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Directional drilling in the oil fields relies particularly on the “on‐fly” measurements of the natural magnetic field (measurements while drilling; MWD); the MWD are eventually used to construct the well path. These measurements are the superposition of the signals from the internal, core and crustal, and external, ionospheric and magnetospheric sources and the noise from magnetic elements in the borehole assembly. The internal signals are mostly constant in time and accounted for through the Earth's internal field models. The signals of external origin give rise to diurnal and irregular spatio‐temporal magnetic field variations observable in the MWD. One of the common ways to mitigate the effects of these variations in the MWD is to correct readings for the data from an adjacent land‐based magnetic observatory/site. This method assumes that the land‐based signals are similar to those at the seabed drilling site. In this paper, we show that the sea level and seabed horizontal magnetic fields differ significantly, reaching up to 30% of sea level values in many oceanic regions. We made this inference from the global forward modeling of the magnetic field using realistic models of conducting Earth and time‐varying sources. To perform such modeling, we elaborated a numerical approach to efficiently calculate the spatio‐temporal evolution of the magnetic field. Finally, we propose and validate a formalism allowing researchers to obtain trustworthy seabed signals using measurements at the adjacent land‐based site and exploiting the modeling results, thus without needing additional measurements at the seabed site.

Knowing the position of existing and new oil wells is vital for economic and safe sub‐horizontal drilling operations. A lower uncertainty in well positions allows for hitting smaller targets and avoiding the risk of well collisions. Determining the well position relies particularly on magnetic sensors installed close to the drill bit. For this, the models of spatially variable but constant in time magnetic field in the region of interest are routinely used. However, spatially and temporally varying space‐weather‐related geomagnetic field disturbances may affect the accuracy of the well position. One of the common ways to mitigate this problem is to correct magnetic field readings at a drill bit for the magnetic field measurements from an adjacent land‐based magnetic observatory. However, this method assumes that the land‐based signals are similar to those at the seabed drilling site. In this paper, we show that this is not the case, that is, the sea level and seabed magnetic fields differ significantly due to the electrical conductivity of the seawater column above the seabed site. Moreover, we propose and justify a numerical scheme which allows researchers to obtain trustworthy seabed signals by still using land‐based data but exploiting the results of dedicated modeling.

We present an approach to efficiently calculate the spatio‐temporal evolution of a magnetic field in a given conductivity model of the Earth

We show that sea level and seabed horizontal magnetic field differ significantly

We propose and justify a formalism allowing us to calculate more accurately seabed magnetic field signals using adjacent land‐based data

Modern directional drillers primarily rely on natural magnetic and gravitational fields to determine the orientation of their borehole assembly (BHA). This is because the GPS signals do not penetrate the underground. Ruggedized versions of vector magnetometers and accelerometers installed in the BHA measure the magnetic and gravity fields at fixed intervals when the drilling is stopped. This process is called “Measurement While Drilling (MWD).” The MWD data are transmitted to the drilling surface via mud pulses through the drilling fluid. By combining the magnetic and accelerometer data, the drilling engineers determine the orientation of the BHA. By combining the distance drilled with the orientation information, they construct the well path of the borehole within an envelope of uncertainty. However, the magnetometers only provide the azimuth of the BHA with respect to the local magnetic field's horizontal direction. Hence, properly converting the magnetic measurements to geographic orientation using a geomagnetic field model is imperative. At the site of the borehole, the local magnetic field is the superposition of magnetic fields primarily arising from the following natural sources: Earth's core, crust and space‐weather‐related electric currents in the magnetosphere and ionosphere. Note that the MWD data might also be influenced by the magnetic signals arising from the BHA. The so‐called “drillstring interference” is mitigated by additional processing such as multi‐station analysis (e.g., Buchanan et al. (2013)) or by using tools manufactured with nonmagnetic materials. The core and crustal fields (being mostly static) can be accounted for through high‐resolution models of the Earth's internal field, such as the High Definition Geomagnetic Model (Nair et al., 2021) and British Geological Survey Global Geomagnetic Model (Beggan et al., 2021). However, electric currents in the ionosphere and magnetosphere and their counterparts induced in the conducting Earth give rise to diurnal and irregular variations observable in the MWD. One of the common ways for MWD engineers to mitigate the space‐weather effects is to correct MWD readings for the magnetic field data from an adjacent land‐based observatory or by interpolating the magnetic variations between a group of adjacent, land‐based observatories (Reay et al., 2005). Poedjono et al. (2015) proposed sea level magnetic measurements using autonomous marine vehicles to support offshore drilling. Both of these methods assume that the signals measured at the sea level are similar to those at the seabed drilling site. Based on electromagnetic (EM) modeling, this paper shows that such approximations cannot be used for an offshore site. The EM induction causes the sea level and seabed magnetic fields to differ significantly at the same lateral location (i.e., at the same geographic latitude and longitude of the sea level and seabed points). Moreover, we propose and verify a numerical formalism allowing researchers to obtain trustworthy seabed signals using magnetic field measurements at the adjacent land‐based site and magnetic fields modeled at land‐based and seabed sites.

The paper is organized as follows. Section 2 discusses a methodology allowing us to calculate accurately and efficiently time‐varying magnetic field in a given conductivity model of the Earth, provided a source of magnetic field variations is also known. The methodology—being general, particularly in terms of the source parameterization—is implemented in this paper to model and analyze the spatio‐temporal evolution of the magnetic field during geomagnetic storms. Global‐scale problem setup advocates the parameterization of the magnetospheric source responsible for the storms using spherical harmonics (SH). Estimation of the corresponding expansion coefficients from hourly‐mean observatory data is also discussed in Section 2. Note that using a large‐scale source model represented by SH and temporally resolved with a relatively low sampling interval (1 hr) precludes analysis of the high‐latitude signals originating from an auroral ionospheric source, typically much smaller‐scale and highly variable in time. Section 3 presents results of time‐domain modeling of the magnetic field at sea level and seabed during the storms. Modeling is performed using three‐dimensional (3‐D) conductivity and (data‐based) source models built as realistic as feasible. Section 3 demonstrates that the sea level horizontal magnetic fields significantly differ from those at the seabed, especially during the main phase of the geomagnetic storms; recall that conventionally they are assumed to be the same in MWD applications. Section 4 compares modeling results with observations at land‐based and seabed sites. Further, Section 5 introduces and justifies a scheme to obtain the seabed signals using the data from the adjacent land‐based observatory and the results of comprehensive modeling discussed in Section 3. A scheme exploits a concept of transfer functions (TFs) that relate—in frequency domain—three components of the magnetic field at the seabed site of interest with those at the adjacent land‐based site. In an example of the seabed observations in the Philippine Sea, we demonstrate the workability of the proposed scheme. Finally, Section 6 summarizes our findings and outlines the paths for further research. The paper also includes three appendices detailing the theoretical results presented in the main text.

We start with the discussion of the problem in the frequency domain. Maxwell's equations govern EM field variations and, in the frequency domain, these equations read as*μ*_{0} is the magnetic permeability of free space, *ω* is angular frequency, **j**^{ext}(**r**, *ω*) is the extraneous (inducing) electric current density, **B**(**r**, *ω*; *σ*) and **E**(**r**, *ω*; *σ*) are magnetic and electric fields, respectively. *σ*(**r**) is the spatial distribution of electrical conductivity, **r** = (*r*, *θ*, *φ*) a position vector, in our case in the spherical geometry. Note that we neglected displacement currents and adopted the following Fourier convention

We will assume that the current density, **j**^{ext}(**r**, *ω*), can be represented as a linear combination of spatial modes **j**_{i}(**r**)

The form of spatial modes **j**_{i}(**r**) (and their number, *L*) varies with application. For example, **j**^{ext}(**r**, *ω*) is parameterized via SH in Püthe and Kuvshinov (2013), Honkonen et al. (2018), Guzavina et al. (2019), Grayver et al. (2021), current loops in Sun and Egbert (2012), eigenmodes from the Principal Component Analysis (PCA) of the physics‐based models in Egbert et al. (2021) and Zenhausern et al. (2021), and eigenmodes from the PCA of the data‐based models in Kruglyakov et al. (2022). In this paper—because we work on a whole sphere—we will use SH parameterization of the source, namely*l* and *m* are degree and order of SH, respectively, expression *δ* is Dirac's delta function, *a* is the mean radius of the Earth, *a*+ means that **e**_{r} is radial unit vector of the spherical coordinate system, ∇_{⊥} = *r*∇_{H}, ∇_{H} is tangential part of gradient,

By virtue of the linearity of Maxwell's equations with respect to the **j**^{ext}(**r**, *ω*) term, we can expand the total (i.e., inducing plus induced) magnetic field as a linear combination of individual fields

The transformation of the Equation 8 into the time domain leads to the representation of the magnetic field as

The reader is referred to Appendix A in Kruglyakov et al. (2022) for more details on the convolution integrals in the latter equation. We note that we use the same notation for the fields in the time and frequency domain. Equation 9 shows how magnetic field can be calculated provided *σ* are given. To make the formula ready for implementation, one also needs to estimate an upper limit of integrals in Equation 9, or, in other words, to evaluate a time interval, *T*, above which

Note that the upper limit in the integrals could be different for different *T* as a maximum from all individual upper limit estimates. Our model experiments (not shown in the paper) advocate that *T* should be taken as half a year.

The details of numerical calculation of the integrals in Equation 10 are presented in Appendix A. In short, assuming that *t* and *T* = *N*_{t}Δ*t*, one calculates **B**(**r**, *t*_{k}; *σ*) at *t*_{k} = *k*Δ*t* as

A few comments are relevant at this point.

Quantities **B**(**r**, *t*_{k}; *σ*) is required. Actual form and estimation of

**r** stands for any location, thus allowing us to calculate magnetic field at satellite altitude, ground or/and seabed.

Calculation of **B**(**r**, *t*_{k}; *σ*) requires knowledge

A numerical scheme for estimating

The conductivity model—as realistic as possible—is known to us;

We work with time series of three components of the magnetic field **B** at *J* geomagnetic observatories with coordinates **r**_{j} = (*a*, *θ*_{j}, *φ*_{j}), *j* = 1, 2, …, *J*. The time series are given with a sampling interval, Δ*t*, which we take as 1 hr, meaning that we will work with hourly‐mean observatory data.

The above‐mentioned time series are available for several years of observations, thus at time instants *t*_{1}, *t*_{1} + Δ*t*, *t*_{1} + 2Δ*t*, ….

With these assumptions in mind, the calculation of *t*_{k} = *k*Δ*t*, *k* = 1, 2, … is performed as follows. Substituting coordinates of observatories into Equation 11 and rearranging the terms, we obtain a system of equations to determine

As we discussed earlier, with *T* as long as a half of a year, *N*_{t} ≈ 24 × 180 = 4,320 provided sampling rate Δ*t* is one hour. If we start with the first time instant, that is, with *t*_{k} = *t*_{1} we do not have

Our model experiments (not shown in the paper) suggest that after a half of year (i.e., for

The expression (13) represents a system of linear equations (SLE) which is overdetermined when the number of unknowns (coefficients), *N*_{c} = *N*_{l} × (*N*_{l} + 2), is smaller than the number of equations, *N*_{o} = *N*_{b} × *J*, where *N*_{b} stands for number of magnetic field components. In practice *N*_{c} (with *N*_{l} = 4 giving *N*_{c} = 24) is always much smaller than *N*_{o} (with *J* near 70 and *N*_{b} = 2 giving *N*_{o} = 140). *N*_{b} = 2 means we use two horizontal magnetic field components (assuming a prior Earth's conductivity model) to estimate *N*_{l} = 4 allows us to represent both magnetospheric and a major part of the mid‐latitude quasi‐periodic (with period of 24 hr) ionospheric source.

Once **B**(**r**, *t*_{k}; *σ*), *k* ≥ 2*N*_{t} using Equation 11 at any location **r**. In our modeling studies to be discussed in the following sections, **r** is either a laterally‐uniform grid at the surface of the Earth (i.e., sea level) and at the seabed or coordinates of the land‐based and seabed sites at which we analyze modeled and experimental results.

We build the 3‐D conductivity model of the Earth, which includes nonuniform oceans and continents (generally with 3‐D conductivity distribution) and a laterally uniform (1‐D) mantle underneath. In this paper, we work with hourly‐mean observatory data, which, in particular, means that we are forced to analyze magnetic field variations with periods of 2 hr and longer due to the Nyquist‐Shannon theorem. For typical values of the Earth's conductivity, the penetration depth of the EM field exceeds a hundred kilometers (even at a period of 2 hr), much larger than the maximum ocean depths and sediment thicknesses (23 km). This fact allows us to shrink the nonuniform layer comprising oceans and continents into a thin shell of laterally‐variable conductance (depth‐integrated conductivity, with the depth taken as 23 km as mentioned earlier); discussion on the adequacy of the thin shell model can be found in Ivannikova et al. (2018).

For the conductance distribution in oceanic regions, we used 0.1° × 0.1° marine map of conductance built by Grayver (2021). The inland conductances are obtained from the global conductivity model of Alekseev et al. (2015) which has a lateral resolution of 0.25° × 0.25°; to make it compatible with oceanic conductances, the resulting inland conductances were interpolated to 0.1° × 0.1° grid. The inland conductance distribution was updated in North America using the recently published contiguous 3‐D conductivity model of US (Kelbert et al., 2019). Figure 1 shows the global distribution of the compiled conductance. As for the 1‐D structure underneath, it is taken from Kuvshinov et al. (2021).

As discussed in Section 2.3, the estimation of

Further, we solved overdetermined system of linear Equation 13 to obtain coefficients *t*_{k} (*t*_{1} stands for 1 January 1997 00:30:00 UTC) taking *N*_{l} = 4 in Equation 6, which gives us *N*_{c} = *N*_{l} × (*N*_{l} + 2) = 24 coefficients per time instant. As mentioned in the previous section, we use only horizontal components of the magnetic field to estimate the coefficients.

We also note that since we consider the long (1997–2019) time series of the magnetic field, we adopt a geographic coordinate system—instead of the usually used geomagnetic coordinates—to avoid possible complications associated with the change of location of the geomagnetic pole during the considered (long) period of time.

During the observational period of 1997–2019, the number of observatories *J* in Equation 13 available at any time varied in the range from 38 to 112 due to the data gaps at specific observatories.

Final comment of this section refers to computation of **r**_{j}.

As an example, Figure 3 shows the time series of the dominant coefficient

After estimating the time series of coefficients

This section presents the global scale results during the main phase of three major geomagnetic storms. The main phase (centered around the peak of the geomagnetic storm) is chosen because the magnitude of the magnetic field reaches its maximum. Moreover, during the main phase, the field has the most complex spatial structure due to the enhanced asymmetry of the magnetospheric ring current, which is the primary source of geomagnetic storm signals. The Figures 4–9 present horizontal magnetic field, **B**_{H} = (*B*_{θ}, *B*_{φ}), at sea level and corresponding differences between seabed and sea level results, *B*_{θ} and *B*_{φ} components. Their difference generally depends on the ocean's depth, but this can be complicated by the spatial structure of the current sources. It is also seen that the spatial patterns of **B**_{H} and *d***B**_{H} differ from storm to storm; however, with visible dominance of spatial structure responsible for the symmetric part of the magnetospheric ring current.

Now we compare storm‐time time series of the modeled and observed magnetic fields in the region where we have both sea level (land‐based) and seabed magnetic field measurements. From November 1999 to July 2000, the seabed magnetotelluric (MT) survey was conducted in the Philippine Sea (Seama et al., 2007) at six locations along a line at water depths between 3,250 and 5,430 m, depicted as OBEM 1, OBEM 2, …, OBEM 6 in Figure 10. Note that OBEM 3, which was installed on the seabed between sites OBEM 2 and OBEM 4, is not shown because it did not provide reliable data.

As for the land‐based site, we have chosen the Kanoya (Intermagnet code: KNY) observatory, the nearest observatory to the seabed MT survey region. Figure 11 presents modeled and observed **B**_{H} at KNY and OBEM 1, OBEM 4, and OBEM 6 sites; OBEM 2 and OBEM 5 sites are excluded from the analysis not to overwhelm the exposition. Note that the baseline and a linear trend were subtracted from the data to remove core and crustal field contributions and possible instrument drift. We show the results for 4 days of the April 2000 storm, which appeared to be the only significant event during the deployment of the seabed MT instruments. We can make several observations from the figure: (a) modeled and observed results agree remarkably well for all (land‐based and seabed) sites and both components; (b) an agreement is slightly better in *B*_{θ} which is three times larger than *B*_{φ}; (c) seabed signals at 20% smaller than land‐based signals which is in agreement with the global results presented in the previous section.

However, during offshore drilling, no seabed magnetic field measurements are performed near the drilling site. Therefore, to correct magnetic field MWD for the disturbing effects from space‐weather events (storms and substorms), one usually uses the magnetic field measurements from the adjacent land‐based site/observatory, assuming that these signals are close to those one could observe at the seabed in the vicinity of the drilling site. But—as demonstrated in this and previous sections—land‐based and seabed time‐varying magnetic fields differ significantly. Considering this fact and encouraged by an agreement between modeled and observed results, we, in the next section, introduce a numerical scheme that allows researchers to obtain offshore seabed signals from the adjacent land‐based observations.

Let us imagine that the drilling is performed at an offshore (seabed) site **r**_{sb}, and one has to estimate (and then account for) magnetic variations at **r**_{sb}. A standard way to estimate these variations is to take magnetic field variations from the nearby land‐based site **r**_{lb}, assuming that variations at **r**_{sb} do not significantly differ from those at **r**_{lb}. However, as we showed in Sections 3 and 4, horizontal components of the magnetic field are substantially different at sea level and seabed.

Below we propose and justify a scheme to more correctly estimate magnetic field variations at an offshore, that is, seabed, drilling point **r**_{sb}. A scheme exploits an assumption that frequency‐domain magnetic fields at locations **r**_{sb} and **r**_{lb} are related through inter‐site 3 × 3 matrix transfer function *T*

It is important to stress that Equation 14 holds approximately, but as we will see later in this section, this approximation works well, especially when the lateral separation between land‐based and seabed sites is not too large. Estimating the elements of *T* at a given frequency *ω* and conductivity model *σ* is performed as follows. First, one calculates the fields **B**^{(i)}, *i* = 1, 2, 3 (see details in Appendix C). Finally, the elements of *T* are estimated row‐wise using the **B**^{(i)} fields. For example, the elements of the first row of *T* are determined as the solution of the following SLE

Note, that in Equation 16 the dependency of all quantities on *ω*, **B**^{(i)} on *σ*, and elements of *T* on *σ*, **r**_{sb} and **r**_{lb} are omitted but implied. Once elements of *T* are estimated at a predefined frequencies, **B** at seabed site **r**_{sb} at a given time instant *t*_{k} = *k*Δ*t* is calculated similarly as it was done in Equation 11, that is,

Before showing results justifying the proposed scheme, one critical comment is relevant here. In Section 3 we stated that the radial component (for the considered variations) is the same on the sea level and the seabed. Thus the question may arise why in Equation 14 we also invoke the radial components? To address this question, we remind the reader that the statement about the similarity of the radial component at sea level and the seabed is indeed valid, provided both signals refer to the same lateral location. However, in the problem setup we consider, the land‐based (sea level) site is usually located at the coast and at a distance from the drilling point (as in our example). Moreover—and in contrast to horizontal components—the coastal radial field is dramatically distorted by the so‐called ocean induction effect originating from large lateral conductivity contrasts between the ocean and land (Olsen & Kuvshinov, 2004; Parkinson & Jones, 1979). Figures 12–14 illustrates this fact by presenting a radial field at a global scale during the main phase of three storms mentioned above. One can see that, indeed, the magnitude of the radial component enhances substantially in coastal regions.

We calculated seabed fields using TF‐based formalism discussed above and compared them with observations. As in Figure 11, Figure 15 shows results for 4 days of the April 2000 storm. It is seen that TF‐based and observed results are in agreement with the observations at all three seabed sites and for both components. As expected, the agreement (generally very good) worsens with the distance from the land‐based (KNY) site, which varies from 478 to 2,146 km (see Table 1). KNY results which are used in Equation 17 are also shown in the figure. Once again, one may notice that land‐based results differ much from seabed results.

As a whole, Figure 15 demonstrates that, indeed, one can obtain trustworthy seabed signals by exploiting TF‐based approach as applied to adjacent land‐based measurements.

In this paper, we presented an approach to efficiently calculating spatio‐temporal evolution of magnetic field at any location (at satellite altitude, ground or seabed) in a given conductivity model of the Earth, provided the source of magnetic field variations is also known. The approach relies on the factorization of the source by spatial modes and time series of respective expansion coefficients and exploits precomputed magnetic field kernels generated by corresponding spatial modes.

The methodology—being general—is implemented in this paper to model and analyze the spatio‐temporal evolution of the magnetic field during geomagnetic storms. Global‐scale problem setup advocates the parameterization of the magnetospheric source responsible for the storms using SH. We also presented a numerical scheme to estimate the time series of corresponding expansion coefficients using the data from the global network of geomagnetic observatories and exploiting precomputed magnetic field kernels.

We implemented the developed approach to model magnetic field behavior during three geomagnetic storms and demonstrated that the sea level horizontal magnetic fields significantly differ from those at the seabed, especially during the main phase of the geomagnetic storms. We then compared modeling results with observations at land‐based and seabed sites and detected remarkable agreement between modeled and observed fields for all (land‐based and seabed) sites and in both components.

Finally, we introduced and justified a scheme to obtain the seabed signals using the data from the adjacent land‐based observatory and the results of comprehensive modeling. A method exploits a concept of transfer functions that relate—in frequency domain—three components of the magnetic field at the seabed site of interest with those at the land‐based site. In an example of the seabed observations in the Philippine Sea, we demonstrate the workability of the proposed scheme.

This paper discussed magnetic fields induced by a large‐scale magnetospheric source that dominates in mid‐ and low‐latitudes. However, it is well known that one can expect much larger signals in polar regions due to substorm geomagnetic disturbances (Pirjola, 2002). The recovery of the spatio‐temporal structure of the auroral ionospheric source, which is responsible for this activity, is more challenging due to the large variability of this source both in time and space. One of the ways to determine realistic auroral currents on a regional scale consists of collecting the data from high‐latitude geomagnetic observatories and polar magnetometer arrays, for example, IMAGE array in Scandinavia (Tanskanen, 2009), CARISMA (Mann et al., 2008) and AUTUMNX (Connors et al., 2016) arrays in Canada, and then reconstructing the auroral current, for instance, by exploiting an approach based on spherical elementary current systems, SECS (Vanhamäki & Juusola, 2020). Note that this approach was used by Kruglyakov et al. (2022), who discussed real‐time 3‐D modeling of the ground electric field due to space weather events.

Once the auroral source is quantified, a similar numerical scheme described in the paper can be implemented with three modifications. One modification concerns the description of the substorm source—instead of using a SH representation, one can approximate the spatio‐temporal evolution of the auroral ionospheric current via spatial modes obtained by PCA of SECS (Kruglyakov et al., 2022). Another modification applies to the 3‐D conductivity model. Since substorm magnetic variations are characterized by periods between seconds and hours, one cannot exploit a model in which the surface layer is approximated by a thin shell, as this paper did. The variable thickness of this layer is essential in this period range; thus, a full 3‐D model (including bathymetry) should be considered. The final modification invokes Cartesian geometry instead of the spherical geometry used in this paper. An application of the proposed scheme to a regional problem setup will be the subject of a subsequent publication.

In this section we provide details how equation*t*, is deduced to equation*t*_{k} is a current time instant. First we notice that with finite *T*, one must account for a possibly non‐negligible linear trend in time series

Substituting Equation A3 into the right‐hand side (RHS) of Equation A1, and considering (for simplicity) only one term in the sum, we obtain

Recall that *T* should be taken large enough to make approximation (A1) valid; particularly, this means that

Following Appendix A of Kruglyakov et al. (2022) the integral in the RHS of the latter equation can be expressed as

Then, Equation A4 can be approximated as

The integrals

Taking into account that we have *t* = *n*Δ*t*, *n* = 0, 1, … , we approximate

Recall that sinc interpolation is a method to construct a continuous band‐limited function from a sequence of real numbers, in our case time series *t* = *n*Δ*t*, *n* = 0, 1, …. Note that in our context, the term “band‐limited function” means that non‐zero values of a Fourier transform of this function are confined to the frequencies

Using the approximation (A9) and taking into account that

Thus, we can write

Further, following the properties of the Fourier transform as applied to sinc function, we obtain that

Finally, substituting Equation A13 in Equation A7, and Equation A7 in the RHS of Equation A1 we obtain

The latter equation can be written in the form of Equation A2 where

Computation of the integrals in the RHS of Equation A15 is performed as follows. First, *t* and 1/(10^{7}Δ*t*), where Δ*t* is one hour for our problem setup. Further, using cubic spline interpolation as applied to calculated

An important note here is that, according to Equation A15, one does not need to compute *P* = 2Δ*t*). This may be obvious, however, this is not the case if one uses piece‐wise constant (PWC) approximation of *t* value; this can pose a problem from the numerical point of view.

Since

Substituting the latter equation into Equation A8 and rearranging the order of integration, we write *ωT*) reads

Integrals in Equation B2 can be efficiently estimated using the digital filter technique. Specifically, one needs to construct a digital filter for the following integral transform

To obtain filter's coefficients for this transform, we exploit the same procedure as in Werthmüller et al. (2019) using the following pair of output and input functions

Here we present a scheme to obtain **B**^{(i)}, *i* = 1, 2, 3 thus allowing us to calculate the desired inter‐site transfer function (cf. Equation 14). Specifically, the scheme includes the following steps:

We estimate *N*_{c} = 24 times series

We perform principal component analysis of these time series to obtain three “major modes” with time series *ν*_{i}(*t*), *i* = 1, 2, 3 and 3 × 24 (time independent) matrix

We apply matrix **B**^{(i)} corresponding to *ν*_{i}(*t*), *i* = 1, 2, 3 as

MK was supported by the New Zealand Ministry of Business, Innovation & Employment through Endeavour Fund Research Programme contract UOOX2002. AK was supported in the framework of Swarm DISC activities, funded by ESA contract no. 4000109587, with the support from EO Science for Society. This research was also supported in part by the NOAA cooperative agreements NA17OAR4320101 and NA22OAR432015. The authors acknowledge British Geological Survey, World Data Center Geomagnetism (Edinburgh), INTERMAGNET and the many institutes around the world that operate magnetic observatories. We are thankful to the editor, Dr Jennifer Gannon, the reviewer Dr Jakub Velimsky, and the anonymous reviewer whose insightful comments helped us improve the original manuscript.

The global map of ocean and sediments conductance by Grayver (2021) is available at

The global conductivity model by Alekseev et al. (2015) is available at

The conductivity model for North America by Kelbert et al. (2019) is available at

1‐D global conductivity model is published in a form of Table 1 in Kuvshinov et al. (2021).

Solver GEMMIE (Kruglyakov & Kuvshinov, 2022) is available at

The data of the ocean bottom survey is available at

The cleaned magnetic field data based on measurements from the global (INTERMAGNET) network of observatories are available by following the instructions at