Dimensionally adaptive hp-finite element simulation and inversion of 2D magnetotelluric measurements

Magnetotelluric (MT) problems often contain different subdomains where the conductivity of the media depends upon one, two, or three spatial variables. Traditionally, when a MT problem incorporates a three-dimensional (3D) subdomain, the numerical method employed for simulation and inversion was 3D over then entire domain. In here, we propose to take advantage of the possibly lower dimensionality of certain subdomains during the inversion process. By doing so, we obtain significant computational savings (up to 75% in some scenarios) and increased accuracy on the results. We numerically illustrate this method by employing two dimensional (2D) computations based on a multi-goal oriented . hp-adaptive Finite Element Method (FEM) that exhibits superior convergence properties. Additionally, we provide a formulation for implementing an efficient adjoint based method for the computation of the derivatives of the impedance, and we show the importance of the (a) proper selection of the inversion variable, and (b) the advantages of using both the Transverse Electric (TE) and Transverse Magnetic (TM) measurements for the proper inversion of MT data

A summary of my twenty years of research according to Google Scholars

I am David Pardo, a researcher from Spain working mainly on numerical analysis applied to geophysics. I am 40 years old, and over a decade ago, I realized that my performance as a researcher was mainly evaluated based on a number called \h-index". This single number contains simultaneously information about the number of publications and received citations. However, dif- ferent h-indices associated to my name appeared in di erent webpages. A quick search allowed me to nd the most convenient (largest) h-index in my case. It corresponded to Google Scholars. In this work, I naively analyze a few curious facts I found about my Google Scholars and, at the same time, this manuscript serves as an experiment to see if it may serve to increase my Google Scholars h-index

A summary of my twenty years of research according to Google Scholars

I am David Pardo, a researcher from Spain working mainly on numerical analysis applied to geophysics. I am 40 years old, and over a decade ago, I realized that my performance as a researcher was mainly evaluated based on a number called \h-index". This single number contains simultaneously information about the number of publications and received citations. However, dif- ferent h-indices associated to my name appeared in di erent webpages. A quick search allowed me to nd the most convenient (largest) h-index in my case. It corresponded to Google Scholars. In this work, I naively analyze a few curious facts I found about my Google Scholars and, at the same time, this manuscript serves as an experiment to see if it may serve to increase my Google Scholars h-index

hp-Adaptive simulation and inversion of magnetotelluric measurements

xlix, 121 p.The magnetotelluric (MT) method is a passive exploration technique that aims at estimating the resistivity distribution of the Earth’s subsurface, and therefore at providing an image of it. This process is divided into two different steps. The first one consists in recording the data. In a second step, recorded measurements are analyzed by employing numerical methods. This dissertation focuses in this second task. We provide a rigorous mathematical setting in the context of the Finite Element Method (FEM) that helps to understand the MT problem and its inversion process. In order to recover a map of the subsurface based on 2D MT measurements, we employ for the first time in MTs a multigoal oriented self adaptive hp-Finite Element Method (FEM). We accurately solve both the full formulation as well as a secondary field formulation where the primary field is given by the solution of a 1D layered media. To truncate the computational domain, we design a Perfectly Matched Layer (PML) that automatically adapts to high-contrast material properties that appear within the subsurface and on the air-ground interface. For the inversion process, we develop a first step of a Dimensionally Adaptive Method (DAM) by considering the dimension of the problem as a variable in the inversion. Additionally, this dissertation supplies a rigorous numerical analysis for the forward and inverse problems. Regarding the forward modelization, we perform a frequency sensitivity analysis, we study the effect of the source, the convergence of the hp-adaptivity, or the effect of the PML in the computation of the electromagnetic fields and impedance. As far as the inversion is concerned, we study the impact of the selected variable for the inversion process, the different information that each mode provides, and the gains of the DAM approachUniversité de Pau et des Pays de l'Adour. bca

Adjoint-based formulation for computing derivatives with respect to bed boundary positions in resistivity geophysics

In inverse geophysical resistivity problems, it is common to optimize for specific resistivity values and bed boundary positions, as needed, for example, in geosteering applications. When using gradient-based inversion methods such as Gauss-Newton, we need to estimate the derivatives of the recorded measurements with respect to the inversion parameters. In this article, we describe an adjoint-based formulation for computing the derivatives of the electromagnetic fields withrespect to the bed boundary positions. The key idea to obtain this adjoint-based formulation is to separate the tangential and normal components of the field, and treat them differently. We then apply this method to a 1.5D borehole resistivity problem. We illustrate its accuracy and some of its convergence properties via numerical experimentation by comparing the results obtained with our proposed adjoint-based method vs. both the analytical results when available and a finite differences approximation of the derivative

A reduced order approach for probabilistic inversions of 3-D magnetotelluric data I: general formulation

Simulation-based probabilistic inversions of 3-D magnetotelluric (MT) data are arguably the best option to deal with the nonlinearity and non-uniqueness of the MT problem. However, the computational cost associated with the modelling of 3-D MT data has so far precluded the community from adopting and/or pursuing full probabilistic inversions of large MT data sets. In this contribution, we present a novel and general inversion framework, driven by Markov Chain Monte Carlo (MCMC) algorithms, which combines (i) an efficient parallel-in-parallel structure to solve the 3-D forward problem, (ii) a reduced order technique to create fast and accurate surrogate models of the forward problem and (iii) adaptive strategies for both the MCMC algorithm and the surrogate model. In particular, and contrary to traditional implementations, the adaptation of the surrogate is integrated into the MCMC inversion. This circumvents the need of costly offline stages to build the surrogate and further increases the overall efficiency of the method. We demonstrate the feasibility and performance of our approach to invert for large-scale conductivity structures with two numerical examples using different parametrizations and dimensionalities. In both cases, we report staggering gains in computational efficiency compared to traditional MCMC implementations. Our method finally removes the main bottleneck of probabilistic inversions of 3-D MT data and opens up new opportunities for both stand-alone MT inversions and multi-observable joint inversions for the physical state of the Earth's interior.Fil: Manassero, María Constanza. Macquarie University; AustraliaFil: Afonso, Juan Carlos. Macquarie University; AustraliaFil: Zyserman, Fabio Ivan. Universidad Nacional de La Plata. Facultad de Ciencias Astronómicas y Geofísicas. Departamento de Geofísica Aplicada; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; ArgentinaFil: Zlotnik, Sergio. Universidad Politécnica de Catalunya; EspañaFil: Fomin, I.. Macquarie University; Australi

Using Covariance Matrix Adaptation Evolutionary Strategy to boost the search accuracy in hierarchic memetic computations

Many global optimization problems arising naturally in science and engineering exhibit some form of intrinsic ill-posedness, such as multimodality and insensitivity. Severe ill-posedness precludes the use of standard regularization techniques and necessitates more specialized approaches, usually comprised of two separate stages - global phase, that determines the problem's modality and provides rough approximations of the solutions, and a local phase, which re fines these approximations. In this work, we attempt to improve one of the most efficient currently known approaches - Hierarchic Memetic Strategy (HMS) - by incorporating the Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) into its local phase. CMA-ES is a stochastic optimization algorithm that in some sense mimics the behavior of population-based evolutionary algorithms without explicitly evolving the population. This way, it avoids, to an extent, the associated cost of multiple evaluations of the objective function. We compare the performance of the HMS on relatively simple multimodal benchmark problems and on an engineering problem. To do so, we consider two con gurations: the CMA-ES and the standard SEA (Simple Evolutionary Algorithm). The results demonstrate that the HMS with CMA-ES in the local phase requires less objective function evaluations to provide the same accuracy, making this approach more efficient than the standard SEA

Fast One-dimensional Finite Element Approximation of Geophysical Measurements

There exist a wide variety of geophysical prospection methods. In this work, we focus on resistivity methods. We categorize these resistivity prospection methods according to their acquisition location as (a) on the surface, such as the ones obtained using Controlled Source Electromagnetics (CSEM) and magnetotelluric, and (b) in the borehole, such as the ones obtained using Logging-While-Drilling (LWD) devices. LWD devices are useful both for reservoir characterization and geosteering purposes, which is the act of adjusting the tool direction to travel within a specific zone. When inverting LWD resistivity measurements, it is a common practice to consider a one-dimensional (1D) layered media to reduce the problem dimensionality using a Hankel transform. Using orthogonality of Bessel functions, we arrive at a system of Ordinary Differential Equations (ODEs); one system of ODEs per Hankel mode. The dimensionality of the resulting problem is referred to as 1.5D since the computational cost to resolve it is in between that needed to solve a 1D problem and a 2D problem. When material properties (namely, resistivity, permittivity, and magnetic permeability) are piecewise-constant, we can solve the resulting ODEs either (a) analytically, which leads to a so-called semi-analytic method after performing a numerical inverse Hankel transform or (b) numerically. Semi-analytic methods are faster, but they also have important limitations, for example, (a) the analytical solution can only account for piecewise constant material properties, and other resistivity distributions cannot be solved analytically, which prevents to accurately model, for example, an Oil-Water Transition (OWT) zone when fluids are considered to be immiscible; (b) a specific set of cumbersome formulas has to be derived for each physical process (e.g., electromagnetism, elasticity, etc.), anisotropy type, etc.; (c) analytical derivatives of specific models (e.g., cross-bedded formations, or derivatives with respect to the bed boundary positions) are often difficult to obtain and have not been published to the best of our knowledge. In view of the above limitations, we propose to solve our forward problems using a numerical solver. A traditional Finite Element Method (FEM) is slow, which makes it unfeasible for our application. To achieve high performance, we developed a multiscale FEM that pre-computes a set of optimal local basis functions that are used at all logging positions. The resulting method is slow when compared to a semi-analytic approach for a single logging position, but it becomes highly competitive for a large number of logging positions, as needed for LWD geosteering applications. Moreover, we can compute the derivatives using an adjoint state method at almost zero additional cost in time. We describe an adjoint-based formulation for computing the derivatives of the electromagnetic fields with respect to the bed boundary positions. The key idea to obtain this adjoint-based formulation is to separate the tangential and normal components of the field, and treat them differently. We then apply this method to a 1.5D borehole resistivity problem. Moreover, we compute the adjoint-state formulation to compute the derivative of the magnetic field with respect to the resistivity value of each layer. We verify the accuracy of our formulations via synthetic examples. When simulating borehole resistivity measurements in a reservoir, it is common to consider an Oil-Water Contact (OWC) planar interface. However, this consideration can lead to an unrealistic model since, in the presence of capillary pressure, the mix of two immiscible fluids (oil and water) often appears as an OWT zone. These transition zones may be large in the vertical direction (20 meters or above), and in context of geosteering, an efficient method to simulate an OWT zone can maximize the production of an oil reservoir. In this work, we prove that by using our proposed 1.5D numerical method, we can easily consider arbitrary resistivity distributions in the vertical direction, as it occurs in an OWT zone. Numerical results on synthetic examples demonstrate significant differences between the results recorded by a geosteering device when considering a realistic OWT zone vs. an OWC sharp interface. As an additional piece of work of this Ph.D. Dissertation, we explore the possibility of using a Deep Neural Network (DNN) to perform a rapid inversion of borehole resistivity measurements. Herein, we build a DNN that approximates the following inverse problem: given a set of borehole resistivity measurements, the DNN is designed to deliver a physically meaningful and data-consistent piecewise one-dimensional layered model of the surrounding subsurface. Once the DNN is built, the actual inversion of the field measurements is efficiently performed in real time. We illustrate the performance of a DNN designed to invert LWD measurements acquired on high-angle wells via synthetic examples

Fast one-dimensional finite element approximation of geophysical measurements

135 p.When inverting Logging-While-Drilling (LWD) resistivity measurements, it is a common practice to consider a one-dimensional (1D) layered media to reduce the problem dimensionality using a Hankel transform. Using orthogonality of Bessel functions, we arrive at a system of Ordinary Differential Equations (ODEs); one systema of ODEs per Hankel mode. The dimensionality of the resulting problem is referred to as 1.5D since the computational cost to resolve it is in between that needed to solve a 1D problema and a 2D problem. When material properties are piecewise-constant, we can solve the resulting ODEs either (a) analytically, which leads to a so-called semi-analytic method, or (b) numerically. Semi-analytic methods are faster, but they also have important limitations, for example, (a) the analytical solution can only account for piecewise constant material properties, and other resistivity distributions cannot be solved analytically, which prevents to accurately model, for example, and OWT zone when fluids are considered to be inmiscible; (b) a specific set of cumbersome formulas has to be derived for each physical process (e.g. electromagnetism, elasticity, etc.), anisotropy type, etc.; (c) analytical derivatives of specific models (e.g. cross-bedded formations, or derivatives with respect to the bed boundary positios) are often diffcult to obtain and have not been published to the best of our knowledge

A numerical 1.5D method for the rapid simulation of geophysical resistivity measurements

In some geological formations, borehole resistivity measurements can be simulated using a sequence of 1D models. By considering a 1D layered media, we can reduce the dimensionality of the problem from 3D to 1.5D via a Hankel transform. The resulting formulation is often solved via a semi-analytic method, mainly due to its high performance. However, semi-analytic methods have important limitations such as, for example, their inability to model piecewise linear variations on the resistivity. Herein, we develop a multi-scale finite element method (FEM) to solve the secondary field formulation. This numerical scheme overcomes the limitations of semi-analytic methods while still delivering high performance. We illustrate the performance of the method with numerical synthetic examples based on two symmetric logging-while-drilling (LWD) induction devices operating at 2 MHz and 500 KHz, respectively