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Simultaneous inversion for source field and mantle electrical conductivity using the variable projection approach
Earth, Planets and Space volume 75, Article number: 83 (2023)
Abstract
Timevarying electromagnetic field observed on the ground or at a spacecraft consists of contributions from (i) electric source currents, such as those in the ionosphere and magnetosphere, and (ii) corresponding fields induced by source currents within the conductive Earth’s interior by virtue of electromagnetic induction. Knowledge about the spatiotemporal structure of inducing currents is a key component in ionospheric and magnetospheric studies, and is also needed in space weather hazard evaluation, whereas the induced currents depend on the Earth’s subsurface electrical conductivity distribution and allow us to probe this physical property. In this study, we present an approach that reconstructs the inducing source and subsurface conductivity structures simultaneously, preserving consistency between the two models by exploiting the inherent physical link. To achieve this, we formulate the underlying inverse problem as a separable nonlinear leastsquares (SNLS) problem, where inducing current and subsurface conductivity parameters enter as linear and nonlinear model unknowns, respectively. We solve the SNLS problem using the variable projection method and compare it with other conventional approaches. We study the properties of the method and demonstrate its feasibility by simultaneously reconstructing the ionospheric and magnetospheric currents along with a 1D average mantle conductivity distribution from the ground magnetic observatory data.
Graphical Abstract
Introduction
Time variations of magnetic field that we observe on the ground or at a spacecraft represent a superposition of the inducing (primary) and induced components. There is substantial interest in knowing both inducing and induced components of the field as accurately as possible. On the one hand, knowledge about spacetime variability of the inducing field constrains the state of source electric currents in the ionosphere and magnetosphere (Yamazaki and Maute 2017; Balasis and Egbert 2006; Tsyganenko 2019), which in turn represents a crucial input for accurate geomagnetic field modelling (Maus and Weidelt 2004; Finlay et al. 2017) and space weather hazard evaluation (Pulkkinen et al. 2003; Kelbert 2020; Juusola et al. 2020). On the other hand, relation between the inducing and the induced field variations, governed by Maxwell’s equations, can be used to probe the electrical conductivity distribution in the Earth’s subsurface (Olsen 1999; Kuvshinov 2012; Kelbert et al. 2009). However, separation of the magnetic field into inducing and induced components is often nontrivial owning to their nonlinear relationship that depends on the 3D distribution of electrical conductivity in the Earth’s interior. Our goal here is to elaborate on this problem further.
To keep the study concise and focused, we make several assumptions that are implied in the derivations and discussions below. First, we concentrate on timevariations with periods longer than a few hours, which is beyond the band where a simple planewave source assumption is valid (this assumption can be used to model external source fields (Kelbert and Lucas 2020) and can be used in the magnetotelluric method (Chave and Jones 2012) for probing the electrical conductivity of subsurface). Second, we assume that field variations are due to the extraneous electric currents and the corresponding electromagnetic response from the conductive Earth’s interior. In other words, the contributions from all other magnetic field sources, such as the crust or the core, were subtracted from the data (it is clear that some residual fields from these sources are always present, but these problems are beyond the scope of our study). Further, the extraneous electric currents are assumed to have their origin in the ionosphere and magnetosphere. By this, we exclude the oceaninduced electromagnetic fields, which require dedicated modelling and inversion approaches (e.g., Velímský et al. 2018).
In the most general form, the extraneous source structure needs to be parameterized with spatially heterogeneous functions and estimated from the data along with the subsurface electrical conductivity distribution by solving a corresponding inverse problem. However, joint estimation of conductivity and external field structures represents a notoriously difficult task. Conventionally, the Gauss method has been used to separate the magnetic field into time series of Spherical Harmonic (SH) coefficients of internal and external origins (Backus et al. 1996). By relating internal and external SH coefficients, one can estimate a transfer function between them and perform the inversion in terms of subsurface electrical conductivity (Olsen 1999; Schmucker 1999; Kuvshinov 2012) or fit the time series of SH coefficients directly (Velímský and Knopp 2021). However, this approach is only applicable to potential fields where inducing field contribution is external to the observer. Moreover, due to sparse measurements, one is typically limited to using a small set of Spherical Harmonic functions to describe the inducing and induced parts of the field (Kuvshinov et al. 2021; Velímský and Knopp 2021).
Recognizing these limitations, a number of recent studies (Koch and Kuvshinov 2013; Sun et al. 2015; Guzavina et al. 2019; Egbert et al. 2021; Zhang et al. 2022) have adopted an alternative strategy where the source structure is estimated given some prior knowledge about the subsurface conductivity. With this estimated source structure, the inversion in terms of subsurface conductivity is subsequently performed and the updated conductivity model can again be used to reestimate the source coefficients. This approach (i) allows for a more general ansatz to describe the source geometry (Zenhäusern et al. 2021; Egbert et al. 2021), (ii) enables derivation of alternative families of transfer functions (Püthe and Kuvshinov 2014; Guzavina et al. 2019), which are not limited to the potential field assumption, and (iii) facilitates incorporation of the prior knowledge on the induction effects due to the ocean and marine sediments (Grayver et al. 2021). Listed points make it possible to mitigate or completely overcome the limitations imposed by the conventional Gauss method. In these aforementioned studies, determination of the inducing source field and the mantle conductivity is performed in an alternating manner on the two separate model spaces (hereinafter, we term this procedure an ”alternating approach”). Such separate estimation of the two model spaces is assumed to result in progressively refined knowledge of both the source and conductivity models.
In this study, we develop this idea further and pose a problem in a general form that allows us to simultaneously estimate the source and subsurface conductivity directly from the data. Since the model space consists of one part (i.e., inducing source currents), upon which the dependence of the observable is linear, and another part (i.e., subsurface electrical conductivity), which enters the objective in a nonlinear manner, the underlying inverse problem (under squared loss) belongs to a class of special optimization problems known as the Separable Nonlinear LeastSquares (SNLS) problem. We will show that the naive ”alternating approach” described above is the simplest way of solving the SNLS problem, although it may lack consistency and suffer from slow convergence. We will explore more efficient ways of solving the SNLS problem. In particular, the variable projection method (hereafter referred to as VP) has been proposed as an optimal method for solving SNLS problems that benefits from both computational efficiency and fast convergence (Golub and Pereyra 1973, 2003). In essence, VP exploits the linear dependency in one part of the model and estimates this part via linear least squares at each iteration, thus optimally (in leastsquares sense) projecting the complete model space onto a reduced subspace for efficient nonlinear optimization.
The advantage of variable projection naturally appeals to a number of geophysical inverse problems where the unknown parameters intrinsically constitute separable least squares. Such behavior is typical of seismic wave propagation and electromagnetic induction, where source characterization is linearly filtered by a medium response that depends nonlinearly on medium properties. In the last decade, this algorithm has been recognized in seismology as an efficient way to invert for velocity structure while simultaneously characterizing the source (Rickett 2013; De Ridder and Maddison 2018), the sourcerelated calibration parameters (Li et al. 2013), or both the source and the receiver factors (Hu et al. 2021). Despite an early conceptualization (Fainberg et al. 1990), this method, to our knowledge, has not yet been elaborated in the context of electromagnetic induction problems, where the merit of VP is potentially much more pronounced: the full model inversion including the source and conductivity, which is prohibitive due to aforementioned high dimension and nonlinearity, becomes tractable thanks to linear variable projection. Here we present the application of the VP method to a problem of electromagnetic induction sounding.
We demonstrate that not only does VP enable simultaneous estimation of the inducing field structure and the electrical conductivity using a natural physical link between them, but it also provides insights into the interplay between determination of inducing field and conductivity models.
Methods
Electromagnetic (EM) field variations are governed by Maxwell’s equations, which in the frequency domain read
where \(\omega\) is the angular frequency, \(\mathbf{r}\) is the position vector, \(\sigma (\mathbf{r}) \in \mathbb {R}\) denotes electrical conductivity of a medium, and \(\mathbf{B}(\mathbf{r}, \omega )\) and \(\mathbf{E}(\mathbf{r}, \omega )\) are the magnetic and electric fields, respectively. \(\mathbf{j}(\mathbf{r}, \omega )\) is the extraneous (impressed) current density. The extraneous currents are assumed to originate within the ionosphere and magnetosphere, separated from the solid Earth by a layer of insulating air. We neglected displacement currents and took \(\mu = \mu _0\) for the magnetic permeability. Here, we adopted the following convention for the Fourier transform:
The solution of Eq. (1) can be found when both the inducing source \(\mathbf{j}\) and the conductivity \(\sigma\) are given. In this case, the magnetic field due to an arbitrary distribution of current density can be formally expressed as
where \(\mathbf{G}\) is the Green’s tensor of the medium (Kuvshinov 2008) and \(\Omega\) is the volume occupied by extraneous currents. A corresponding timedomain counterpart contains a temporal convolution, and has the form
Independent of the domain where we operate, the modelling process can always be expressed in the operator form
where \(\mathbf{L}(\sigma ): (V_{\mathbf{j}}(\Omega _{\mathbf{j}}))^3 \mapsto (V_{\mathbf{B}}(\Omega _{\mathbf{B}}))^3\) is a linear operator, mapping an electric current field to a vector magnetic field, and \(\mathbf{L}(\cdot ): V_\sigma (\Omega _\sigma ) \mapsto \mathcal {L}((V_{\mathbf{j}}(\Omega _{\mathbf{j}}))^3, (V_{\mathbf{B}}(\Omega _{\mathbf{B}}))^3)\) is a nonlinear function that maps the conductivity distribution into a linear operator. Here, \(\Omega _\mathbf{j}\), \(\Omega _\sigma\) and \(\Omega _\mathbf{B}\) are the domains of inducing currents, induced currents, and observed magnetic fields, respectively. \(V_\mathbf{j}\), \(V_\sigma\), and \(V_\mathbf{B}\) are function spaces on the corresponding domains, and \(\mathcal {L}(U, V)\) denotes the linear space of the linear maps from U to V.
Equation 5 shows that the magnetic field is related to the source by a linear operator, which is a nonlinear functional of the electrical conductivity. The equivalent for the electric field is straightforward, but we omit it because we only consider magnetic field observations in this study. We can thus express the forward modelling in a concise algebraic form as
where \(\mathbf{d}^{\text{mod}}\) is the modelled data vector, \(\mathbf{m}\) is a parameterization of the conductivity model, \(\mathbf{c}\) is the inducing source vector, and \(\mathbf{F}(\mathbf{m})\) is a functional of \(\mathbf{m}\) that links the field to the extraneous currents. The specific form of \(\mathbf{F}(\mathbf{m})\) depends on the adopted discretization and parameterization of \(\sigma\) and \(\mathbf{j}\), but the stated general algebraic form accommodates the full set of modelling approaches. Our goal is to estimate the unknown variables consisting of electrical conductivity model \(\mathbf{m}\) and extraneous currents \(\mathbf{c}\) from observations of the magnetic field taken at specified locations and times. To achieve this goal, we seek a combination of \(\mathbf{m}\) and \(\mathbf{c}\) that minimizes the data misfit
where \(\mathbf{d}^{\text{obs}}\) is the observational data vector, given by magnetic field observations in our case, and \(d(\cdot , \cdot )\) denotes the distance metric induced by the corresponding Banach space. A popular choice for such metric in EM induction soundings is the distance induced by the vector norm weighted by the data covariance
where \(\mathbf{C}_d\) is the data covariance matrix. Note we use the superscript H to denote the Hermitian transpose of the matrix or vector, as the data vector may be complex. In the absence of covariances, the data samples are assumed to be mutually independent, in which case \(\mathbf{C}_d = {\text{diag}}\left( s_i^2\right)\), where \(s_i^2\) is the variance of the ith datum. Introducing \(\mathbf{W} = \mathbf{C}_d^{1/2} = {\text{diag}}(s_i^{1})\), the data misfit can be rewritten as the squared \(\ell _2\) norm of the weighted residual
Here, \(\Vert \cdot \Vert _2\) denotes the \(\ell _2\) norm, \(\mathbf{r} = {\mathbf{d}^{\text{obs}}}  {\mathbf{F}}({\mathbf{m}}) {\mathbf{c}}\) is the residual vector, and \(\mathbf{r}_w = \mathbf{W}\mathbf{r}\), \(\mathbf{d}_w = \mathbf{W} \mathbf{d}\), and \(\mathbf{F}_w = \mathbf{W} \mathbf{F}\) are the weighted forms of the residual vector, the data vector, and the linear operator, respectively. To mitigate the inherent nonuniqueness of the problem, we add a regularization term \(\lambda R(\mathbf{m})\), where \(R(\cdot )\) is the penalty function, and \(\lambda\) is the regularization strength. Here, we consider the penalty function that penalizes the \(\ell _2\) norm of model structure, given by \(R(\mathbf{m}) = \frac{1}{2}\Vert \varvec{\Gamma }\mathbf{m}\Vert _2^2\), where \(\varvec{\Gamma }\) is known as the Tikhonov matrix. In the experiments that follow, we shall use a firstorder difference operator as \(\varvec{\Gamma }\) to enforce the smoothness of the model. Similarly, we can also regularize the linear parameters directly, i.e., \(\frac{\lambda _c}{2}\left\ \varvec{\Gamma }_c \, \mathbf{c}\right\ _2^2\). The full optimization problem is then given by
Regularization on the source parameters \(\mathbf{c}\) allows the use of prior knowledge on the source geometry (Sun et al. 2015; Laundal et al. 2021), e.g., while using a highdimensional parameter space for the source. This would become beneficial or even necessary when an accurate complex source estimation is required. In this study, however, we shall proceed by defining \(\lambda _c = 0\), as our experiments all assume a lowdegree source structure. Generalization to include the regularization on \(\mathbf{c}\) is mathematically straightforward but further complicates the formulas, and hence is only documented for completeness in Appendix A.
Seeking a solution to the stated problem directly in the joint model space of \(\mathbf{m}\) and \(\mathbf{c}\) induces a fully nonlinear leastsquares problem in a highdimensional space. We note again that the magnetic field is a linear functional of the extraneous currents represented by \(\mathbf{c}\), but a nonlinear functional of the subsurface electrical conductivity expressed through \(\mathbf{m}\). This particular structure of the inverse problem with the data misfit defined in Eq. 9 makes it an example of the socalled separable nonlinear least squares (SNLS). While the problem in Eq. 10 can be linearized and solved in the full model space, this ”naive” approach is inefficient and prohibitive for problems of interest. Fortunately, the particular structure of an SNLS problem allows us to adopt more efficient solution strategies.
Variable projection approach
Variable projection (VP) has been first proposed by Golub and Pereyra (1973) as an optimization method for solving SNLS problems. Exploiting the linear dependency on \(\mathbf{c}\), at each given conductivity model \(\mathbf{m}\), the bestfitting linear part can be obtained via a linear regression \(\mathbf{c} = \hat{\mathbf{c}}(\mathbf{m}) = \mathbf{F}_w^{\dagger }(\mathbf{m}) \, \mathbf{d}_w^{\text{obs}}\), where \(\mathbf{F}_w^{\dagger }\) denotes the Moore–Penrose pseudoinverse of \(\mathbf{F}_w\). We use \(\hat{\mathbf{c}}\) to explicitly denote the dependency of the leastsquares solution of \(\mathbf{c}\) on \(\mathbf{m}\). With the linear regression at each iteration, the optimization is then optimally (in statistical sense) constrained to the nonlinear part of the model space
where \(\mathbf{P}_{\mathbf{F}_w}^\perp = \mathbf{I}  \mathbf{F}_w\mathbf{F}_w^{\dagger }\) is a projector onto the orthogonal complement of the range of \(\mathbf{F}_w(\mathbf{m})\). Note we used \(\mathbf{d} = \mathbf{d}^{\text{obs}}\) for brevity.
A minimum to the nonlinear leastsquares problem in Eq. 11 can be found using either a gradientbased or a Newtonbased optimization method. In both cases, the update on the nonlinear model involves evaluation of the Fréchet derivatives with respect to the nonlinear parameters. In turn, this requires us to incorporate the implicit dependency of \(\mathbf{c}\) on \(\mathbf{m}\). In what follows, we will use \(\textsf{D} \mathbf{A}\) to denote the derivative of \(\mathbf{A}\) with respect to \(\mathbf{m}\), where \(\mathbf{A}\) is a functional of \(\mathbf{m}\). In its discrete form where \(\mathbf{A} \in \mathbb {C}^{i_1\times i_2\times \cdots i_l}\), the result \(\textsf{D} \mathbf{A}\) is a tensor of order \(l+1\), and the last dimension denotes the differentiation component. More explicitly
For \(l\ge 2\), matrix multiplications involving \(\textsf{D} \mathbf{A}\) are always assumed to be performed on the leading 2 dimensions. Golub and Pereyra (1973) derives the expression for the gradient of the objective function and the Jacobian of the residual vector in terms of pseudoinverses and derivatives of the linear operator \(\mathbf{F}_w\). We adopt the notations used in Hong et al. (2017), and introduce the following two partial Jacobians, as derivatives taken explicitly on the original data misfit without variable projection (Eq. 9)
Now, the linear projection can also be stated as \(\hat{\mathbf{c}} =  \mathbf{J}_c^{\dagger } \, \mathbf{d}_w\), and the orthogonal projector is given by \(\mathbf{P}_{\mathbf{F}_w}^\perp = \mathbf{I}  \mathbf{J}_c \mathbf{J}_c^{\dagger } = \mathbf{P}_{\mathbf{J}_c}^\perp\). We note that the two explicit Jacobians are coupled in the model space (i.e., \(\mathbf{J}_m\) and \(\mathbf{J}_c\) are dependent upon \(\mathbf{c}\) and \(\mathbf{m}\), respectively). This will be clearly seen in the case of VP, where the complete Jacobian of the variableprojected misfit term (Eq. 11) is given by
Invoking the derivative of pseudoinverse (see Golub and Pereyra 1973 for derivation details)
the complete Jacobian of the variableprojected system can hence be reiterated and expressed solely in terms of \(\mathbf{J}_m\), \(\mathbf{J}_c\) together with its derivative and pseudoinverse
The last step uses the fact that \(\mathbf{A} \mathbf{A}^{\dagger } (\mathbf{A}^{\dagger })^H = \left( \mathbf{A}\mathbf{A}^{\dagger }\right) ^H \left( \mathbf{A}^{\dagger }\right) ^H = \left( \mathbf{A}^{\dagger } \mathbf{A} \mathbf{A}^{\dagger }\right) ^H = \left( \mathbf{A}^{\dagger }\right) ^H\) and \(\mathbf{A} (\mathbf{P}_{\mathbf{A}^H}^\perp )^H \equiv \mathbf{0}\). Part of the dependency of \(\mathbf{c}\) upon \(\mathbf{m}\), namely the 3rd term in Eq. 15, has no contribution to the complete Jacobian, since it is perpendicular to \(\mathbf{J}_c\). The complete Jacobian reads
Note that if an inverse problem were posed solely in the space of conductivity model, then only the first term, namely \(\mathbf{J}_m\), would be present. The trailing two terms involve the dependency of the source estimate on the change in the subsurface conductivity, confining the model updates of \(\mathbf{m}\) to the hyperplane defined by the regression of \(\mathbf{c}\). Reintroducing linear operators via Eq. 13, we arrive at the expression for the Jacobian of the residual vector
Accordingly, the gradient of the misfit function reads
The second line uses the fact that \((\mathbf{P}_{\mathbf{F}_w}^\perp )^2 = \mathbf{P}_{\mathbf{F}_w}^\perp\), and \(\mathbf{F}_w^{\dagger } \mathbf{P}_{\mathbf{F}_w}^\perp = \mathbf{F}_w^{\dagger } (\mathbf{I}  \mathbf{F}_w \mathbf{F}_w^{\dagger }) = \mathbf{0}\), and the last equality uses \(\mathbf{J}_m= \textsf{D} \mathbf{F}_w \hat{\mathbf{c}}\) and \(\mathbf{r}_w = \mathbf{P}_{\mathbf{F}_w}^\perp \mathbf{d}_w\). Equations 1819 define the first order Fréchet derivatives with the variable projection. To avoid higher order derivatives, we use the Gauss–Newton algorithm to update conductivity model, where the Hessian is approximated as \(\mathbf{H} \approx {\text{Re}}[\mathbf{J}^H \mathbf{J}]\). The model update thus takes the form
We refer to the inversion scheme that calculates Jacobian via Eq. 18 as the fullVP scheme. In the case of a 1D radial conductivity model, the calculation of \(\textsf{D} \mathbf{F}_w\) is cheap and can often be obtained semianalytically. For a general 3D conductivity model, the explicit evaluation and storage of \(\textsf{D} \mathbf{F}_w\) is often prohibitive. In this case, the adjoint method (Pankratov and Kuvshinov 2010; Egbert and Kelbert 2012) can be used to efficiently calculate the gradient and create a lowrank representation of the \(\textsf{D} \mathbf{F}_w\) (Egbert 2012) or solve Eq. (20) for the model update using Krylov subspace methods, both avoiding storage and evaluation of large matrices (e.g., Jacobian). However, even without explicit evaluation of Jacobian, the fullVP algorithm entails additional calculations due to interactions between the linear and nonlinear parts of the model space. It is therefore desirable to explore approximations that allow for fewer evaluations of \(\textsf{D} \mathbf{F}_w\).
Two such approximations have been proposed by Ruhe and Wedin (1980). One option is to drop the last term in Eq. 17, effectively dropping the 2nd term in Eq. 18, yielding
We adopt the terminology used by Hong et al. (2017) and hereinafter refer to this as the VPRW2 scheme. The dropped term is considered a higher order refinement. This scheme retains high convergence rate and accuracy, while outperforming the fullVP in terms of computational efficiency (Ruhe and Wedin 1980; O’Leary and Rust 2013). The second option is to drop both the 2nd and the 3rd terms in Eq. 17, leading to the very simple form
hereinafter referred to as the VPRW3 scheme. This is equivalent to assuming fixed inducing currents (i.e., \(\textsf{D} \mathbf{c} = 0\)) at each iteration while searching for updates on the conductivity structure. This is in contrast to both fullVP and VPRW2 schemes, where Jacobian contains additional information on the implicit feedback of the source. These three variants are closely related in the scope of VP but have different levels of approximation, and will be compared in the context of EM induction sounding. Despite poor performance of the VPRW3 scheme previously reported by Hong et al. (2017) for matrix factorization problems, we chose to consider this scheme here, particularly because of its resemblance to what we call the ”alternating approach”, which we will revisit later under the framework of variable projection.
As a final remark, we observe that the gradient \(\textsf{D} \chi ^2\) always has the same expression as in Eq. 18, regardless of the approximation used for constructing the Jacobian. This is due to the fact that as \(\hat{\mathbf{c}} = \mathbf{F}_w^{\dagger } \mathbf{d}_w\) guarantees that the source parameters minimize the leastsquares misfit of the data, the residual inevitably lives in the orthogonal complement of the linear operator, and only manifest itself through \(\mathbf{J}_m\). In other words, as long as the current source estimation minimizes the data misfit, gradients do not ”sense” the implicit feedback of the source, but always view the source as truly fixed (as if it were the ground truth model of the source), as has been noticed by Aravkin and van Leeuwen (2012). Therefore, purely gradientbased optimization schemes are not affected by the choice of the variant of VP. Optimization schemes utilizing higher order information, such as Gauss–Newton method and Levenberg–Marquardt algorithms, are however different for different VP variants.
Alternating approach
Conventionally, models of magnetospheric/ionospheric current systems and models of the mantle electrical conductivity are estimated separately, using dedicated approaches. Combining these procedures, Koch and Kuvshinov (2013) proposed a scheme where, starting from an initial model of subsurface conductivity, one first obtains a preliminary estimate of inducing currents, then recalculates the conductivity model with the estimated source, and then goes back to refining the source with the ”updated” mantle conductivity. This procedure can in principle be repeated several times, until model estimates or data misfits reach certain convergence criteria. The same alternating method has most recently been utilized by Zhang et al. (2022) to invert for the conductivity in the mantle transition zone (MTZ), in combination with their physicsbased representation of the inducing currents.
Similar to variants of the variable projection, the alternating approach also offers a way to optimize on external currents and mantle conductivity simultaneously, without resorting to fully nonlinear inversion schemes. It can be viewed as a conglomeration of successive inversions, conventionally carried out independently with respect to external currents and mantle conductivity. The major difference from VP is that in the case of a naive alternating approach, once one part of the model is estimated, inversion on the other part is carried out in a complete standalone stage to minimize the objective. This behavior is especially pronounced during inversion of the electrical conductivity, where a significant number of iterations are usually needed to capture the nonlinear dependence of the predicted data on the conductivity model. In VP, estimate on the source is projected and updated at each iteration step and is only used for one update, whereas for alternating approaches, all iterations on the conductivity model in one inversion phase are conducted under a fixed source. Such approach may potentially lead to high redundancy in iterations and result in biased model estimates.
In this study, we revisit and generalize the idea of the alternating approach, by implementing a flexible version of the inversion scheme for our problem. Our implementation is based on nonlinear model updates: at each iteration, update on the nonlinear model is generated using the Gauss–Newton method, while the source is kept fixed. At iterations predefined by certain criteria (referred to as linear update criteria), the inducing source is updated. The scheme can be summarized by the following pseudocode:
By varying the linear update criterion, this implementation can potentially incorporate a spectrum of inversion schemes. For instance, by disabling update on the linear model until the inversion on the nonlinear part has converged (or stagnated), one obtains one endmember scenario, which is exactly the approach described in Koch and Kuvshinov (2013). This scenario contains the least frequent linear model updates. In contrast, by forcing linear model regression at each iteration, one obtains the other endmember, a scheme equivalent to the VPRW3 (Eq. 22). A customized linear update criterion allows for intermediate solutions between these two endmembers.
Both VP and alternating approaches provide alternative means to solve the joint model space inversion (Eq. 10). Although beyond the scope of this work, it can be further shown that the linear system for nonlinear updates resulting from VP/alternating approaches at each iteration is also closely related to that obtained in the joint model space inversion (see Appendix B). Therefore, these surrogate methods all sample subsets of the manifold describing the objective function in the higher dimensional joint model space.
As a final remark, we discuss in brief the computational aspects of different variants of VP and alternating approaches assuming a common scenario where evaluation of \(\textsf{D} \mathbf{F}_w(\mathbf{m})\) is the most resourcedemanding part in computation of Fréchet derivatives. Specifically, each evaluation of the matrix–vector multiplication \(\mathbf{u}^H (\textsf{D}\mathbf{F}_w \mathbf{v})\) or \(\mathbf{v}^H \textsf{D}\mathbf{F}_w^H \mathbf{u}\) would incur a forward or an adjoint solution of the electromagnetic modelling problem, respectively, with \(\mathbf{u}\) and \(\mathbf{v}\) being arbitrary vectors of matching dimensions. Following Eqs. 18–22, evaluation of the Jacobians in VPRW2 and VPRW3 involves only one evaluation of \(\textsf{D} \mathbf{F}_w\), while fullVP incurs two evaluations. Therefore, the computational cost per iteration for VPRW2 and VPRW3 is roughly half of that for fullVP. Compared to VPRW3, VPRW2 involves an additional linear projection in calculation of the Jacobian. Since the cost for linear regression is often marginal in comparison to evaluation of derivatives, the difference between the cost per iteration between VPRW2 and VPRW3 will not play a significant role. There is virtually no difference between the evaluation of Fréchet derivatives in VPRW3 and alternating approaches, and hence, the two approaches should be considered equal in terms of cost per iteration, except for extra linear regressions required at each iteration in VPRW3. As was already mentioned in the previous section, in cases where explicit (e.g., for a 3D conductivity parameterization) storage of the Jacobian is prohibitive, it can be avoided for all variants of VP or alternating approaches by evaluating Jacobianvector products on the fly. Obviously, this still preserves the relative cost of different methods discussed above.
Forward modelling
We remind the reader that the separation of the joint model space into linear and nonlinear parts is an innate property of EM induction sounding stemming from the governing Maxwell’s equations. Therefore, the formulation provided above is general and will apply to any electromagnetic imaging problem where both source and physical properties are unknown. To test different inversion approaches, we need to choose a specific form of inducing source parameterization \(\mathbf{c}\) and a forward modelling operator \(\mathbf{F}(\sigma )\). We limit the experiment in this study to a simple scenario satisfying the following two assumptions. First, we consider only observations made within a currentfree space between the inducing source and the induced currents. In other words, the observed magnetic field is assumed to be potential (\(\mathbf{B} = \nabla V\)), where the potential V can be expanded using Spherical Harmonic (SH) functions in the frequency domain as
where \(\sum _{n,m} \equiv \sum _{n=1}^N \sum _{m=n}^n\); \(Y_n^m(\theta , \varphi ) = P_n^{m}(\cos \theta )e^{\text {i}m \phi }\) is a complex SH function of degree n and order m, with \(P_n^{m}\) being Schmidt quasinormalized associated Legendre functions, \(\mathbf{r} = (r, \theta , \phi )\) is the position vector in spherical coordinates, and a is the Earth radius; \(\varepsilon _n^m\) and \(\iota _n^m\) are the external and internal SH coefficients, respectively. These assumptions will facilitate the comparison of our methods with conventional Gaussbased workflows.
Second, we assume a 1D radial conductivity structure of the Earth (that is, \(\sigma (\mathbf{r}) \equiv \sigma (r)\)). This assumption allows us to use a Qresponse to describe the induction in the model (Olsen 1999). Qresponse is a frequencydependent global transfer function (TF) that is independent of the SH order m for a 1D radially symmetric conductivity, and is formally defined as the ratio between the internal and the corresponding external Gauss coefficients
Then, the forward operator that links magnetic field (\(\mathbf{B}\)) with model parameters (external coefficients \(\varepsilon\) and conductivity \(\sigma\)) can be stated as follows:
Equation 26 gives the magnetic field at a position \(\mathbf{r}\) and at frequency \(\omega\) in terms of unknown variables \(\sigma\) and \(\varepsilon _n^m\), which can be written in the vector form as
where \(\mathbf{B}(\mathbf{r}, \omega ) \in \mathbb {C}^3\) is the vector magnetic field in the frequency domain, and \(\mathbf{B}_{n}^m (\mathbf{r}, \omega ; \sigma ) \in \mathbb {C}^3\) is the transfer function related to mode \(\varepsilon _n^m\) for a given \(\mathbf{r}\) and \(\omega\), whose detailed expression is given in Eq. 26.
While the SH coefficients \(\varepsilon _n^m\) appear as coefficients of spherical harmonic expansion for the potential magnetic field, they can also be used for representing the inducing current. To this end, consider an extraneous sheet current floating at an altitude h, then the sheet current density can be written as \(\mathbf{j}(\mathbf{r}, \omega )\, =  \delta (r  b) \hat{\mathbf{e}}_r \times \nabla _H \Psi ^{\text{ext}}(\theta , \phi )\), where \(b=a+h\), and the external current stream function can be expanded in SH using \(\varepsilon _n^m\) as
It follows that the coefficients \(\varepsilon _n^m(\omega )\) give the parameterization of the inducing currents, and constitute the aforementioned source vector \(\mathbf{c}\).
We stress here that forward operators with other parameterizations of source currents, not limited to a potential representation (Egbert et al. 2021; Zenhäusern et al. 2021), and a general 3D conductivity distribution (Grayver et al. 2021) are possible and can be incorporated in the formalism of Eq. 6, but this leads to a rather lengthy and technically cumbersome implementation. Choosing a simplified forward operator here allows us to concentrate on studying the properties of the SNLS problem and variable projection method, which we consider to be the main contribution of this study.
To capture the temporal behavior of the external field as well as its properties in the frequency domain, the forward operator and the inversion are both established in the windowed Fourier domain, where each window is considered a realization of the source. For a given frequency \(\omega\) and a time window \(\tau\), the magnetic field is related to the source coefficients via
where \(\mathbf{d}\) denotes the data vector, \(\mathbf{c}\) denotes the vector of external spherical harmonic coefficients, and their subscripts \(\tau\) and \(\omega\) indicate the time window and frequency, respectively. As a common practice, we always inverted for logarithmic electrical conductivity directly in this study, in which case \(\mathbf{m} = \log (\sigma )\) denotes the logarithmic conductivity (in S/m) of the subsurface. By concatenating time windows and periods, the forward operator with respect to the complete set of observations can be recast to the algebraic form given by Eq. 6.
Data
Our original dataset consists of hourly mean observations of the magnetic field at 163 observatories shown in Fig. 1. Both synthetic and real data experiments were carried out based on these observations. Synthetic data were generated using a realistic external field, and a twolayer mantle electrical conductivity model, following the procedures outlined below. We took observatory magnetic field hourly means for years 2014–2018 and subtracted the core, crust, and ionospheric contributions as given by the Comprehensive Inversion within the ESA Swarm data processing chain (Olsen et al. 2013). Then, hourly time series of the external and internal coefficients up to degree and order three were estimated using SHA and robust regression. Only midlatitude observatories (observatories with geomagnetic latitudes from 5° to 56° north and south) were used in the process. The purpose of this step is to obtain time series of the external field that are representative of the real largescale magnetospheric/ionospheric currents in terms of spatial and temporal characteristics. Using the estimated external field coefficients and a predefined 1D Earth electrical conductivity model, synthetic magnetic field time series at real observatory locations were obtained using Eq. 26. In this synthetic test, we set up a simple but realistic twolayer mantle conductivity model, with an upper mantle (surface to 660km depth) electrical conductivity of 0.01 S/m, and a lower mantle (660km to 2900km depth) electrical conductivity of 1.0 S/m. Finally, we contaminated the synthetic time series with a realization of an independent and identically distributed Gaussian white noise with a standard deviation of 1nT.
Since our implementations of the VP methods and the alternating approach are posed in the frequency domain, the data vector \(\mathbf{d}\) is prepared by transforming the time series of magnetic field observations using the windowed spectral transform with tapering, which is defined in discrete form as
where \(t_n\) marks time points within a time window, \(N_\tau\) is the number of points within a time window \(\tau\), and \(w_n\) is a weighting coefficient associated with the nth point for a tapering window function. Choice of normalization does not affect the inversion scheme, but adopting this specific convention preserves the amplitude during the transform (i.e., a monochromatic oscillating field of period T and amplitude A will be transformed to a peak of amplitude A at frequency 1/T), yielding physical meaning to the recovered inducing field.
Finally, we comment on the uncertainty of the data. Recall that the data misfit in Eq. 9 is normalized by the standard deviation. Therefore, the uncertainty of the data in the Fourier domain affects weighting and alters the topography of the objective function. We identify two contributions of the uncertainty of the windowed spectrum. First, since measurements are made in the time domain, the noise in the time domain propagates to the Fourier domain, which we term the propagated spectral uncertainty, denoted as \({s_\text{prop}}\). For an arbitrary time series with a Gaussian white noise of variance \(s_0^2\) in time domain, the corresponding uncertainty in windowed Fourier domain following transform defined in Eq. 30 is given by
Therefore, the propagated spectral uncertainty is not merely proportional to its temporal counterpart, but is also dependent on the length of the time window. The proof of this property is given in Appendix C. For this to hold, the noise is assumed to be independent identically distributed with zero mean. The noise we add to the synthetic data indeed satisfies this assumption, and thus, our estimation of this uncertainty is optimal.
In addition to the propagated spectral uncertainty, modelling in the frequency domain in short time windows and using tapering functions introduce extra errors to the problem. While such segmentation and windowing enhance robustness of frequencydomain inversion, it also introduces spectral leakage (see Appendix D for details). This error varies depending on the shape of the spectrum in the vicinity of the frequency, but we assume that it has similar magnitudes for different frequencies. Denoting this error as \(s_{\text{spec}}\) and assuming \(s_{\text{spec}}\) and \(s_{\text{prop}}\) are independent, we can write
The windowed transformed data are then inversely weighted by the complete variance \(s^2(\tau , \omega )\). In cases where \(s_{\text{prop}}\) is very small (e.g., at long periods), \(s_{\text{spec}}\) serves as the baseline (or error floor) for the overall uncertainty. We note that the choice of \(s_{\text{spec}}\) is generally problem dependent. By comparing the windowed spectrum of a time series modelled with a long time window and the spectrum modelled windowwise, we chose \(s_{\text{spec}}\approx 0.05\)nT for our experiments. This would have only marginal effect for real data because the noise \(s_0\) is large, but it can play a role for our synthetic experiments with the idealized white noise model.
Results
We conducted both synthetic and real data experiments. In both cases, we implemented a conventional approach for field separation and subsurface conductivity inversion. The conventional inversion scheme includes separating the internal and external fields using the Gauss method, estimating an EM transfer function, and then inverting the transfer function for subsurface electrical conductivity. For consistency, the same transfer function as is used in our forward operator, i.e., the Qresponse, is chosen, and the nonlinear optimization problem for a subsurface conductivity distribution is stated as
where \(Q^{\text{obs}}_n\) gives the Q response estimated from \(\varepsilon _n^m\) and \(\iota _n^m\), and \(dQ_n\) is the formal uncertainty. This method will be referred to as Qresponse inversion. The model vector \({\mathbf{m}}\) is again the logarithmic electrical conductivity, same as our forward operator in VP/alternating approach. The Tikhonov matrix \(\varvec{\Gamma }\) in Eqs. 10 and 33 is equivalent. A solution that minimizes the objective function (33) is sought using a Newtonbased algorithm. The regularization strength as a hyperparameter in inversion is identified using the Lcurve analysis (Hansen and O’Leary 1993).
We note that the absolute values of the data misfit for Qresponse inversion and VP/alternating methods are not directly comparable. Even when the data misfit is normalized by the number of data samples, Qresponse inversion and VP/alternating methods perform inversion with respect to different data. The latter directly works on the magnetic field data in the frequency domain, the uncertainties of which are propagated from the timedomain estimates, while the former is conducted on estimates of transfer functions and their formal uncertainties obtained from regression in spectral domain. We therefore do not intend to directly compare the magnitude of the objective function between Gauss and VP/alternating methods. It naturally follows that the respective suitable regularization strengths are also not directly comparable in magnitude, as they would vary between these two types of scheme. It is, however, worth mentioning that VP and alternating methods, regardless of their variants, share the same data misfit evaluation, and are thus comparable in terms of data misfit as well as regularization strengths. We stick to the standard procedure of choosing regularization for each scheme, but such choices may be general across different variants of VP and alternating approaches, as is indeed the case in the experiments shown later.
Synthetic experiment
Since both Qresponse inversion and the VP/alternating approaches are done in the frequency domain, we choose the same discrete frequencies for all inversions. For the synthetic experiment, we chose 15 periods logspaced between 1 and 100 days. The inducing field to be determined is parameterized using SH functions up to SH degree and order 3, and the electrical conductivity model is parameterized as a 15layer 1D profile. Although the coherence of the Qresponses obtained by Gauss method is adequately high for all modes and frequencies due to the ideal synthetic data, we chose to invert only the degree one response \(Q_1\), as is done in practice. VP/alternating approaches, on the other hand, automatically try to fit all modes and frequencies simultaneously.
For VP methods, all three variants were tested on the synthetic dataset. For alternating approaches, we tested four different linear update rules: (1) the external field is estimated once at the beginning using some initial conductivity model, and never updated afterwards; (2) the external field is reestimated every ten iterations; (3) the external field is reestimated every five iterations, and (4) the external field is updated following the Fibonacci sequence (that is, at iterations 1, 2, 3, 5, 8,...). The alternating methods with these four linear update rules will be abbreviated as alt\(\infty\), alt10, alt5, and altFibonacci, respectively. These variants of VP and alternating approaches can be uniformly described by the two controlling parameters, i.e., the incorporation of linear constraints, which varies for different variants of VP, and the frequency of linear update, which varies for different variants of alternating approaches. The relation of these inversion schemes is schematically summarized in Fig. 2.
Most of the synthetic data inversions produce satisfactory results of the conductivity profile within \(\sim 20\) iterations. In particular, we first examine two representative cases, namely fullVP and altFibonacci, in addition to the reference Qresponse inversion. These include the bestperforming schemes within VP variants and alternating approach variants, as will become clear in the results and discussions that follow. In fullVP and altFibonacci schemes, the converged models yield normalized RMS values (\(\chi _{\text{rms}}\)) of \(\approx 0.9\), indicating successful data fitting. In addition, the RMS misfits are roughly at a uniform level across the considered range of frequencies (Fig. 3). The recovered mantle conductivity profiles are shown in Fig. 4. The conductive lower mantle is recovered almost perfectly, especially in the case of fullVP and altFibonacci, while the inverted upper mantle conductivity follows a gradual decrease from 600km depth upwards, and, in these cases, experience a mild reverse trend at lithospheric depths, mostly appearing as a result of regularization and low sensitivity to these depths.
Variable projection inversions as well as alternating approaches simultaneously produce an estimate of the linear model, i.e., the inducing field in this context, along with an estimate of the mantle electrical conductivity. Since the inversions are carried out in the frequency domain, the linear model is estimated in the form of windowed spectrum \(\varepsilon _n^m(\tau , \omega )\). In the synthetic test, the ground truth external coefficients are known, and hence can be used to validate the windowed spectrum of the inducing field SH coefficients inverted using these approaches. A comparison for three period bands of \(\varepsilon _2^1(\tau , \omega _i)\) with \(\omega _i = 2\pi /T_i\) and \(T_i=\)1, 10 and 100 days obtained using fullVP is presented in Fig. 5. From visual inspection, our synthetic tests for VP yield almost perfect recovery of the windowed spectrum of the external field. This is not limited to frequency bands or spherical harmonic components with strong external signals (e.g., the daily band \(\sim 1\) day for \(\varepsilon _2^1\), Fig. 5 left panel), but also applies to less energetic frequency bands and SH modes (e.g., Fig. 5 right panel). Similar results are observed for alternating scheme with the Fibonacci linear update rule (Fig. 22).
We have hence demonstrated that with appropriate hyperparameters and specific variants, all types of inversion methods are able to yield satisfactory solution on the synthetic dataset. Furthermore, we observe similar convergence behavior across these representative inversion schemes (Fig. 6). All of these inversion cases converged to the stable solution within 20 iterations. We stress that, at least for this experiment, the fullVP scheme and altFibonacci scheme exhibit convergence rates that are at least as good as or slightly outperform the Qresponse inversion, even though the SNLS problem has seemingly more ”work” to do, because it also estimates the source structure for all 15 SH coefficients at all frequencies.
In all cases, we observe an initial upsurge in the model roughness, followed by a gradual decrease, accompanied by an almost monotonic decrease of the data misfit. This behavior is expected as the inversions start from a uniform model with zero roughness, thus they first attempt to fit the data at the cost of expanding model complexity, and then stabilize by settling for a smoother model after the data misfit reaches a certain level.
Since the inducing field model is known for the synthetic study, we also examine how the inducing field estimates converge towards the ground truth solution for VP and alternating methods. We introduce the frequencywise relative error for the SH coefficients, defined as
The evolutions of \(\epsilon _{\text{SH}^{nm}}(\omega )\) as a function of iterations for fullVP and altFibonacci are shown in Fig. 7. For VP, the inducing field monotonically approaches the ground truth except for marginal oscillations around the converged solution, and eventually converges at iteration 6, hence slightly earlier than the conductivity model (Fig. 4). For the altFibonacci scheme, the inducing field does not converge until the 13th iteration. Since alternating approaches do not update the inducing currents at each iteration, only five linear projections have been made by iteration 13, following the Fibonacci linear update rule. Slow convergence of the inducing field solution is especially observed at long periods when using the altFibonacci scheme. For instance, the estimated inducing field at iteration 8 has the same level of relative error with the field at iteration 5 for the 100day period band (Fig. 7, middle column). This shows the importance of maintaining consistency between the source and conductivity models, which is most consistently done in the fullVP method.
Our tests using different variants of VP show that all variants of VP converge to almost the exact same electrical conductivity model that is satisfactorily close to the ground truth for the synthetic data (Fig. 8). However, we do observe that VPRW3 exhibits slower convergence for both linear (Fig. 9) and nonlinear parts of the model space (Fig. 8). Most period bands of the external field take 8–10 iterations to converge. Deterioration of the solution, i.e., increased error in the external field at later iterations, is also observed for VPRW3 for periods longer than \(10^6\) s, while such increase in error is absent in fullVP and VPRW2. As is shown in Eqs. 1822, implicit feedback of the inducing field estimate is utilized by both fullVP and VPRW2, but absent in VPRW3. The observed slower convergence as well as deterioration is hence the result of omitting the relevant terms in VPRW3. We anticipate that the performance of VPRW3 (and alternating approaches) will further deteriorate for more complex and highdimensional models.
The conductivity model recovery for different variants of the alternating approach is shown in Fig. 10. As was shown above, updating the inducing field coefficients at iterations using a Fibonacci sequence still allows the inversion to reach a solution that is fairly close to the VP solution (Fig. 4) within 20 nonlinear iterations with only six linear updates, but as soon as the frequency of linear updates is reduced to every five iterations, considerable deterioration of the electrical conductivity recovery occurs, particularly in the lower mantle (Fig. 10). Not only is the inversion taking longer to reach a stationary point, but the scheme fails to locate the bestfitting nonlinear solution within 20 iterations as well, proving to be at best only half as efficient as the VP or altFibonacci.
Interestingly, despite the deteriorated recovery of the mantle conductivity, alt5 produces an estimate of the external field that is almost the same in accuracy to VP or altFibonacci scheme (Fig. 11), with slightly increased error only in some long period bands. However, as seen in Fig. 12, the misfits and roughness values for alt5 have already stagnated at the final stage, and the convergence criterion is satisfied at iteration 18, indicating that the optimization has already converged. Therefore, the misfit between the ground truth and the inverted conductivity models can only be attributed to the marginal difference in external field, and the final inverted conductivity profile should be considered the optimizer of the manifold constrained by the slightly incorrect external field. Alt5 provides such a clear example where a relatively small error in the inducing source field leads to significant artifacts in conductivity model. When the number of linear updates are cut even further, both the external field estimation and the mantle conductivity recovery deteriorate further, as in the case of alt10 and alt\(\infty\) (Figs. 10 and 11).
It is worth mentioning that we observe two types of ”stagnation” behavior in our inversions. In one scenario, the model estimates along with the diagnostic parameters, such as \(\chi _{\text{rms}}\) and roughness, either fulfill the convergence criterion and terminate the inversion, or oscillate mildly in the vicinity of a stationary value. The model is considered converged in this scenario, whether to the vicinity of the ground truth (as in VP variants and altFibonacci scheme) or to a local optimum (undoubtedly the case in alt5) of the manifold in the joint model space. In another scenario (observed in the case of alt\(\infty\)), the objective is not improved for more than eight iterations. This behavior indicates that the trustregion Newton method we employed for optimization repeatedly rejected all update proposals, likely because of a poor local quadratic approximation and severe ill conditioning far from the optimum.
Real data inversion
We applied the VP method to the real ground geomagnetic observatory data measured between years 2014–2018 to simultaneously reconstruct mantle conductivity and external field spectrum. The data come from 120 geomagnetic observatories within the midlow geomagnetic latitude range of \(5^\circ\)\(56^\circ\). Several amendments were introduced to the workflow used in numerical experiments to adjust the method to realEarth sounding. First, a new frequency band (12 hours) is added to the 15 frequency bands spanning 2 decades. This is used for improving constraints to the asthenospheric conductivity. Second, we expanded the parameterization of the mantle conductivity to a 45layer 1D profile. Following Grayver et al. (2017), we used a fixed surface layer with the conductance of 6600 S that represents an average oceansediments conductance over the globe.
First, we obtained an estimate of the Qresponses by applying conventional Gauss method and robust spectral stacking. The responses can be converted to and visualized as a global Cresponse (e.g., Olsen 1999), via the equation
Having the dimension of length, the real part of the Cresponse corresponds to the central depth of induced currents, and is hence indicative of the penetration depth of the EM field at a certain frequency (Weidelt 1972). The available data and frequency bands used in this study are most sensitive to a depth range of 500–1500 km, as can be seen from the \(C_1\) responses (Fig. 13). Based on squared coherences, we use the following Qresponses as the input data for subsequent conductivity inversion: \(Q_1\) estimated from the SH coefficient (1, 0) at periods longer than 1 day, \(Q_2\) estimated from the SH coefficient (2, 1) at diurnal band (24 hours), and \(Q_3\) estimated from the SH coefficient (3, 2) at semidiurnal band (12 h). Variations in the daily band are mostly driven by ionospheric current systems.
For VP, we took only observatories with at least \(99\%\) of valid data in each time window and linearly interpolated the missing observations before conducting the windowed spectral transform (Eq. 30). For real time series of the magnetic field, no information on either spatiotemporal properties or magnitude of the noise is available. It is apparent that when assuming a Gaussian white noise, the assumed noise magnitude serves as a mere normalizing factor for the data misfit (Eq. 31), but would not alter the topography of the objective function. It then follows that the properties for inversions (convergence, results, etc.) are preserved except for corresponding normalization factors for regularization terms and uncertainties. Meanwhile, the spectral and temporal characteristics of the unknown noise pose a much greater threat. Without the Gaussian white noise assumption, our estimation of the Fourierdomain uncertainty would be invalidated. Nevertheless, we proceed by assuming a Gaussian white noise with the standard deviation of 1nT for real observations.
A series of VP inversions were conducted with varying regularization strengths, and the desired hyperparameter was chosen based on the Lcurve (Fig. 24). The inversion results of the mantle conductivity profile using VP as well as conventional Qresponse inversion are shown in Fig. 14. Both conductivity models show a resistive upper mantle with a conductivity monotonically increasing from \(10^{3}\)S/m at lithospheric depths to 1S/m at the bottom of the MTZ. Due to the limited sensitivity of our data to the upper mantle (Fig. 13), however, the magnitude and the detailed shape of conductivity in this region are less reliable. The mantle conductivity at 750km depth just beneath the MTZ is well constrained by our data at 2 to 3S/m, characterized by a conductive peak, a feature that appears quite robust when using weaker regularizations. Beneath this conductive layer, the lower mantle is characterized by a resistive kink, followed by mildly increasing conductivity from \(\sim 1\)S/m at 1200 km depth to 2S/m at 1600.
Despite resemblance between the conductivity profiles produced using Qresponse inversion and VP inversion, these models show considerable discrepancies compared to previous 1D conductivity models, e.g., the 1D profile inverted from \(C_1\) in Grayver et al. (2017), shown in Fig. 15. Compared to the previous models, the conductive peak and the resistive kink in our inverted models are much more pronounced, and the depth within the MTZ in our model is considerably more resistive. These discrepancies can be reconciled, however, if we used only the (1, 0) SH mode in the inversion. Instead of combining data with different SH degrees (i.e., \(n \le 3\)), by inverting only the \(Q_1\) responses estimated from (1, 0), or by parameterizing the source structure using only the first zonal harmonic (controlled by the magnetospheric ring current) in the VP inversion, the obtained conductivity profiles match well with the previous profiles within the depth range where our data have adequate sensitivity (Fig. 15). It is not the focus of this study to discuss the differences between models in Figs. 14 and 15. Whether these differences are dictated by source effects or induced by subsurface differences that become more pronounced in other than \(P_1^0\) terms, they imply that the adopted parameterization of the inducing field has a strong impact on the retrieved conductivity model.
As in the synthetic tests, we obtained the windowed spectra of the external field SH coefficients up to degree and order 3 from the VP method. The estimated external field is close to that obtained by the Gauss method (Fig. 16), but the misfit between the two is generally dependent on energy of the mode. For energetic spatial modes, e.g., \(\varepsilon _1^0\) at period bands longer than 1 day (Fig. 23), \(\varepsilon _2^1\) at 1 day period (Fig. 16 left column), the results are very close. Modes with low power also show correlated trends, but estimated magnitudes disagree. Similar to the relative error defined in Eq. 34, a relative measure of the discrepancy of the estimated SH coefficients can be introduced as
We observe that the relative difference between the inducing fields derived from the Gauss and VP methods is strongly correlated with coherence in the corresponding Qresponse estimation. For a spherical harmonic mode, frequency bands with higher Qresponse coherence are associated with lower relative differences in the inducing field estimates from different methods, as is clearly shown in the case of mode (2, 1) (Fig. 17). The correlation between the transfer function coherence and the inducing source consistency does not come as a surprise, but is a natural consequence of the physical connection between the inducing field and its induced counterpart. The coherence in transfer function estimation is used to describe how much of the induced field can be causally explained by the transfer function, which in our case is the Qresponse, whereas such physical link is explicitly incorporated in VP. Therefore, in frequency bands and SH modes where such physical connection is appropriate for explaining the data (i.e., high coherence), the inducing source estimate in the Gauss method should be more consistent with that obtained in VP.
Discussion
Separability of inducing source modes
We have emphasized that VP/alternating approaches are not limited to potential field and can accommodate diverse measurements and sources in contrast to inversions based on TFs, such as the Qresponse, estimated based on SH coefficients obtained from the Gauss method. In addition, VP and alternating approaches also benefit from more reliable source estimate when compared to the Gauss method. This is because the Gauss method needs to estimate both internal and external coefficients together, while the linear operator in VP or alternating approaches involves only variables for the inducing source.
To illustrate the effect, we explored the dependency of the condition number of linear regression operators upon different combinations of internal and external field parameterizations. High condition number implies nearly colinear columns, which leads to poorly separable parameters and catastrophic amplification of data errors (Heath 2018). In the Gauss method, the internal and external fields are coestimated; different truncation degrees of internal field give rise to linear systems of different dimensions. In VP/alternating approaches, the internal field is modelled. Varying the maximum degree of modelled internal field changes the columns in the matrix, but the column dimension of the linear system which corresponds to the external field parameters remains the same. We constructed linear systems for field estimation for both Gauss method and VP, assuming a real observatory layout with 68 observatory locations (red circles in Fig. 1), a configuration taken from the distribution of available observatories in the dataset on Nov 29, 2019. The result is shown in Fig. 18. While the condition number for the Gauss method increases substantially with both internal and external SH degrees, the condition number for VP depends mostly only on the maximum external SH degree, but hardly changes over one order of magnitude for different degrees of modelled internal field. As a result, source determination within the VP method remains well conditioned (e.g., condition number \(\approx 20\) for maximum external SH degree of 5), whereas the corresponding matrix for the Gauss method may already be very illconditioned (condition number \(> 10^3\) for the same external field parameterization). This experiment reveals severe limitations of the conventional Gauss method for 3D scenarios, where the intention is to go for higher degrees in internal field parameterization to capture lateral conductivity variations at desired length scales. In these settings, VP and alternating approaches offer a definite advantage.
Effect of linear update and derivative approximation
Both VP and alternating approaches provide efficient means to find a suitable solution in the joint model space by introducing pointdependent constraints. Whereas variable projection variants reestimate the local ”optimal” constraint at each iteration, alternating approaches allow a user to delay the next reprojection of the linear model and reestimation of the constraint, implicitly assuming that the constraint remains valid at each subsequent iteration that reuses the initial projection. Although this saves resources and time, we observe that such assumption may not be valid and can potentially cause considerable deterioration of the solution when utilizing the alternating concept. Excessive iterations using a fixed external field model push a conductivity model away from an optimal solution, which in turn projects the inaccuracy back to the external field at the next iteration (Fig. 7, Alt.  Fibonacci). In real settings, without information about the ground truth, detecting such behavior is practically impossible. Therefore, as appealing as the alternating approach might be due to its simple nature, insufficiently frequent updates of the linear model create a risk of obtaining biased solutions, as was demonstrated in this study. On the other hand, we found that more elaborate update rules, such as altFibonacci, succeed in locating the optimal solution. This process is facilitated by more frequent linear updates at early stages of the inversion. Therefore, alternating approaches should be used with care; in particular, for a given source model, the nonlinear inversion on the conductivity model should not be run until it stagnates, by which time the conductivity model (and with it, the estimate of the external field in the next stage) is probably already biased. In contrast, it is beneficial to alternate between conductivity inversion and inducing field estimation as often as possible initially, with more rare updates to be permitted at later stages.
For our rather simple synthetic tests, we observe no significant difference between the models obtained with different VP variants. However, we observe a slower convergence as well as deterioration of the models in intermediate iterations for the VPRW3 variant (Fig. 9). Different levels of approximation of the Fréchet derivatives seem to work almost equally well for the synthetic experiment, and in practice, it might be beneficial to adopt either VPRW2 or VPRW3 variants for the sake of computational efficiency, especially when evaluation of \(\textsf{D}\mathbf{F}\) is expensive (to be the case once full 3D forward operator is required).
Interplay between conductivity model and external field
The external field model and the mantle conductivity model are mutually dependent in the optimization problem defined in Eq. 10. In particular, the mantle conductivity model is sensitive to perturbations in the external field, as was evident in our experiment where alternations between linear and nonlinear models are done at varied frequencies (altFibonacci, alt5, alt10, and alt\(\infty\)). Note that the application of the VP method completely eliminates this problem, and preserves consistency between the linear and nonlinear model unknowns. It does not mean that the VP is less ambiguous than the alternating approach, but it allows one to attain the best (in leastsquares sense) possible tradeoff between source and conductivity models.
In turn, mantle conductivity has a nonnegligible feedback on source reconstruction. To quantify and illustrate the effect, we compared the quality of inducing field reconstruction from synthetic data using simplistic subsurface conductivity models, such as a uniform mantle conductivity of 0.1S/m, which is used as the starting model for all our inversions, and a simplistic twolayer Earth model consisting of a 1200kmthick perfectly insulating mantle and a perfectly conducting core, hereinafter referred to as the bilayer model. For the perfect insulator–conductor bilayer model, the Qresponse of degree n degenerates to a frequencyindependent algebraic form
where z is the thickness of the overlying perfectly insulating layer. Due to its simplicity, the bilayer model is often used in space weather and geomagnetic field modelling to get a firstorder estimate of the induction effect.
Figure 19 shows estimates of the external field windowed spectra obtained with different conductivity models. There are considerable discrepancies between the inducing field estimates and the ground truth field when using overly simplistic models. For instance, note the amplitude discrepancies of the field recovered using the bilayer model, especially in short (e.g., 1 day) and long periods (e.g., 100 days). The discrepancies are also obvious from the relative field errors evaluated in energies (Fig. 20), calculated from Eq. 34. While the inversion result from VP gives an external field \(\varepsilon _3^1\) that is \(1\%\) to \(5\%\) different from the ground truth in most frequency bands, the aforementioned simplistic models yield external fields that typically exhibit over \(15\%\) error. The initial uniform conductivity model, for instance, gives relative inducing field errors of \(15\%\) in short periods, which increases to values of \(90\%\) at periods of about 1 month. For the simplistic twolayer model, the relative error of the external field increases from \(20\%\) in the diurnal band, to \(45\%\) in the period bands of 1–3 months. This large discrepancy is partially attributed to the low energies in these modes, but the patterns are the same in more energetic modes.
In short, source estimates calculated using overly simplistic or wrong conductivity models are prone to additional errors (up to \(90\%\) in our experiment), even in the scenario where the response is given by a simple 1D model. For a realistic 3D earth, the conductivity model might have a more pronounced effect, especially in the vertical component of the magnetic field, when strong lateral variations are present (Grayver et al. 2021).
Conclusions
We addressed the problem of inverting for the inducing source and subsurface conductivity through the solution of a Separable Nonlinear LeastSquares (SNLS) problem. By exploiting the inherent property whereby observations depend on source coefficients linearly, whereas the dependency on the subsurface electrical conductivity is nonlinear, we proposed a novel inversion scheme that solves the underlying SNLS problem using the variable projection to determine source and conductivity structures simultaneously and retain consistency between them. We applied this method to both synthetic tests and real observations. Although our experiments and inversions were limited to ground magnetic field observations and a rather simple 1D conductivity model parameterization, the provided method derivations are general in both the observational data and the model parameterization aspects. Our derivations in the Methods section present a versatile generic framework for exploiting the VP and show how conventional inversion schemes that often already implement Jacobians for separate source and conductivity estimation can be reformulated into the SNLS form and solved using the VP or alternating approaches. To gain additional insight into the problem, we studied several variants of the VP and showed its relation to the full joint inversion as well alternating inversion approaches.
Alternating approach provides a simple alternative to the VP method that can be used to solve SNLS. However, one important aspect that was not identified in the previous studies is that alternating approach with too rare source model updates can result in deteriorated model estimates along iterations, which eventually undermines the convergence and model recovery. To avoid this, alternating inversions need to perform frequent reestimation of the inducing source, especially at early stages. We also observed a slower convergence of the alternating approaches compared to the fullVP method, although this can be compensated in practice by a lower computational cost per iteration.
We demonstrated that by introducing additional constraints to the joint model space, variable projection methods and alternating approaches are capable of recovering both external field and mantle conductivity simultaneously. They show comparable performance to the Gauss method and transfer function inversion on our (simple) test cases, where potential field assumption is applicable. However, unlike approaches that invoke the Gauss method, the SNLS problem solved by the VP method is not limited to the potential field scenarios. In particular, it can accommodate arbitrary source geometries at arbitrary locations [e.g., current loops, dipoles, spherical elementary current systems (SECS)]; incorporate electric field data as well as both ground and satellite observations. Importantly, VP methods make the explicit use of the physical link (through Maxwell’s equations) between the source and conductivity, which ensures that consistency between both model spaces is preserved (at least to an extent for which data coverage and quality allow). This is in contrast to the conventional approaches where the source and conductivity are estimated independently and it is often (implicitly) assumed in subsequent transfer function estimation and inversion that the external source estimate is ”noisefree”. Our synthetic tests showed that even small inconsistencies in a source model can lead to significant artifacts in subsurface conductivity. We also showed that inadequate modelling of the induced field leads to a biased estimate of the external field structure.
Data availability
Time series of the hourly means at ground magnetic observatories were taken from the AUX_OBS ESA Swarm product https://earth.esa.int/eogateway/missions/swarm/productdatahandbook/auxiliaryproductdefinitions.
Abbreviations
 EM:

Electromagnetic
 GDS:

Geomagnetic depth sounding
 MT:

Magnetotellurics
 RMS:

Rootmeansquare
 SH:

Spherical harmonics
 SHA:

Spherical harmonic analysis
 SNLS:

Separable nonlinear least squares
 VP:

Variable projection method
 VPRW2:

Second algorithm of variable projection, proposed by Ruhe and Wedin (1980)
 VPRW3:

Third algorithm of variable projection, proposed by Ruhe and Wedin (1980)
References
Aravkin AY, van Leeuwen T (2012) Estimating nuisance parameters in inverse problems. Inverse Problems 28(11):115016. https://doi.org/10.1088/02665611/28/11/115016
Backus G, Parker RL, Constable C (1996) Foundations of geomagnetism. Cambridge University Press, Cambridge
Balasis G, Egbert GD (2006) Empirical orthogonal function analysis of magnetic observatory data: further evidence for nonaxisymmetric magnetospheric sources for satellite induction studies. Geophys Res Lett 33:2006GL025721. https://doi.org/10.1029/2006GL025721
Chave AD, Jones AG (2012) The magnetotelluric method: theory and practice. Cambridge University Press, Cambridge
De Ridder SA, Maddison JR (2018) Full wavefield inversion of ambient seismic noise. Geophys J Int 215:1215–1230. https://doi.org/10.1093/GJI/GGY328
Egbert GD (2012) Hybrid conjugate gradientOccam algorithms for inversion of multifrequency and multitransmitter EM data. Geophys J Int 190:255–266. https://doi.org/10.1111/j.1365246X.2012.05523.x
Egbert GD, Kelbert A (2012) Computational recipes for electromagnetic inverse problems. Geophys J Int 189(1):251–267. https://doi.org/10.1111/j.1365246X.2011.05347.x
Egbert GD, Alken P, Maute A, Zhang H (2021) Modelling diurnal variation magnetic fields due to ionospheric currents. Geophys J Int 225(2):1086–1109. https://doi.org/10.1093/gji/ggaa533
Fainberg EB, Kuvshinov AV, Mishina LP, Singer BS (1990) The new approach to global deep sounding. Pure Appl Geophys 134:527–531. https://doi.org/10.1007/BF00878016
Finlay CC, Lesur V, Thébault E, Vervelidou F, Morschhauser A, Shore R (2017) Challenges handling magnetospheric and ionospheric signals in internal geomagnetic field modelling. Space Sci Rev 206(1):157–189. https://doi.org/10.1007/s1121401602859
Golub G, Pereyra V (2003) Separable nonlinear least squares: the variable projection method and its applications. Inverse Problems 19(2):R1. https://doi.org/10.1088/02665611/19/2/201
Golub GH, Pereyra V (1973) The differentiation of pseudoinverses and nonlinear least square problems whose variables separate. SIAM J Numer Anal 10:413–432. https://doi.org/10.1137/0710036
Grayver AV, Munch FD, Kuvshinov AV, Khan A, Sabaka TJ, TøffnerClausen L (2017) Joint inversion of satellitedetected tidal and magnetospheric signals constrains electrical conductivity and water content of the upper mantle and transition zone. Geophys Res Lett 44:6074–6081. https://doi.org/10.1002/2017GL073446
Grayver AV, Kuvshinov A, Werthmüller D (2021) Timedomain modeling of threedimensional earth’s and planetary electromagnetic induction effect in ground and satellite observations. J Geophys Res Space Phys 126:e2020JA028672. https://doi.org/10.1029/2020JA028672
Guzavina M, Grayver A, Kuvshinov A (2019) Probing upper mantle electrical conductivity with daily magnetic variations using globaltolocal transfer functions. Geophys J Int 219:2125–2147. https://doi.org/10.1093/gji/ggz412
Hansen PC, O’Leary DP (1993) The use of the Lcurve in the regularization of discrete illposed problems. SIAM J Sci Comput 14:1487–1503. https://doi.org/10.1137/0914086
Heath MT (2018) Scientific Computing. Society for Industrial and Applied Mathematics, Philadelphia, PA, 10(1137/1):9781611975581
Hong JH, Zach C, Fitzgibbon A (2017) Revisiting the variable projection method for separable nonlinear least squares problems. In: 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp 5939–5947. https://doi.org/10.1109/CVPR.2017.629
Hu X, Wei G, Song J, Yang Z, Lu M, Gao L (2021) Fullwaveform inversion with source and receiver coupling effects correction. Geophysics 86(4):R463–R470. https://doi.org/10.1190/geo20200331.1
Juusola L, Vanhamäki H, Viljanen A, Smirnov M (2020) Induced currents due to 3D ground conductivity play a major role in the interpretation of geomagnetic variations. Annales Geophysicae 38:983–998. https://doi.org/10.5194/angeo389832020
Kelbert A (2020) The role of global/regional earth conductivity models in natural geomagnetic hazard mitigation. Surveys Geophys 41:115–166. https://doi.org/10.1007/s1071201909579z
Kelbert A, Lucas GM (2020) Modified GIC estimation using 3D earth conductivity. Space Weather 18(8):e2020SW002467. https://doi.org/10.1029/2020SW002467
Kelbert A, Schultz A, Egbert G (2009) Global electromagnetic induction constraints on transitionzone water content variations. Nature 460:1003–1006. https://doi.org/10.1038/nature08257
Koch S, Kuvshinov A (2013) Global 3D EM inversion of Sq variations based on simultaneous source and conductivity determination: concept validation and resolution studies. Geophys J Int 195(1):98–116. https://doi.org/10.1093/gji/ggt227
Kuvshinov A (2008) 3D global induction in the oceans and solid earth: recent progress in modeling magnetic and electric fields from sources of magnetospheric, ionospheric and oceanic origin. Surveys Geophys 29:139–186. https://doi.org/10.1007/s107120089045z
Kuvshinov A (2012) Deep electromagnetic studies from land, sea, and space: progress status in the past 10 years. Surveys Geophys 33:169–209. https://doi.org/10.1007/s1071201191182
Kuvshinov A, Grayver A, TøffnerClausen L, Olsen N (2021) Probing 3D electrical conductivity of the mantle using 6 years of Swarm, CryoSat2 and observatory magnetic data and exploiting matrix Qresponses approach. Earth Planets Space 73:67. https://doi.org/10.1186/s40623020013419
Laundal KM, Yee JH, Merkin VG, Gjerloev JW, Vanhamäki H, Reistad JP, Madelaire M, Sorathia K, Espy PJ (2021) Electrojet estimates from mesospheric magnetic field measurements. J Geophys Res Space Phys 126:e2020JA028644. https://doi.org/10.1029/2020JA028644
Li M, Rickett J, Abubakar A (2013) Application of the variable projection scheme for frequencydomain fullwaveform inversion. Geophysics 78:R249–R257. https://doi.org/10.1190/geo20120351.1
Maus S, Weidelt P (2004) Separating the magnetospheric disturbance magnetic field into external and transient internal contributions using a 1D conductivity model of the earth. Geophys Res Lett. https://doi.org/10.1029/2004GL020232
O’Leary DP, Rust BW (2013) Variable projection for nonlinear least squares problems. Comput Optim Appl 54:579–593. https://doi.org/10.1007/s1058901294929
Olsen N (1999) Induction studies with satellite data. Surveys Geophys 20:309–340. https://doi.org/10.1023/a:1006611303582
Olsen N, FriisChristensen E, Floberghagen R, Alken P, Beggan CD, Chulliat A, Doornbos E, da Encarnação JT, Hamilton B, Hulot G, van den IJssel J, Kuvshinov A, Lesur V, Lühr H, Macmillan S, Maus S, Noja M, Olsen PEH, Park J, Plank G, Püthe C, Rauberg J, Ritter P, Rother M, Sabaka TJ, Schachtschneider R, Sirol O, Stolle C, Thébault E, Thomson AWP, TøffnerClausen L, Velímský J, Vigneron P, Visser PN (2013) The Swarm satellite constellation application and research facility (SCARF) and Swarm data products. Earth Planets Space 65:1189–1200. https://doi.org/10.5047/eps.2013.07.001
Pankratov OV, Kuvshinov AV (2010) Fast calculation of the sensitivity matrix for responses to the earth’s conductivity: general strategy and examples, Izvestiya. Phys Solid Earth 46:788–804. https://doi.org/10.1134/S1069351310090089
Pulkkinen A, Amm O, Viljanen A (2003) Ionospheric equivalent current distributions determined with the method of spherical elementary current systems. J Geophys Res Space Phys https://doi.org/10.1029/2001JA005085
Püthe C, Kuvshinov A (2014) Mapping 3D mantle electrical conductivity from space: a new 3D inversion scheme based on analysis of matrix Qresponses. Geophys J Int 197(2):768–784. https://doi.org/10.1093/gji/ggu027
Rickett J (2013) The variable projection method for waveform inversion with an unknown source function. Geophys Prospect 61:874–881. https://doi.org/10.1111/13652478.12008
Ruhe A, Wedin PÅ (1980) Algorithms for separable nonlinear least squares problems. SIAM Rev 22:318–337. https://doi.org/10.1137/1022057
Schmucker U (1999) A spherical harmonic analysis of solar daily variations in the years 1964–1965: Response estimates and source fields for global inductioni. methods. Geophys J Int 136:439–454. https://doi.org/10.1046/j.1365246X.1999.00742.x
Sun J, Kelbert A, Egbert GD (2015) Ionospheric current source modeling and global geomagnetic induction using ground geomagnetic observatory data. J Geophys Res Solid Earth 120(10):6771–6796. https://doi.org/10.1002/2015JB012063
Tsyganenko NA (2019) Secular drift of the auroral ovals: How fast do they actually move? Geophys Res Lett 46:3017–3023. https://doi.org/10.1029/2019GL082159
Velímský J, Knopp O (2021) Lateral variations of electrical conductivity in the lower mantle constrained by Swarm and CryoSat2 missions. Earth Planets Space 73:4. https://doi.org/10.1186/s40623020013348
Velímský J, Grayver A, Kuvshinov A, Šachl L (2018) On the modelling of M2 tidal magnetic signatures: effects of physical approximations and numerical resolution. Earth Planets Space 70:192. https://doi.org/10.1186/s4062301809675
Weidelt P (1972) The inverse problem of geomagnetic induction. J Geophys 38:257–289
Yamazaki Y, Maute A (2017) Sq and EEJa review on the daily variation of the geomagnetic field caused by ionospheric dynamo currents. Space Sci Rev 206:299–405. https://doi.org/10.1007/s112140160282z
Zenhäusern G, Kuvshinov A, Guzavina M, Maute A (2021) Towards probing earth’s upper mantle with daily magnetic field variations: exploring a physicsbased parametrization of the source. Earth Planets Space 73:136. https://doi.org/10.1186/s40623021014558
Zhang H, Egbert GD, Huang Q (2022) A relatively dry mantle transition zone revealed by geomagnetic diurnal variations. Sci Adv https://doi.org/10.1126/sciadv.abo3293
Acknowledgements
AG is thankful to Malcolm Sambridge for fruitful discussions about SNLS problems. Constructive reviews by two anonymous reviewers helped improve the original draft substantially.
Funding
Open Access funding enabled and organized by Projekt DEAL. AG was supported by the ESA Swarm DISC (Contract No. 4000109587) and the Heisenberg Grant from the German Research Foundation, Deutsche Forschungsgemeinschaft (Project No. 465486300). JM is grateful for funding from the European Research Council (Agreement No. 833848UEMHP) under the Horizon 2020 programme.
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JM: methodology, software, data analysis, and writing—original draft; AG: conceptualization, methodology, data analysis, and writing—review and editing. Both authors read and approved the final manuscript.
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Appendices
Appendix A: VP with regularization on the linear part
For the sake of completeness, we present formulas for VP assuming a scenario where the source estimation requires regularization. This would be the case when a rich parameter space is needed to represent a source with a complex geometry. In accordance with the metric used in the rest of this paper, we only consider the \(\ell _2\) regularization. Regularization on the linear parameters does not change the general pipeline of inversion, but would alter the form of the Fréchet derivatives, as will become clear in this following derivations.
We shall start from the general form of the joint optimization problem (Eq. 10). To exploit the preexisting derivations for VP without the regularization term on linear parameters, we introduce the augmented residual vector, defined as
where we use the notation \(\widetilde{\varvec{\Gamma }}_c = \lambda ^{1/2}\varvec{\Gamma }_c\). The explicit Jacobians on the augmented residual vector are linked to the original Jacobians (Eq. 13) through
The augmented formulation allows us to rewrite Eq. 10 as
Since this is the same form as the VP formulation with \(\lambda _c=0\), all previous formulas apply, except for replacing the original derivatives with the derivatives associated with the augmented residual. Furthermore, with Eqs. A.1 and A.2, the estimated linear model, the Jacobian, and the gradient of the residual vector can be expressed explicitly in terms of the original quantities. The linear regression will be given by
where \(\varvec{\Lambda }_c = \lambda _c \varvec{\Gamma }_c^H \varvec{\Gamma }_c\) is the regularization matrix, and \(\mathbf{J}_c^{\Lambda _c} = (\mathbf{J}_c^H \mathbf{J}_c + \varvec{\Lambda }_c)^{1} \mathbf{J}_c^H\) is the regularized version of the pseudoinverse. If \(\mathbf{J}_c^H\mathbf{J}_c\) is itself invertible, and \(\varvec{\Lambda }_c=\mathbf{0}\), the matrix \(\mathbf{J}_c^{\Lambda _c}\) defined in this manner would be exactly the Moore–Penrose pseudoinverse \(\mathbf{J}_c^{\dagger }\). The complete Jacobian on the augmented residual vector is
The equation above gives the Jacobian for the fullVP scheme. Different variants of VP can now be achieved by choosing different approximations for \(\widetilde{\mathbf{J}}\). For VPRW2, only the first two terms on the second line of Eq. A.5 is kept, while for VPRW3, only the first one is retained. The gradient of the augmented data misfit is once again independent of the variants of VP, and takes the form
where \(\widetilde{\phi }(\mathbf{m}) = \frac{1}{2} \left\ \widetilde{\mathbf{r}}_w(\mathbf{m}, \hat{\mathbf{c}}(\mathbf{m}))\right\ _2^2\). For the optimization problem (Eq. A.3), the update on the model parameter yielded by the Gauss–Newton algorithm satisfies
Appendix B: Linking VP and joint model space optimization
We have alluded to the the role of VP and alternating approaches as surrogate methods to the joint model space inversion. Here, we shall more explicitly state how they are related. We consider optimization of the same problem (Eq. 10), but in joint model space. The model vector is then represented as a concatenated vector \(\widetilde{\mathbf{m}} = [\mathbf{m}{^\text{T}}, \mathbf{c}{^\text{T}}]{^\text{T}}\). The Jacobian and the Hessian of the misfit (under the same Gauss–Newton algorithm) in the full model space are given by
where \(\mathbf{J}_m\) and \(\mathbf{J}_c\) are defined in Eq. 13. Adding the regularization term yields the model update equation for the joint model space, given by the linear system
where \(\mathbf{m}\) is the current estimate of the conductivity model. The linear system has a total dimension of \(M_m + M_c\), where \(M_m\) and \(M_c\) are the dimension of the models \(\mathbf{m}\) and \(\mathbf{c}\), respectively. In geomagnetic deep sounding problems the inducing field parameterization usually occupies a much more higher dimensional subspace than the conductivity model, and the resulting system can be formidable to tackle. To compare the model update obtained using the joint model space optimization, we use Schur complement to extract the model update on \(\mathbf{m}\), which is given by
Let us now consider the model update induced by VP and its variants. Following Eq. 19, the gradient retains its form regardless of the adopted approximation. Using the Gauss–Newton algorithm with Jacobians given by Eqs. 18, 21 and 22, the model updates are expressed as
We can see that the system induced by the joint model space optimization is very similar to the model update within the VPRW2 method. In particular, expressing the current model in the joint model space as \(\widetilde{\mathbf{m}} = [\mathbf{m}, \mathbf{c}] = [\mathbf{m},  \mathbf{J}_c^{\dagger } \mathbf{d}_w]\), we can apply the orthogonal property of the residual vector with respect to \(\mathbf{J}_c\)
The linear system for joint model space optimization (Eq. B.3) is then effectively identical to VPRW2 (Eq. B.4). We therefore conclude that given the same conductivity model \(\mathbf{m}\) and an optimized source model \(\mathbf{c} = \mathbf{J}_c^{\dagger }(\mathbf{m}) \, \mathbf{d}_w\), the conductivity model update proposed by VPRW2 is exactly the same as that proposed by the joint model space inversion. However, we note that the proposed linear update in joint model space inversion is given by
and does not yield optimized update on \(\mathbf{c}\). In contrast, the VP methods conduct regression at every iteration, which guarantees that the choice of \(\mathbf{c}\) is optimal (in leastsquares sense). Therefore, starting from the same \(\mathbf{m}, \mathbf{c}= \mathbf{J}_c^{\dagger } \mathbf{d}_w\) combinations, VPRW2 is guaranteed to propose a linear model that yields smaller data misfit compared to joint model space inversion, without resorting to a more complex nonlinear model.
As a simple illustration, we consider the wavelet fitting problem of a Ricker wavelet, which takes the form
where c controls the amplitude and enters the observation linearly, and \(\alpha\) controls the width of the wavelet and enters the data nonlinearly. The wavelet fitting problem with squared misfit is an SNLS problem, and can be solved using the aforementioned methods. Different variants of VP were conducted, together with a joint model space inversion. The synthetic data were generated with \(\alpha _*=1\) amd \(c_*=1\), and all inversion schemes started from an initial guess \(\alpha _0 = 6\). For VP, no initial guess on the linear model is needed; for the joint model space inversion, we took the linear regression result \(\hat{c}_0\) at \(\alpha _0=6\).
The convergence of different inversion schemes in the 2D model space is shown in Fig. 21. In this toy example, all inversion schemes converged to the ground truth, but we observe the convergence slows down as we go from fullVP, VPRW2 to VPRW3. Comparison between the trajectories of VPRW2 (blue line) and the joint model space inversion (red line) shows that within the first four iterations, the update on the nonlinear parameter \(\alpha\) is very close, while the linear model c in the joint model space inversion slowly drifts away from the optimal linear leastsquares solution, a phenomenon predicted by our derivations above. The nonoptimal linear model for the joint model space inversion results in a detour of the trajectory after the fifth iteration, undermining the convergence of the inversion scheme.
Appendix C: Variance propagation for windowed Fourier transform
Here, we present a derivation for Eq. 31. To this end, we consider a time series \(x(t_n)\), which is a sum of a deterministic signal \(x^{(0)}(t_n)\), and a white noise \(\epsilon (t_n) \sim \mathcal {N}(0, s)\). Thus, we have
We shall use X, \(X^{(0)}\) and \(\mathcal {E}\) to denote the windowed spectrum of x, \(x^{(0)}\) and \(\epsilon\), respectively. Following the windowed Fourier transform (Eq. 30), we write the expectation of the windowed spectrum:
Therefore, the windowed spectrum X is the unbiased estimate of \(X^{(0)}\). On the other hand, the variance of the windowed spectrum X is determined by the variance of the windowed spectrum of the noise
The expectation of \(\mathcal {E}^2\) is in turn given by
The last two steps use the fact that \(\epsilon (t_n)\) and \(\epsilon (t_m)\) are independent random variables with zero mean. This completes the proof for Eq. 31. For a boxcar window function, i.e. \(w_n\equiv 1\), we have the following:
Appendix D: Modelling in windowed Fourier domain
In our synthetic test, we observe that although \(\chi _{\text{rms}}\approx 1\) can be obtained for the entire dataset, it is not the case with every frequency band (Fig. 3). This phenomenon should be attributed to the spectral leakage and, as a consequence, the inevitable imperfect nature of windowedFourierdomain modelling. In particular, we consider two time series y(t) and x(t), which are related in the frequency domain via
where X, Y are the spectra of x and y, respectively, and \(H(\omega )\) is the transfer function. In the general formulation of VP/alternating methods, X and Y correspond to the inducing current parameterization \(\mathbf{c}\) and the data vector \(\mathbf{d}\), and \(H(\omega )\) corresponds to the forward operator \(\mathbf{F}(\sigma )\) (Eq. 6); in the formulation of Qresponse estimation, X and Y are \(\varepsilon _n^m\) and \(\iota _n^m\), respectively, while \(H(\omega )\) is nothing but the \(Q_n(\omega )\) response (Eq. 25). Without loss of generality, we limit ourselves to scalars \(X, Y, H \in \mathbb {C}\) in this appendix. The goal here is to show that the windowed spectra of x and y, given by
with the transforms defined in Eq. 30 do not strictly satisfy the same relation in the frequency domain. In other words, in general, we have
In this appendix, we strictly distinguish between the term windowed Fourier domain and the Fourier domain. The former is defined in Eq. 30, and the latter is defined in its continuous form in Eq. 2, and in its discrete form as the following convention for the discrete Fourier transform (DFT) and its inverse (iDFT):
where \(t_k = k\Delta t\) are the sampling time points, and \(\omega _q = q/N\Delta t\) are the frequency points, \(k, q = 0, \cdots N1\). The windowed spectral transform of x in time window \(\tau\) at frequency \(\omega\) can be written as
Here, \(k_{\tau 0}\) denotes the first time index of the time window \(\tau\), and \(K_\tau\) denotes the total number of time points in the time window \(\tau\). Defining the normalized spectrum of the window function within the time window as
the windowed spectrum of x can be reiterated as
The trailing factor \(e^{i\omega _q t_{k_{\tau 0}}}\) shifts the phases at respective frequencies to the beginning of the time window. The spectrum \(\widetilde{W}\) is normalized as such, so that \(\widetilde{W}(0) \equiv 1\). In the extreme case of infinite length time series and time windows, we will have \(\widetilde{W}(\omega ) = \delta (\omega )\), leading to \(X(\tau , \omega ) \propto X(\omega )\). This is, however, never the case for finite length time series and time windows, where the windowed spectrum at frequency \(\omega\) always contains the spectrum at adjacent DFT frequencies, i.e., \(X(\omega _q)\), a phenomenon known as spectral leakage. Appropriate choices of the window function yield \(\widetilde{W}\) that can suppress the leakage, but this phenomenon still exists. The ”imperfection” of the forward modelling in windowed Fourier domain becomes clear when we also write the windowed spectrum of Y in a similar form
From Eqs. D.7 and D.8, we see that the \(X(\tau , \omega )\) is not simply linked to \(Y(\tau , \omega )\) through a product with the transfer function, i.e., Eq. D.3. Only under one specific condition, i.e., \(H(\omega ) \equiv H_0\), do the two quantities follow the same relation as their Fourierdomain counterparts, as
In other words, only when the transfer function has a flat spectrum (the impulse response is impulsive) is the modelling in windowed Fourier domain exactly the same as the modelling in the Fourier domain. Otherwise, the forward modelling as \(Y(\tau , \omega ) = H(\omega ) \, X(\tau , \omega )\) cannot explain the scattering of the data \(Y(\tau , \omega )\). This phenomenon should be perceived as an imperfection that affects both the TF estimation, as well as the VP/alternating approaches when using the specific form of forward modelling (Eq. 29). In Qresponse estimation, this indicates that even for perfect synthetic data, there will be residuals in the fitting of \(\iota(\tau, \omega)\) that cannot be explained by Eq. 25. In our implementation of VP/alternating approaches combined with the forward operators (Eq. 29), this imperfection is included in the uncertainty by the spectral transform error in Eq. 32.
Appendix E: Additional figures
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Min, J., Grayver, A. Simultaneous inversion for source field and mantle electrical conductivity using the variable projection approach. Earth Planets Space 75, 83 (2023). https://doi.org/10.1186/s40623023018165
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DOI: https://doi.org/10.1186/s40623023018165
Keywords
 Geomagnetism
 Electromagnetic induction
 Variable projection