to-receiver offset of the WSS acquisition, make the GA inversion particularly crucial as it provides a starting model for the gradient-based FWI that contains the large-medium wavelengths of the seismic velocity field. The following gradient-based FWI yields Vp and Vs models characterized by an improved resolution with respect to the outcomes of GA-FWI. The match between the observed and the predicted seismic data proves the reliability of our predictions.

Using well log data and statistical Gaussian simulations to estimate the crack density value within a geothermal reservoir located in hard rocks

To identify the most promising targets when exploring geothermal reservoirs it is crucial to infer the crack density value (the fracture density per volume) in the subsurface. In this work a geophysical forward modeling approach and statistical techniques were used to estimate the crack density within the deepest geothermal reservoir of the Larderello-Travale field where fractured intrusive/metamorphic rocks constitute the main drilling targets of geothermal exploration.Waveform sonic recording and circumferential borehole imager log, acquired in the investigated area, evidenced the presence of several vertically aligned fractures with a preferential orientation NNW-SSE at the depth of the productive levels, whereas the encasing rocks appeared to be quite isotropic. This characteristic permitted to approximate the target level as a transverse isotropic medium with a horizontal axis of symmetry (HTI medium). Then, basing on borehole data, several models were developed that keep the encasing medium and the strike of the fractures within the target constant, but change the crack density from 0 (no fractures) to 0.1 (highly fractured). For each given crack density value, the associated P-wave and S-wave velocity values within the fractured zone were derived. Then, comparing the probability distribution of the simulated velocities with the logged velocities it was possible to estimate the most likely crack density value.The proposed methodology applied to two wells returned plausible and similar results for the crack density value.