CSEM Optimization Using the Correspondence Principle

Abstract

Traditionally, 3D modeling of marine controlled-source electromagnetic (CSEM) data (in the frequency domain) involves high-memory demand, requiring solving a large linear system for each frequency. To address this problem, we propose to solve Maxwell’s equations in a fictitious dielectric medium with time-domain finite-difference methods, with the support of the correspondence principle. As an advantage of this approach, we highlight the possibility of its implementation for execution with GPU accelerators, in addition to multi-frequency data modeling with a single simulation. Furthermore, we explore using the correspondence principle to the inversion of CSEM data by calculating the gradient of the least-squares objective function employing the adjoint-state method to establish the relationship between adjoint fields in a conductive medium and their counterparts in the fictitious dielectric medium, similar to the approach used in forward modeling. We validate this method through 2D inversions of three synthetic CSEM datasets, computed for a simple model consisting of two resistors in a conductive medium, a model adapted from a CSEM modeling and inversion package, and the last one based on a reference model of turbidite reservoirs on the Brazilian continental margin. We also evaluate the differences between the results of inversions using the steepest descent method and our proposed momentum method, comparing them with the limited-memory BFGS (Broyden–Fletcher–Goldfarb–Shanno) algorithm (L-BFGS-B). In all experiments, we use smoothing by model reparameterization as a strategy for regularizing and stabilizing the iterations throughout the inversions. The results indicate that, although it requires more iterations, our modified momentum method produces the best models, which are consistent with results from the L-BFGS-B algorithm and require less storage per iteration.

1. Introduction

The controlled source electromagnetic method (CSEM) is a consolidated method in the off-shore industry [1,2]. It is used not only to reduce the exploratory risk [3], helping to decide well placement, but also in applications aimed at reservoir monitoring [4,5], given the significant role of electrical resistivity in characterizing rock porosity and fluid saturation.

Based on measurements of the electric and magnetic fields made by receivers on the seabed (while a ship towed the submerged electromagnetic source), CSEM allows for obtaining a model of electrical conductivity in the subsurface through an inversion process, which iteratively minimizes the misfit between observed and modeled data.

Several possible approaches exist to perform forward modeling in the context of the inversion of CSEM data [6]. Some of these approaches are based on solving Maxwell’s equations for conductive media directly in the frequency domain and require solving a linear system for each frequency. Furthermore, the Gauss–Newton approximation for the model update is common during inverse problem iterations [7,8], which usually require solving large linear systems, especially in 3D problems. Another critical aspect of CSEM inversion is the regularization strategy applied to stabilize the iterative process [9,10,11,12]. We propose modeling CSEM data through the numerical time-domain finite-difference solution of Maxwell’s equations for a fictitious dielectric medium, following the prescription of [13]. The correspondence principle allows the evaluation of the fields for the corresponding conductive medium in the frequency domain [14]. In the context of inversion, we present the formulation for gradient calculation, obtained using the adjoint-state method [15,16], based on adjoint modeling following the same formulation as the forward modeling. This is the approach presented by [17] but with the correction to the equations that define the mapping between the fictitious and diffusive domains. To stabilize the inverse problem, we use model reparameterization smoothing [18,19], which does not require trial-and-error selection of a regularization parameter.

We also evaluate two optimization algorithms: the steepest descent (using a simple line search strategy) and a momentum method [20], parameterized according to the Adam algorithm [21], known as a low memory demand method. We also compared the results with those obtained using L-BFGS-B [22,23], which is traditionally used in minimization problems. Our numerical experiments show that both algorithms successfully fit the observations and correctly localize the inversion targets, but our momentum approach has low storage demand.

2. Methodology

2.1. Csem Forward Modeling

Maxwell’s equations (in the frequency domain) for a conductive medium at low frequencies are as follows:

$$-\nabla \times \mathbf{H}(\mathbf{x},\omega )+\mathsf{\sigma }\left(\mathbf{x}\right)\mathbf{E}(\mathbf{x},\omega )=-\mathbf{J}(\mathbf{x},\omega ),$$

(1)

$$\nabla \times \mathbf{E}(\mathbf{x},\omega )-i\omega \mu \mathbf{H}(\mathbf{x},\omega )=\mathbf{0},$$

(2)

where $\mathbf{E}(\mathbf{x},\omega )$ and $\mathbf{H}(\mathbf{x},\omega )$ are the electric and magnetic field vectors, respectively, $\mathsf{\sigma }\left(\mathbf{x}\right)$ represents the conductivity tensor, $\mu$ is the magnetic permeability, and $\mathbf{J}(\mathbf{x},t)$ is the electric current density vector, which represents the injection source distribution.

In the same context, Maxwell’s equations in the time domain are given as follows:

$$-\nabla \times \mathbf{H}(\mathbf{x},t)+\mathsf{\sigma }\left(\mathbf{x}\right)\mathbf{E}(\mathbf{x},t)=-\mathbf{J}(\mathbf{x},t),$$

(3)

$$\nabla \times \mathbf{E}(\mathbf{x},t)+\mu {\partial }_t\mathbf{H}(\mathbf{x},t)=\mathbf{0},$$

(4)

since we can obtain Equations (1) and (2) from Equations (3) and (4) using the Fourier transform:

$$F\left(\omega \right)={\int }_{-\infty }^{+\infty }dtf\left(t\right)e^{i\omega t},$$

(5)

$$f\left(t\right)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{+\infty }d\omega F\left(\omega \right)e^{-i\omega t}.$$

(6)

In a non-conductive dielectric medium, Maxwell’s equations in the time domain are:

$$-\nabla \times \mathbf{H}(\mathbf{x},t)+\mathsf{ϵ}\left(\mathbf{x}\right){\displaystyle \frac{\partial \mathbf{E}(\mathbf{x},t)}{\partial t}}=-\mathbf{J}(\mathbf{x},t),$$

(7)

$$\nabla \times \mathbf{E}(\mathbf{x},t)+\mu {\displaystyle \frac{\partial \mathbf{H}(\mathbf{x},t)}{\partial t}}=\mathbf{0},$$

(8)

and in the frequency domain they are the following:

$$-\nabla \times \mathbf{H}(\mathbf{x},\omega )-i\omega \mathsf{ϵ}\left(\mathbf{x}\right)\mathbf{E}(\mathbf{x},\omega )=-\mathbf{J}(\mathbf{x},\omega ),$$

(9)

$$\nabla \times \mathbf{E}(\mathbf{x},\omega )-i\omega \mu \mathbf{H}(\mathbf{x},\omega )=\mathbf{0},$$

(10)

where $\mathsf{ϵ}\left(\mathbf{x}\right)$ is the dielectric permittivity tensor.

Defining the fictitious dielectric permittivity,

$$\sigma \left(\mathbf{x}\right)\equiv 2{\omega }_0{ϵ}^{\prime }\left(\mathbf{x}\right),$$

(11)

we can rewrite Equations (1) and (2):

$$-\nabla \times {\mathbf{H}}^{\prime }(\mathbf{x},{\omega }^{\prime })-i{\omega }^{\prime }{\mathsf{ϵ}}^{\prime }\left(\mathbf{x}\right){\mathbf{E}}^{\prime }(\mathbf{x},{\omega }^{\prime })=-{\mathbf{J}}^{\prime }(\mathbf{x},{\omega }^{\prime }),$$

(12)

$$\nabla \times {\mathbf{E}}^{\prime }(\mathbf{x},{\omega }^{\prime })-i{\omega }^{\prime }\mu {\mathbf{H}}^{\prime }(\mathbf{x},{\omega }^{\prime })=\mathbf{0},$$

(13)

where

$${\omega }^{\prime }\equiv (1+i)\sqrt{\omega {\omega }_0},$$

(14)

$${\mathbf{E}}^{\prime }(\mathbf{x},{\omega }^{\prime })\equiv \mathbf{E}(\mathbf{x},\omega ),$$

(15)

$${\mathbf{H}}^{\prime }(\mathbf{x},{\omega }^{\prime })\equiv \sqrt{{\displaystyle \frac{-i\omega }{2{\omega }_0}}}\mathbf{H}(\mathbf{x},\omega ),$$

(16)

$${\mathbf{J}}^{\prime }(\mathbf{x},{\omega }^{\prime })\equiv \sqrt{{\displaystyle \frac{-i\omega }{2{\omega }_0}}}\mathbf{J}(\mathbf{x},\omega ).$$

(17)

Comparing with Equations (9) and (10), we obtain the equations in the fictitious time domain:

$$-\nabla \times {\mathbf{H}}^{\prime }(\mathbf{x},t^{\prime })+{\mathsf{ϵ}}^{\prime }\left(\mathbf{x}\right){\displaystyle \frac{\partial {\mathbf{E}}^{\prime }(\mathbf{x},t^{\prime })}{\partial t^{\prime }}}=-{\mathbf{J}}^{\prime }(\mathbf{x},t^{\prime }),$$

(18)

$$\nabla \times {\mathbf{E}}^{\prime }(\mathbf{x},t^{\prime })+\mu {\displaystyle \frac{\partial {\mathbf{H}}^{\prime }(\mathbf{x},t^{\prime })}{\partial t^{\prime }}}=\mathbf{0}.$$

(19)

The mapping defined in Equations (14)–(17) warrants this construction, which establishes a direct relationship between the electromagnetic fields in a conductive medium and their corresponding in a fictitious dielectric model, as established by the correspondence principle [14]. We solve Equations (18) and (19) using finite difference, following the prescriptions of [13], and obtain the components $E_i$ of the electric field in the conductive medium using Equation (15):

$$\begin{array}{c}\hfill E_i(\mathbf{x},\omega )={\int }_0^{T}d{t}^{\prime }{E}_i^{\prime }(\mathbf{x},t^{\prime })e^{i{\omega }^{\prime }{t}^{\prime }}={\int }_0^{T}d{t}^{\prime }{E}_i^{\prime }(\mathbf{x},t^{\prime })e^{-\sqrt{\omega {\omega }_0}{t}^{\prime }}{e}^{i\sqrt{\omega {\omega }_0}{t}^{\prime }},\end{array}$$

(20)

whose numerical evaluation is stable in consequence of the exponential decay imposed by the term $e^{-\sqrt{\omega {\omega }_0}{t}^{\prime }}$, regardless of the duration of the signal recorded during the simulation in the fictitious time domain.

The injection of the source component $J^{\prime }(\mathbf{x},t^{\prime })$ uses the band-limited spatial Delta function approximation:

$$J^{\prime }({\mathbf{x}}_s,t^{\prime })=\delta (\mathbf{x}-{\mathbf{x}}_s)J^{\prime }\left(t^{\prime }\right),$$

(21)

with a signature spectrum given as follows:

$$\begin{array}{c}\hfill J\left(\omega \right)=\sqrt{{\displaystyle \frac{-2{\omega }_0}{i\omega }}}{\int }_0^{T}d{t}^{\prime }{J}^{\prime }\left(t^{\prime }\right)e^{-\sqrt{\omega {\omega }_0}{t}^{\prime }}{e}^{i\sqrt{\omega {\omega }_0}{t}^{\prime }}.\end{array}$$

(22)

In practice, the choice of the source signature for modeling in the fictitious domain is flexible since the data obtained after performing the CSEM survey corresponds to the field normalized by the source at each frequency.

2.2. Gradient Calculation for CSEM Inversion

The estimate of subsurface resistivity is an ill-posed non-linear problem and inversion of CSEM data usually applies interactive algorithms, where update of the resistivity model is a step along the gradient of the objective function relative to the model parameters. The adjoint state method allows for the efficient computation of the gradient [16].

For a conductive model, $\mathsf{\sigma }$, the objective function for inversion of CSEM data is the least squares mismatch between the observed and modeled data:

$$\begin{array}{c}\hfill \chi (\mathbf{E},\mathsf{\sigma })=\sum _s\sum _{\omega }\sum _r{W}^E({\mathbf{x}}^r,{\mathbf{x}}^s){∥{\mathbf{E}}^{obs}({\mathbf{x}}^r,\omega ;{\mathbf{x}}^s)-\mathbf{E}({\mathbf{x}}^r,\omega ;{\mathbf{x}}^s|\mathsf{\sigma })∥}^2,\end{array}$$

(23)

where index s indicates the summation over all sources, index r indicates the summation over the receiver position, and index $\omega$ indicates the summation over all recorded frequencies. The weighting factors $W^E$ penalize the residuals between the modeled and observed field components [24]. The observed electric field at the receiver position, denoted by ${\mathbf{x}}^r$, due to a source located at ${\mathbf{x}}^s$, for a given frequency, $\omega$, is represented by the vector ${\mathbf{E}}^{obs}({\mathbf{x}}^r,{\mathbf{x}}^s,\omega )$. The field vector $\mathbf{E}({\mathbf{x}}^r,{\mathbf{x}}^s,\omega )$ represents the corresponding modeled field for a given conductivity model, $\mathsf{\sigma }$.

We derived the gradient of $\chi (\mathbf{E},\mathsf{\sigma })$ relative to the conductivity model through the adjoint state method, which is an extension of the Lagrange multipliers method. The starting point is the extended Lagrangian, containing the sum of the least-squares data misfit plus the state equations that enforce the modeled fields and medium properties should honor the Maxwell equations, i.e.,

$$\begin{array}{cc}\hfill \mathcal{L}(\mathbf{E},\mathbf{H},\mathsf{\sigma },{\mathsf{\lambda }}_E,{\mathsf{\lambda }}_H)=& \sum _s\sum _{\omega }\sum _r{W}^E({\mathbf{x}}^r,\omega ;{\mathbf{x}}^s){∥{\mathbf{E}}^{obs}({\mathbf{x}}^r,\omega ;{\mathbf{x}}^s)-\mathbf{E}({\mathbf{x}}^r,\omega ;{\mathbf{x}}^s|\mathsf{\sigma })∥}^2\hfill \\ \hfill & +\sum _s\sum _{\omega }\Re \left({\int }_{\mathsf{\Omega }}d\mathsf{\Omega }{\mathsf{\lambda }}_E^{*}\left(\mathbf{u}\right)\left[-\nabla \times \mathbf{H}\left(\mathbf{u}\right)+\mathsf{\sigma }\left(\mathbf{x}\right)\mathbf{E}+\mathbf{J}\left(\mathbf{u}\right)\right]\right)\hfill \\ \hfill & +\sum _s\sum _{\omega }\Re \left({\int }_{\mathsf{\Omega }}d\mathsf{\Omega }{\mathsf{\lambda }}_H^{*}\left(\mathbf{u}\right)\left[\nabla \times \mathbf{E}\left(\mathbf{u}\right)-i\omega \mu \mathbf{H}\left(\mathbf{u}\right)\right]\right)\hfill \end{array}$$

(24)

where $\Re \left(z\right)$ indicates the real part of its complex argument and $\mathbf{u}\equiv (\mathbf{x},\omega ;{\mathbf{x}}^s)$. ${\mathsf{\lambda }}_E(\mathbf{x},\omega ;{\mathbf{x}}^s)$, and ${\mathsf{\lambda }}_H(\mathbf{x},\omega ;{\mathbf{x}}^s)$ are the adjoint-state fields.

The stationarity of the first variation of the Lagrangian functional relative to the electromagnetic fields produces the system of equations for the adjoint-state fields [15,16], ${\mathsf{\lambda }}_E$ and ${\mathsf{\lambda }}_H$,

$$\begin{array}{cc}\hfill \nabla \times {\mathsf{\lambda }}_H^{*}\left(\mathbf{u}\right)+& \mathsf{\sigma }\left(\mathbf{x}\right){\mathsf{\lambda }}_E^{*}\left(\mathbf{u}\right)=\sum _r\delta (\mathbf{x}-{\mathbf{x}}^r)W^E\left(\mathbf{u}\right)\left[{\mathbf{E}}^{obs}\left(\mathbf{u}\right)-\mathbf{E}\left(\mathbf{u}\right)\right],\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & -\nabla \times {\mathsf{\lambda }}_E^{*}\left(\mathbf{u}\right)-i\omega \mu {\mathsf{\lambda }}_H^{*}\left(\mathbf{u}\right)=\mathbf{0},\hfill \end{array}$$

(26)

and the gradient of the objective function relative to the conductivity tensor,

$${\left.{\displaystyle \frac{\partial \chi }{\partial \mathsf{\sigma }}}\right|}_{\mathbf{x}}=\sum _s\sum _{\omega }\Re \left\{{\mathsf{\lambda }}_E^{*}(\mathbf{x},\omega ;{\mathbf{x}}^s)\mathbf{E}(\mathbf{x},\omega ;{\mathbf{x}}^s)\right\}.$$

(27)

Therefore, to evaluate the gradient of the objective function, we need to solve the direct electromagnetic fields equation’s and the adjoint-state fields equation’s.

To solve the system Equations (25) and (26), we apply the correspondence principle proposed by [13].

Our derivation demonstrates that the corresponding adjoint-field equations in the fictitious dielectric medium in the frequency domain are as follows:

$$\begin{array}{cc}\hfill -\nabla \times {\mathsf{\lambda }}_H^{\prime }(\mathbf{x},{\omega }^{\prime })+(-i{\omega }^{\prime }){\mathsf{ϵ}}^{\prime }\left(\mathbf{x}\right){\mathsf{\lambda }}_E^{\prime }(\mathbf{x},{\omega }^{\prime })& =-\sum _r\delta (\mathbf{x}-{\mathbf{x}}^r){\mathbf{J}}^{\prime }(\mathbf{x},{\omega }^{\prime }),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \nabla \times {\mathsf{\lambda }}_E^{\prime }(\mathbf{x},{\omega }^{\prime })+(-i{\omega }^{\prime })\mu {\mathsf{\lambda }}_H^{\prime }(\mathbf{x},{\omega }^{\prime })& =\mathbf{0}.\hfill \end{array}$$

(29)

The relationship between the adjoint-state equations in a conductive medium and the corresponding adjoint-state equations in a fictitious dielectric medium are:

$$\begin{array}{cc}\hfill {\omega }^{\prime }& \equiv (1+i)\sqrt{\omega {\omega }_0},\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\mathsf{ϵ}}^{\prime }\left(\mathbf{x}\right)& \equiv \mathsf{\sigma }\left(\mathbf{x}\right)/\left(2{\omega }_0\right),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\mathsf{\lambda }}_E^{\prime }(\mathbf{x},{\omega }^{\prime })& \equiv {\mathsf{\lambda }}_E^{*}(\mathbf{x},\omega ),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\mathsf{\lambda }}_H^{\prime }(\mathbf{x},{\omega }^{\prime })& \equiv -\sqrt{{\displaystyle \frac{-i\omega }{2{\omega }_0}}}{\mathsf{\lambda }}_H^{*}(\mathbf{x},\omega ),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\mathbf{J}}^{\prime }(\mathbf{x},{\omega }^{\prime })& \equiv -\sqrt{{\displaystyle \frac{-i\omega }{2{\omega }_0}}}\Delta {\mathbf{E}}^{*}(\mathbf{x},\omega ).\hfill \end{array}$$

(34)

The Equations from (30) to (34) map the time-domain adjoint-state fields to the corresponding frequency-domain adjoint-state fields in the conductive medium. The system of Equations (28) and (29) in the time domain is:

$$\begin{array}{cc}\hfill -\nabla \times {\mathsf{\lambda }}_H^{\prime }(\mathbf{x},t^{\prime })+{\mathsf{ϵ}}^{\prime }\left(\mathbf{x}\right){\partial }_t^{\prime }{\mathsf{\lambda }}_E^{\prime }(\mathbf{x},t^{\prime })& =-\sum _r\delta (\mathbf{x}-{\mathbf{x}}^r){\mathbf{J}}^{\prime }(\mathbf{x},t^{\prime }),\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill \nabla \times {\mathsf{\lambda }}_E^{\prime }(\mathbf{x},t^{\prime })+\mu {\partial }_t^{\prime }{\mathsf{\lambda }}_H^{\prime }(\mathbf{x},t^{\prime })& =\mathbf{0}.\hfill \end{array}$$

(36)

To solve these equations through finite-difference, we use the algorithm proposed by [13]. The current density vector at each receiver position is as follows (34):

$$\begin{array}{c}\hfill {\mathbf{J}}^{\prime }({\mathbf{x}}^r,\omega )={\int }_0^{T}d{t}^{\prime }\mathbf{J}({\mathbf{x}}^r,t^{\prime })e^{-\sqrt{\omega {\omega }_0}{t}^{\prime }}{e}^{i\sqrt{\omega {\omega }_0}{t}^{\prime }}.\end{array}$$

(37)

Finally, the adjoint-field ${\mathsf{\lambda }}_E^{*}(\mathbf{x},\omega )$ is evaluated on the fly at each finite-difference time update, applying the correspondence relation in Equation (32):

$$\begin{array}{c}\hfill {\mathsf{\lambda }}_E^{*}(\mathbf{x},\omega )={\int }_0^T{\mathsf{\lambda }}_E^{\prime }(\mathbf{x},t^{\prime })e^{-\sqrt{\omega {\omega }_0}{t}^{\prime }}{e}^{i\sqrt{\omega {\omega }_0}{t}^{\prime }}d{t}^{\prime }.\end{array}$$

(38)

2.3. Adjoint Source

The source injected at a receiver during adjoint modeling in the fictitious time domain must satisfy the following equation:

$$\begin{array}{c}\hfill \Delta E^{*}({\mathbf{x}}_r,\omega )=-\sqrt{{\displaystyle \frac{-2{\omega }_0}{i\omega }}}{\int }_0^{T}d{t}^{\prime }{J}^{\prime }({\mathbf{x}}_r,t^{\prime })e^{-\sqrt{\omega {\omega }_0}{t}^{\prime }}{e}^{i\sqrt{\omega {\omega }_0}{t}^{\prime }}.\end{array}$$

(39)

Obtaining the adjoint source (in fictitious time domain) from $\Delta E^{*}({\mathbf{x}}_r,\omega )$ is an underdetermined problem, which allows a high degree of freedom in the choice of $J^{\prime }({\mathbf{x}}_r,t^{\prime })$ [13]. For example, one can assume an estimated pulse, ${\widehat{J}}^{\prime }({\mathbf{x}}_r,t^{\prime })$, formed from the sum of pulses corresponding to the second derivative of a Gaussian:

$$\begin{array}{c}\hfill {\widehat{J}}^{\prime }({\mathbf{x}}_r,t^{\prime })=\sum _{m=1}^{N_{\omega }}{A}_m{\partial }_{t^{\prime }}^2\mathsf{\Gamma }(t^{\prime }-{\tau }_m),\end{array}$$

(40)

$$\begin{array}{c}\hfill \mathsf{\Gamma }(t^{\prime }-{\tau }_m)=\sqrt{{\displaystyle \frac{{\beta }_m}{\pi }}}{e}^{-{\beta }_m{(t^{\prime }-{\tau }_m)}^2},\end{array}$$

(41)

where $N_{\omega }$ is the number of frequencies, and ${\beta }_m={\pi }^3/{\tau }_m^2$.

With this choice, the adjoint source depends on the $2N_{\omega }$ parameters, $A_m$ and ${\tau }_m$, determined by minimizing the functional defined by the difference between the spectra of the residuals and the sources (calculated from Equation (40)) in the least-squares sense.

Following another strategy, we chose to estimate the adjoint source by solving the linear system corresponding to the discrete form of Equation (39), with equations given as follows:

$$\begin{array}{c}\hfill \Delta E^{*}({\mathbf{x}}_r,{\omega }_q)=\sum _{n=1}^{N_t}{A}_{qn}{J}^{\prime }({\mathbf{x}}_r,t_n^{\prime }),\end{array}$$

(42)

$$\begin{array}{c}\hfill A_{qn}=-\Delta t^{\prime }\sqrt{{\displaystyle \frac{-2{\omega }_0}{i{\omega }_q}}}{e}^{-\sqrt{{\omega }_q{\omega }_0}{t}_n^{\prime }}{e}^{i\sqrt{{\omega }_q{\omega }_0}{t}_n^{\prime }},\end{array}$$

(43)

where q is the index associated with the frequencies, i.e., $q\in \{1,\cdots ,N_{\omega }\}$, and $N_t$ is the number of time samples. To stabilize the system solution, we added the minimum norm regularization with coefficient $\lambda$ [25], and we also used preconditioning via reparameterization:

$${\mathbf{J}}^{\prime }=\mathbf{P}\mathbf{m},$$

(44)

where the preconditioning operator $\mathbf{P}$ has the form of a Ricker pulse with peak frequency $f_p$, expressed as follows:

$$P_{nk}=\left(1-2{\pi }^2{f}_p^2{\left(t_n-t_k\right)}^2\right)e^{-{\pi }^2{f}_p^2{\left(t_n-t_k\right)}^2}.$$

(45)

Finally, we obtain the adjoint source (in the fictitious time domain) by solving the linear system:

$$\begin{array}{c}\hfill \left(\begin{array}{c}\mathbf{AP}\\ \lambda \mathbf{I}\end{array}\right)\mathbf{m}=\left(\begin{array}{c}\Delta {\mathbf{E}}^{*}\\ \mathbf{0}\end{array}\right).\end{array}$$

(46)

We obtain the solution through the conjugate gradient algorithm [19].

2.4. Gradient Preconditioning

The gradient calculated according to Equation (27) is subjected to a preconditioning process [26], used to compensate for its attenuation with depth. We implement two different approaches. The first approach, preconditioning 1, involves calculating the ratio of the gradient to the squared modulus of the electric field in each model sample, requiring calibration to avoid amplifying the gradient at the edges. In practice, at each frequency, the division is performed by the sum of the normalized field with a constant:

$$g_{precond}={\displaystyle \frac{g}{\left(\frac{E}{\max \left(E\right)}+\delta \right)}},$$

(47)

where $E=E_x^2+E_y^2+E_z^2$, and $\delta$ can be chosen by evaluating the square modulus of the field in the region of interest.

The second strategy (referred to as preconditioning 2) involves essentially applying a depth gain factor by multiplying the gradient by an exponential [27]:

$$g_{precond}=g\times e^{kz/\delta },$$

(48)

where k is a dimensionless parameter (close to $1.0$), z is the depth related to the seabed, and $\delta$ is the skin-depth [28] at the frequency of the gradient calculation.

2.5. Model Regularization

We used the model reparameterization method proposed by [18] to promote the model’s smoothness. In this case, we assume the following:

$$\mathsf{\sigma }={\mathbf{S}}_2{\mathbf{S}}_1\mathbf{p},$$

(49)

where $S_1$ and $S_2$ are exponential filters [29] that depend on the size of the previously defined smoothing window and operate in the directions where the model parameters vary.

We can obtain the gradient relative to the parameter vector, $\mathbf{p}\left(\mathbf{x}\right)$, using the chain rule:

$${\displaystyle \frac{\partial \chi }{\partial \mathbf{p}}}={\displaystyle \frac{\partial \chi }{\partial \mathsf{\sigma }}}{\displaystyle \frac{\partial \mathsf{\sigma }}{\partial \mathbf{p}}}=S_2{S}_1{\displaystyle \frac{\partial \chi }{\partial \mathsf{\sigma }}}.$$

(50)

2.6. Steepest Descent and Momentum Methods

CSEM inversion aims to obtain a subsurface conductivity model representative of the investigated region and honors the measurements of electromagnetic fields collected by receivers on the seabed. It is an iterative process that seeks to minimize the objective function defined in Equation (23) through successive steps in the parameter space, using the direction opposite to the gradient of the objective function calculated from the model obtained in each iteration. This update form is the main feature of steepest descent methods [20], which leads to the following expression:

$${\mathsf{\sigma }}_{k+1}={\mathsf{\sigma }}_k+\alpha {\mathbf{d}}_k,$$

(51)

where $\alpha$ is the step size, determined by a simple backtracking line search, and the update direction, ${\mathbf{d}}_k$, is given by the preconditioned gradient, ${\mathbf{g}}_k$, of $\chi \left(\mathsf{\sigma }\right)$ relative to the model conductivity:

$${\mathbf{d}}_k=-{\displaystyle \frac{{\mathbf{g}}_k}{\parallel {\mathbf{g}}_k\parallel }}.$$

(52)

We use the most straightforward line search strategy, starting with a step size equal to $1.0$ in Equation (51). We calculate the value of the objective function using the updated model, and if there is a decrease, we proceed to the next iteration. Otherwise, we discard the update and compute a new model with a step size that is half of the previous step. This process iterates until the objective function decreases or we reach the maximum number of iterations allowed, in which case the inversion stops. In our experiments, this maximum number is 20.

Another inversion strategy used involves updating the model with a momentum vector, $\mathbf{s}$, defined from the gradient of the objective function and the momentum calculated in the previous iteration as follows:

$$\begin{array}{cc}\hfill {\mathsf{\sigma }}_{k+1}& ={\mathsf{\sigma }}_k-\alpha {\widehat{\mathbf{s}}}_k,\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill \widehat{\mathbf{s}}={\displaystyle \frac{\sqrt{1-{\beta }_2^k}}{1-{\beta }_1^k}}{\mathbf{s}}_k& ;{\mathbf{s}}_k={\beta }_1{\mathbf{s}}_{k-1}+\left(1-{\beta }_1\right){\mathbf{d}}_k,\hfill \end{array}$$

(54)

where we adopt the parameters ${\beta }_1=0.9$ e ${\beta }_2=0.999$, according to the usual prescription of the Adam algorithm [21,30], formulated initially with a second momentum (which we do not use here). Indeed, our momentum update is very similar to the formulation of the Adam algorithm, where we essentially remove the term involving the second momentum from the equation. Furthermore, although it is not common practice, we combine momentum with the line search strategy already adopted in the context of steepest descent in an attempt to obtain more accurate models.

3. Numerical Experiments

3.1. Gradient Calculation

We validate our implementation to calculate the gradient of the objective function using the data from a 3D isotropic model measuring 18 km in the x direction, 10 km in the y direction, and 3 km deep (in the z direction), inspired by [31]. This model simulates a 2D distribution of conductivities consisting of a resistive body with a resistivity of 25 ohm.m, in a homogeneous background medium with 1 ohm.m, and a water layer of 800 m, with its central section shown in Figure 1. We also use a uniform grid with cell dimensions of 100 m (vertical) × 200 m × 200 m (in this work, we use a 3D modeling code, and to ensure compatibility with the 2D inversion codes, we adopt the strategy of replicating the models obtained during the optimization process along the third dimension).

With the model illustrated in Figure 1, we simulate, using reciprocity, the survey along its central section to use the horizontal component of the electric field (at frequencies of 0.25 Hz, 0.50 Hz, and 0.75 Hz) measured by the receivers on the seafloor with a regular spacing of 1 km, from coordinates 6 km to 12 km. The sources, located 100 m above the receivers, are distributed at intervals of 200 m, between 1.1 km and 16.9 km. Figure 2 illustrates the data associated with the central receiver at the three frequencies (since we use reciprocity, which allows the interchange between sources and receivers, the data illustrated in Figure 2 correspond to the field measured at the source positions for a source attributed to the position of the central receiver).

The estimation of the adjoint source involves solving the system of Equation (46) for the residual (right-hand side of the equation) at each offset, such that at the end of the process, the conjugate gradient routine provides a pulse in the (fictitious) time domain (vector $\mathbf{m}$) as the solution. An important aspect of the approach we adopted is related to the need to fit the residuals at each frequency separately, in contrast to the more intuitive strategy of simultaneously fitting the residuals at the frequencies where the gradient is to be calculated, as we can see in [17]. The adoption of this approach is essential to obtain the results that will be presented below.

Next, we analyze the accuracy of our adjoint source estimation, considering the residuals between the observed data and the data calculated with a model consisting only of the 1 ohm.m background present in the model shown in Figure 1. Figure 3 illustrates the estimated pulses (at each frequency) for source 60 and the data from receiver number 4 and their amplitude spectra, along which we indicate the residuals (in the frequency domain) adjusted during obtaining adjoint sources.

We also illustrate the gradient calculation relative to the vertical component of the conductivity tensor based on data for receiver 1 at a frequency of 0.50 Hz. Figure 4 shows the absolute value (in log scale) of the $E_z$ component of the electric field normalized by the source and the field ${\lambda }_{E_z}$, computed using forward and adjoint modeling, respectively, as well as their product, calculated according to Equation (27). To improve the visualization, we apply muting in a depth range corresponding to five samples below the seafloor.

The “complete” gradient is obtained by summing the gradients calculated from the data of each receiver. Figure 5 shows the results for frequencies of 0.25, 0.50, and 0.75 Hz, which have similar characteristics in shape but different depth ranges. Figure 6 illustrates the effect of the preconditioning strategies discussed in Section 2.4, where we show the result of their application to the gradient obtained with the 0.25 Hz data. As we can see, precondition 1 (performed with $\delta ={10}^{-4}$) positions the anomaly deeper while precondition 2 (performed with $k=1.0$) increases the depth range of the anomaly.

3.2. Inversion Results

In this section, we will discuss the inversion results of the data obtained under three different scenarios. The first is similar to the model used in Section 3.1, consisting of two resistors located at varying depths in a homogeneous background medium. The second is an adaptation of one of the demonstration models from the MARE2DEM modeling and inversion package (available at http://mare2dem.ucsd.edu acessed on 11 December 2023) [32], and the third corresponds to an adaptation of the synthetic model of the Marlim field (on the Brazilian continental margin) [33], constructed from seismic sections and information from the main horizons in the area. The optimizers used will be the steepest descent and the momentum method described in Section 2.6, as well as L-BFGS-B [22] (computed using a limited-memory matrix using the previous five gradients).

3.2.1. Model with Two Resistors

The model with two resistors has the same background as in Section 3.1, with the same resistivity and water layer thickness. The difference is that there are two resistors (not just one) located at depths of 1.8 km and 2.1 km (Figure 7a). The data correspond to the simulated horizontal electric field measured between 3 and 15 km, with sources every 200 m, in the interval of 1.1 km to 16.9 km. The data from each receiver were restricted to offsets between 1 and 6 km. Figure 7b illustrates the gradient at the frequency of 0.25 Hz, relative to the vertical conductivity, computed in a homogeneous medium with a resistivity of 1 ohm.m. As we can see, there is consistency in the regions where the gradient is prominent, considering the position of the resistors in the model used to generate the observed data. The higher value in the left portion is a consequence of the shallower depth of the resistor located in that region.

The inversions used data at a frequency of 0.25 Hz on a mesh with a grid interval of 100 m (along the vertical) and 200 m (along the horizontal) and a homogeneous initial model with a resistivity of 1 ohm.m. Below, we present the best results obtained with each tested method. In inversions with the steepest descent and momentum, we used preconditioning 1 ($\delta ={10}^{-4}$), and in inversion with L-BFGS-B, we used preconditioning 2 ($k=1.0$). Figure 8 shows the curves of the objective function’s evolution during the inversions, highlighting the greater efficiency of the steepest descent method, which takes just over 20 iterations to reach a value comparable to that obtained by the momentum inversion after 60 iterations. The L-BFGS-B method is similar to the steepest descent, although it takes more iterations to reach the minimum of the objective function value.

The models obtained from the inversions are similar (Figure 9). They are indicated by the two resistive bodies at different depths, consistent with the resistors’ positions in the model used to produce the observed data. In the cases of the steepest descent and the momentum method, we observe that the deeper resistor is slightly above the desired position, unlike the model obtained with the L-BFGS-B method, in which this resistor is better positioned.

3.2.2. MARE Model

The model we will call MARE consists of an adaptation of the isotropic conductivity model available in one of the versions of the MARE2DEM package for modeling and inversion (the MARE2DEM package [32] is a CSEM 2.5D data modeling and inversion software suite based on finite elements. The code was initially developed as part of the SEMC consortium (Seafloor Electromagnetic Methods Consortium) at Scripps Institution of Oceanography, University of California, San Diego, USA). It simulates the presence of two bodies: a smaller one with a resistivity of 50 ohm.m and a larger one, to the left of the first, of 4 ohm.m. These resistors are embedded in a medium formed by conductive layers and an extended structure with 100 ohm.m, located in the lower portion of the model. The adaptation involved replacing the bathymetry present in the original model with a horizontal seabed at a depth of 1.2 km, as we can see in Figure 10.

The acquisition used in the simulation of observed data consisted of 25 receivers positioned on the seabed, regularly spaced between 1 and 25 km, and sources located 50 m above the receivers, from coordinate 0 to coordinate 26 km, with sampling every 200 m. We computed the synthetic data using the MARE2DEM modeling code [34]. The field used corresponds to the inline electric field. The receiver data were limited to offsets greater than 1 km and those with field amplitude greater than ${10}^{-15}$ V/m. The inversions used 0.25 Hz data.

To perform the inversions, we limited the vertical extension of the model to a depth of 6 km on a grid with a spacing of 100 m (vertical) by 200 m (horizontal), as seen in the initial model (shown in Figure 11a), constructed by smoothing the background formed by the conductive layers present in the reference model. Figure 11b illustrates the gradient relative to vertical conductivity at a frequency of 0.25 Hz, calculated with the model presented in Figure 11a. There is a strong correlation between the regions where the gradient stands out and the locations where the resistors are present in the reference model that produced the observed data.

Next, we present the best results obtained with each methodology presented. In the case of the steepest descent with line search (Figure 12a) and the inversion with momentum (Figure 12b), we used the gradient with precondition 2. In contrast, in the inversion with L-BFGS-B (Figure 12c), the best result was obtained using precondition 1. Figure 13 shows the convergence curves for the objective function. As we can see, the steepest descent and L-BFGS-B present the best convergence behavior compared to the inversion with the momentum algorithm. The estimated models are very similar, especially concerning the smaller and more resistive body on the right, which is well-defined by the three inversions (with resistivity values slightly closer to the reference model in the results obtained with momentum and L-BFGS-B). Regarding the larger resistor on the left, it is possible to notice that there is an overestimation of resistivity in the three results, with a slightly pronounced effect in the result of the inversion with the steepest descent and L-BFGS-B.

3.2.3. Marlim Model

The model we will refer to as Marlim is an adaptation of the MR3D resistivity model constructed by [33] from 3D seismic and well-log data from the Marlim field (Brazilian continental margin). The features of interest in using electromagnetic (EM) methods include a heterogeneous reservoir embedded in a conductive background and a salt body with a resistivity of 1000 ohm.m. It is a good representation of the type of scenario where the use of the CSEM method might be required, which was also described by [35] when discussing the forward modeling performed from the MR3D model.

The modifications made relative to the original MR3D model consist of removing the effects of bathymetry and using only the vertical resistivity of the original model (in this case, we considered the model to be isotropic). Figure 14 illustrates the central section of the model (Figure 14a) from which we simulated the observed data through modeling that used a grid of 20 m (vertical) by 100 m (horizontal) and recorded the inline electric field measured by 20 receivers regularly arranged along the line that coincides with the section in Figure 14a (in the interval from 11 km to 30 km). The sources are distributed between 0 to 41 km every 200 m. We discarded data with offsets smaller than 1 km and amplitudes smaller than ${10}^{-15}$ V/m, and the inversions used single-frequency data at 0.25 Hz. The initial model is homogeneous, with a resistivity of 2.8 ohm.m (which corresponds to the average background resistivity in the region where the reservoir is in the Marlim model), the same used in calculating the gradient relative to vertical conductivity (Figure 14b) on a mesh with a grid spacing of 100 m (along the vertical) and 200 m (along the horizontal). Note the asymmetry in the gradient shape associated with the location of the resistor in the reference model.

Using the data and the initial model described, the inversions performed produced the results that will be illustrated below. Using the steepest descent method with gradient precondition 1, we obtained the model shown in Figure 15a. In the inversion with momentum, where we used precondition 2, we obtained the model in Figure 15b, and the inversion with L-BFGS-B (performed using the gradient with precondition 1), resulting in the model shown in Figure 15c. All results indicate the presence of a resistor in the region where the reservoir is located in the Marlim model, each of them presenting its own characteristics. The models obtained with the steepest descent and L-BFGS-B are similar, as they are characterized by an anomaly with higher resistivity in the case of the L-BFGS-B inversion. The result obtained with momentum is better distributed, covering a larger reservoir extent. The curves showing the evolution of the objective function during the inversions are shown in Figure 16, indicating a slightly more effective reduction rate in the case of L-BFGS-B, especially in the first iterations. In contrast, the others show similar behavior, with the momentum method reaching a slightly lower residual value.

4. Discussion

The results illustrated in the previous section consolidate the use of the adjoint-state method and the correspondence principle to obtain the gradient (relative to the model conductivities) of the objective function defined in Equation (23). The fictitious domain modeling strategy, already established in the context of solving the forward problem, also proved efficient in obtaining the adjoint fields. For the calculation of the gradient consistent with the perturbation observed in the data due to the presence of a resistor in the model, we opted for the approach based on obtaining the adjoint sources for each frequency separately, unlike other authors who address the topic, as seen in the work of [36].

We validated the quality of the gradient calculated with the proposed methodology, preconditioning, and regularization using model parameterization through the inversions of simulated data from three different models: the model consisting of two resistors in a conductive background, the MARE model, an adaptation of the example model from the MARE2DEM package, and the Marlim model, adapted from the MR3D model, constructed by [33]. We used three optimization methods: steepest descent, our proposed modified momentum algorithm, and L-BFGS-B.

Regardless of the dataset we used and the inversion methodology employed, we observed interesting aspects in the inversion results, which produced models with well-positioned resistors at depth and without artifacts near the receiver positions, commonly seen in CSEM inversion results. We attribute these achievements to the use of gradient preconditioning and smoothing by reparameterization.

For the model with two resistors, all three methods successfully obtained results that included the bodies at different depths, with the positioning of the deeper resistor being slightly better in the case of L-BFGS-B (Figure 9). From the perspective of the number of iterations needed for the objective function to reach its minimum value, L-BFGS-B was slightly slower than the steepest descent, with the momentum inversion requiring the most iterations (Figure 8).

The inversions for the MARE model were obtained using synthetic data produced with the MARE2DEM package, and the analysis of the objective function evolution curves during the inversions indicated slower convergence for the momentum optimization, which reached its lowest level after iteration 30, in contrast to the performance of steepest descent and L-BFGS-B, which did so at around iterations 20 and 15, respectively (Figure 13). The results obtained were similar but presented a subtle difference in the resistivity of the larger resistor, which was slightly overestimated in some cases (Figure 12). This feature was somewhat more pronounced in the model obtained with the steepest descent and less noticeable in the model obtained with momentum.

In the inversions for the Marlim model, we observed similar behavior in the convergence curve of the objective function during the first iterations with the steepest descent and momentum; the L-BFGS-B showed a sharper reduction followed by stagnation that lasted for about ten iterations, after which the objective function began to decrease again (Figure 16). The models obtained with the three optimizers differed (Figure 15). While the steepest descent result indicated the presence of a modest and localized resistor, the result obtained with momentum was more consistent with the overall geometry of the reservoir. The result produced with L-BFGS-B was somewhat similar to that obtained with the steepest descent but slightly better as it identified a larger and more resistive resistor.

The analysis of the results produced with the different optimization methodologies indicates that the momentum method tends to fit the observed data slightly better compared to the steepest descent method (as observed in the experiments conducted with the data from the two-resistor model and the MARE model). In the inversion for the Marlim model, this approach not only fit the data better but also recovered a model that was more consistent with the reference model.

5. Conclusions

In this work, we implemented CSEM inversion without solving large linear systems, which can be extremely useful in the context of 3D problems, where this task might become prohibitive. Our implementation used the correspondence principle to derive the forward modeling and gradient calculation. To improve the conditioning of the inversion, we used gradient preconditioning and model reparameterization smoothing. As a result, we obtained very satisfactory models from the inversion of data produced from three different scenarios, using the steepest descent and the momentum algorithm and comparing them to the results from L-BFGS-B. In the two-resistor and the MARE models, the superior performance of the steepest descent and L-BFGS-B was evident, as they required fewer iterations to achieve results similar to those obtained by the momentum method (which needed more iterations). In the case of the Marlim model, the performances were comparable, with better models achieved by the momentum method and L-BFGS-B. Our proposal for using momentum is based on its update from two parameters, similar to what is done in the formulation of the Adam algorithm. However, unlike this one, we used only one momentum term. As we observed, the results from the algorithm we proposed are comparable to the models produced with L-BFGS-B, with the advantage of requiring less memory consumption and fitting the observed data slightly better.