Deep learning (DL) algorithms have shown incredible potential
in many applications. The success of these data-hungry methods is largely
associated with the availability of large-scale datasets, as millions of
observations are often required to achieve acceptable performance levels.
Recently, there has been an increased interest in applying deep learning
methods to geophysical applications where electromagnetic methods are used
to map the subsurface geology by observing variations in the electrical
resistivity of the subsurface materials. To date, there are no standardized
datasets for electromagnetic methods, which hinders the progress,
evaluation, benchmarking, and evolution of deep learning algorithms due to
data inconsistency. Therefore, we present a large-scale electrical
resistivity model database (RMD) with a wide variety of geologically plausible and
geophysically resolvable subsurface structures for the commonly deployed
ground-based and airborne electromagnetic systems. Potentially, the
presented database can be used to build surrogate models of well-known
processes and to aid in labour-intensive tasks. The geophysically
constrained property of this database will not only achieve enhanced
performance and improved generalization but, more importantly,
incorporate consistency and credibility into deep learning models. We show the
effectiveness of the presented database by surrogating the forward-modelling
process, and we urge the geophysical community interested in deep learning
for electromagnetic methods to utilize the presented database. The dataset
is publicly available at 10.5281/zenodo.7260886
(Asif et al., 2022a).
Recent years have witnessed the success of many deep learning (DL)
applications. Although DL emerged in 1982 in the form of neural networks
(Hopfield, 1982), it started to gain attention in 2012 due to its notable
performance for image classification tasks (Krizhevsky et al., 2017,
2012). Since then, it has been applied successfully to many applications
including object detection (Asif et al., 2019; Redmon et al., 2016; Ren
et al., 2015), image super-resolution (Dong et al., 2016; Zhang et
al., 2018), speech recognition (Zhang et al., 2017), and stock market
predictions (Pang et al., 2020). The revival of DL was mainly influenced
by the availability of cheap computing resources, deeper network
architectures, and large-scale publicly available datasets. Deeper network
architectures and an increased number of samples in the training datasets
are key factors for improved performance and better generalization of DL
models (Wang et al., 2016).
Geophysics is a branch of earth sciences, and geophysical methods are often
used to infer information about the subsurface geology by mapping physical
properties. The integration of neural networks in geophysics started several
decades ago and has covered many domains of geophysics (Baan and Jutten,
2000; Dramsch, 2020), including seismic (Röth and Tarantola, 1994;
Zhang et al., 2020), magneto-telluric (Conway et al., 2019; Liu et al.,
2020; Zhang and Paulson, 1997), geo-mechanical (Feng and Seto, 1998;
Khatibi and Aghajanpour, 2020), and electromagnetic domains (Birken and Poulton,
1999; Birken et al., 1999; Bording et al., 2021; Kwan et al., 2015; Poulton
et al., 1992; Zhu et al., 2012). Interestingly, the last few years have seen
a significant increase in interest in applying DL to electromagnetic (EM)
methods (see Table 1), where the artificially
generated EM fields are used to map variations in the electrical resistivity
properties of the subsurface. For more details regarding the EM methods,
readers are referred to the literature (e.g. Kirsch, 2006). The increasing
interest in applying DL to EM methods is mainly influenced by the increased
ability of the EM methods to collect huge datasets in short amounts of
time, which make the subsequent processes extremely laborious and time
consuming. Therefore, a DL method could be beneficial in surrogating
well-known EM processes, e.g. forward modelling where the propagation of the
EM fields is simulated, resulting in the forward responses (Xue et al.,
2020), and inverse modelling (inversion) where the electrical resistivity
properties of the subsurface are deduced from observed EM data (Zhdanov,
2015). DL methods can also assist with manual tasks, which may require
considerable time when performed manually, such as anomaly detection in EM
data. Further opportunities may lie in other tasks, e.g. data de-noising.
To apply a DL algorithm to EM methods for various applications, subsurface
resistivity models and/or the corresponding EM responses are often required.
To achieve optimal performance, a DL method should be trained on a large
number of geologically realistic subsurface models. Evident from
Table 1, the recently developed DL methods either
use subsurface resistivity models acquired from field data or generate the
models randomly or in a pseudorandom manner for training. However, a method
trained on random models, where the resistivity of each geological layer is
chosen from a probability distribution, would not result in optimal
performance, as many of the training samples would be geologically
unrealistic. A good solution is to use either resistivity models inverted
from field data or pseudorandom resistivity models where the resistivity of
the training models is based on some prior geological information to reflect
various characteristics of field data (Bai et al., 2020). However, a
DL method trained on such training samples would only be effective for
specific geological conditions and would result in an unsatisfactory
performance for significantly different geological settings (Bording et
al., 2021), as bias in the training data can affect generalizability
substantially. Additionally, the unavailability of a standard benchmark
database hinders the progress, evaluation, benchmarking, and evolution of DL
algorithms due to data inconsistency (Bergen et al., 2019; Reichstein
et al., 2019).
Recent publications (2019–2021) of DL applications in EM which
show the number of training samples and type of training dataset (random,
pseudorandom, or field data).
ReferenceNo. of samplesTraining observation typeApplicationin training setWu et al. (2021a)80 000Pseudorandom resistivity models and forward responsesInversionColombo et al. (2021a)5000Pseudorandom resistivity models and forward responsesInversionColombo et al. (2021b)20 000Random resistivity models and forward responsesInversionWu et al. (2021b)16 800Forward responses of random resistivity modelsDe-noisingBording et al. (2021)93 500Field data and inversion modelsForward modellingPuzyrev and Swidinsky (2021)512 000Random resistivity models and forward responsesInversionAsif et al. (2021a)100 000Field data and inversion modelsForward modellingMoghadas et al. (2020)20 000Random resistivity models and forward responsesForward modellingBai et al. (2020)12 000Pseudorandom resistivity models and forward responsesInversionLi et al. (2020)1 000 000Pseudorandom resistivity models and forward responsesInversionBang et al. (2021)25 173Pseudorandom resistivity models and forward responsesInversionNoh et al. (2020)20 000Random resistivity models and forward responsesInversionMoghadas (2020)20 000Random resistivity models and forward responsesInversionColombo et al. (2020a)235 620Pseudorandom resistivity models and forward responsesInversionColombo et al. (2020b)88Pseudorandom resistivity models and forward responsesInversionLin et al. (2019)2400Field data and inverted model forward responsesDe-noisingGuo et al. (2019)10 000Pseudorandom resistivity models and forward responsesInversionPuzyrev (2019)20 000Pseudorandom resistivity models and forward responsesInversionQin et al. (2019)50 000Random resistivity models and forward responsesInversion
To have an inclusive DL solution for various applications in EM, we present
a physics-driven large-scale model database (∼ 1 million models) of
geologically plausible and EM-resolvable 1-D subsurface resistivity models
spanning the resistivity range from 1 to 2000 Ωm and to a
depth of 500 m. This model database is suitable for ground-based and
airborne EM systems in a DL context. We use broad-banded von Kármán
covariance functions to generate geologically constrained resistivity
models. Geophysical constraints are imposed by calculating the EM forward
data of the initial resistivity models followed by inversion of the EM
forward data to obtain the final resistivity models. This allows us to create a
comprehensive resistivity model database (RMD) that may not only improve
performance and generalization but also incorporate consistency and
reliability into the DL models. We believe that the presented RMD will be a
valuable resource to accelerate the inter- and trans-disciplinary research
of earth and data sciences. The presented DL-RMD will also provide
uniformity in training and benchmarking for DL methods in EM. Therefore, we
urge the geophysical community interested in DL for EM methods to use the
DL-RMD.
The rest of this paper is organized as follows. Section 2 describes the
general methodology of generating the subsurface resistivity models, while
specific settings for the DL-RMD for the three EM system categories are
specified in Sect. 3. Section 4 provides details for training a DL method
to surrogate the forward-modelling problem and shows the effectiveness of
the DL-RMD. Discussion, code and data availability, and concluding remarks
are given in Sects. 5, 6, and 7, respectively.
Methodology
Geological processes do not result in random structures, nor are the
subsurface resistivity structures random, as some spatial correlation is
generally present (Tacher et al., 2006). Therefore, it is reasonable
that the training of a DL method is based on subsurface structures that are
geologically plausible and, in an EM context, overall resolvable by the EM
method. Additionally, the scale of the resistivity structure in the models
should reflect the resolution capability of the EM methods, as training a DL
method to resolve structures that are not evident in the input data is not
possible. EM methods are diffusive methods with significantly decreasing
resolution with depth, and the electrical conductivity contrast plays an
important role for the resolution capability; hence, a metric number for a
given EM method's resolution capability and the depth of investigation cannot be given.
To obtain geologically realistic models, we use the broad-banded von
Kármán covariance functions (Møller et al., 2001) to generate
geologically plausible models (von Kármán models). The suite of von
Kármán models consists of fine geological structures and contain
some resistivity variations and patterns that are unlikely to be resolved,
due to the resolution limitation of the EM method. To replicate the
resolution capability of the EM method, we generate EM forward responses of
the initially over-detailed von Kármán models and invert these
forward responses to obtain the final resistivity models. Since we aim at
generating 1-D resistivity models, we are only concerned about the
resistivity (ρ) variations in the vertical direction (z) from surface
to some depth in our model generation.
Initially, we base the spatial variation character of (z, log10(ρ)) for our von Kármán models on the broad-banded von Kármán
covariance functions (Christiansen and Auken, 2003; Møller et al.,
2001).
Cz,A,v=A2C0zLνKνzL,
where A becomes the amplitude of the logarithmic resistivity, C0 is a
scaling constant, z is the spatial (vertical) distance, L characterizes the
maximum correlation length accounted for, and Kν is the modified
Bessel function of the second kind and order ν. In the model generation,
L is fixed to a high number (1800 m) which gives us strong correlation for
z≪L (Maurer et al., 1998). By using combinations of
ν, C0, and resistivity and compiling several realizations of the
stochastic von Kármán process, we generate a variety of resistivity
models on multiple scales. Table 2 summarizes the
L, ν, C0, and resistivity values used.
Parameters used in all combinations to generate the initial von
Kármán resistivity models.
ParameterValuesResistivity1 to 2000 Ωm, log-spaced,20 values per decadeLFixed: 1800 mv[0.6, 0.7, 0.8, 0.9, 1.0]C0[0.5, 1, 2, 4]No. of sharp boundaries[1, 2, 3, 4, 5]
Examples of this are shown in Fig. 1a–c where
the von Kármán models (in black curves) are generated with a
combination of the extreme values of ν and C0 for an initial
resistivity value of 30 Ωm. Low ν and high C0 produce models
with fine- and large-scale variations (Fig. 1a),
while high ν and high C0 values produce a relatively smooth
model (Fig. 1b) but still with resistivity
variations spanning 2–3 decades of resistivity. The combination of low
ν and C0 values ensures that the simple and close-to-half-space
models are also represented (Fig. 1c).
Sharp layering in the subsurface is plausible, and large resistivity
amplitudes and short correlation lengths in the von Kármán functions
will form layering in the models. To include more models with a sharp
layering, we stitch 2–6 randomly selected depth intervals of the initially
generated von Kármán models from a uniform distribution. An example
of a stitched model is shown in Fig. 1d. These
stitched models also ensure that different combinations of ν and C0
are represented within one model.
Examples of von Kármán models and the result after the
forward and inversion process, where black curves show von Kármán
models (re-discretized to 90 layers), and the red curve shows the final model.
Panels (a) to (c) are for the combination of ν and C0 stated in the title; panel (d) is for a
stitched, layered model (green arrows mark the imposed sharp layer
boundaries). The red curves show the obtained model from inversion of the
forward response of the black model.
Prior to the EM forward calculation, the von Kármán models are
re-discretized to 90 layers for faster forward computation and easier
handling. The top-layer thickness and depth to the last layer boundary for
the re-discretized layers are detailed in
Table 3 for three generic EM systems with different
depths of investigations (see Sect. 3 for further details). For the
forward calculation, the geometric mean of the last 5 m of the
re-discretized von Kármán models is assigned to the last model layer
that continues to infinite depth. In order to avoid making assumptions on
the acquisition conditions, on the specific instrument setups, etc., the calculated
forward data are pragmatically assigned a uniform uncertainty of 5 % to
take noise into account and are inverted with a 30-layer model with a minimum
structure (smooth) regularization scheme (Viezzoli et al., 2008). The
layer thicknesses for the 30-layer models are fixed, and they are
listed in Table 3. The red model curves in
Fig. 1 represent the resistivity models after the
forward and inversion process and represent the models that enter the DL-RMD. As
seen from Fig. 1, the von Kármán models
hold structures that are not resolved by the inverted resistivity models, so
the models obtained after the forward and inversion process result in
structures resolvable by the EM method. A total of ∼ 95 % of
the inverted resistivity models explain (fit) the forward data within the
assumed data uncertainty. In other words, the inverted models are explaining
the more complex von Kármán models to a very high degree.
The forward and inverse modelling is carried out for three different generic
time-domain EM (TEM) systems spanning different depth ranges using the
AarhusInv modelling code (Auken et al., 2015). The specific DL-RMD
settings for different TEM systems are summarized in Sect. 3.
Deep learning resistivity model database (DL-RMD)
EM systems for subsurface exploration have existed since the 1950s, and
nowadays a large variety of airborne and ground-based time-domain
electromagnetic (TEM) and frequency-domain electromagnetic (FEM) systems
exist. Both TEM and FEM methods map the electrical resistivity of the
subsurface by inducing EM fields. TEM methods record the decay of the
secondary EM field in the absence of the transmitted EM field in the
time domain, while FEM methods record the secondary EM field in the
frequency domain in the presence of the transmitted EM field
(Christiansen et al., 2006). TEM and FEM methods also differ in
resolution and depth of investigation, depending on the TEM system
configuration, e.g. transmitter turn-off time, transmitter moment, and
airborne or ground-based. For the DL-RMD to be compatible for different TEM
systems, we have compiled three model databases with ∼ 1 million models in each for three generic TEM systems with different depths
of investigation as their primary differences. We refer to the three DL-RMDs
as shallow, intermediate, and deep, with the initialisms S-RMD, I-RMD, and D-RMD, respectively. S-RMD mimics
a shallow-focusing ground-based TEM system, initiated by a short transmitter
turn-off time. For S-RMD, the models are discretized down to 125 m with a
top-layer thickness of 0.5 m. I-RMD and D-RMD mimic airborne TEM systems with
different depths of investigation and are hence discretized down to depths of 350
and 500 m and top-layer thicknesses of 3 and 5 m, respectively. The
calculation of depth of investigation follows Christiansen and Auken (2012).
Model discretization and key specifications of the generic TEM
systems for three resistivity model databases. The generic TEM systems are
all central loop configurations.
TypeParameterS-RMDI-RMDD-RMDVon Kármán modelsMax depth (m)125355505 mDiscretization (m)0.10.10.1Re-discretization (m)0.2–120 m, 90-layer log-spaced1–350 m, 90-layer log-spaced2–500 m, 90-layer log-spacedDatabase resistivitymodelsModel discretization0.5–120 m, 30-layer log-spaced3–350 m, 30-layer log-spaced5–500 m, 30-layer log-spacedGeneric TEMTurn-off time (µs)41240configuration*Gate time start (µs)51350*Gate time end (ms)11032Modelling height (m)0 – ground-based40 – airborne40 – airborne
* Gate start/end times have zero-time
reference at the beginning of turn-off time.
Statistical insights into the DL-RMD. (a–c) Resistivity distributions
of the S-RMD, I-RMD, and D-RMD, respectively. (d–f) Distributions of depth of
investigation of models in the S-RMD, I-RMD, and D-RMD, plotted
as a cumulative sum.
The model discretization details for the three DL-RMDs for the initial von Kármán
models and for the final resistivity models entering the RMD are summarized
in Table 3. Table 3 also
holds the key specifications of the three generic TEM systems. The settings
for the generation of the von Kármán models are specified in
Table 2 and are common for the three DL-RMDs. Each of
the three DL-RMDs holds ∼ 1 million models spanning the
resistivity interval 1–2000 Ωm, where 1/6 of the models originate
from the initially generated von Kármán models and where 5/6 of the models
come from the stitched, layered von Kármán models.
Some insights into the three DL-RMDs are given in Fig. 2, where Fig. 2a–c show the layer resistivity
distribution of the three DL-RMDs. The resistivity distributions of the von
Kármán models were generated uniformly, but the forward and
inversion process makes the resistivity distribution slightly skewed towards
the lower-resistivity end, due to the lower sensitivity/resolution in the
high-resistivity end for the EM method (Christiansen et al., 2006;
Jørgensen et al., 2005). The larger start and end bins compared to the
neighbouring bins in Fig. 2a–c are due to the 1 and 2000 Ωm resistivity truncation. The estimated depths
of investigation for the three DL-RMDs are shown in Fig. 2d–f. We observe
that approximately 70 % of the models have depths of investigation that are less
than the depth to last layer boundary of the given DL-RMD. Notably, a
thick conductive layer near the surface will significantly limit the depth
of investigation for a given TEM configuration. The uneven and in some cases
limited depth of investigation does not pose a problem for a deep learning
algorithm, as the EM method will compromise a similar depth of investigation
limitation for the given resistivity model (see the Discussion section for more
details).
Example of an EM application using the DL-RMD
EM methods can benefit from the presented DL-RMD in many ways. For example,
the DL-RMD can be used to surrogate the computationally expensive numerical
forward modelling by using a computationally efficient DL method, which
would speed up the whole inversion process. It can also be used to develop a
DL algorithm to replace the calculation of the partial derivatives in
deterministic inversion methods, where the subsurface resistivity model is
updated iteratively by using the partial derivatives of the model
parameters. Detecting anomalies in the EM data by using a DL approach using
the DL-RMD can significantly speed up the EM data processing and limit the
involvement of human-centric manual workflows. Additionally, EM data
de-noising also becomes plausible.
As an example in this paper, we use the DL-RMD to surrogate the forward
modelling problem for a ground-based TEM system using a fast DL method,
since a significant number of forward calculations are required during the
inversion process, when either deterministic or stochastic inversion methods
are used. By replacing the computationally expensive numerical forward
modelling approach, the whole inversion process may be accelerated without
further modification to a standard inversion workflow (Asif et al.,
2021b). However, it is crucial that the performance of the DL method
balances the numerical precision and increased speed of computation. If the
prediction accuracy is not sufficiently high, the application in an
inversion framework may result in spurious subsurface features and erroneous
geological interpretations of the geophysical EM mapping results.
Deep learning (DL) setup
We design the surrogate model for the tTEM system (Auken et al., 2018).
The tTEM system is a ground-based towed TEM system with a maximum depth of
investigation of 120 m based on the data time interval from ∼ 5 µs to ∼ 1 ms, which matches the specification of
S-RMD; therefore, we use it to train our DL method.
The input to the DL algorithm becomes the 30-layer resistivity model
m in S-RMD, where the layer thickness of each resistivity layer is
fixed. The target outputs are the numerical TEM forward responses, i.e.
dB/dt, for the
corresponding inputs. A standard EM modelling code (Auken et al., 2015)
is used to generate the TEM forward responses for the resistivity models
m with fixed layer thicknesses. We generate the responses from
∼ 1 ns to ∼ 10 ms by exponentially increasing
gate widths sampled at 14 gates per decade.
Prior to the training of a DL method, inputs and the corresponding target
outputs are normalized. Each resistivity model m is normalized,
where the logarithmic variations in the model parameters can take both
positive and negative values.
mn=log10m-μlog10(mmax)+log10mmin2,
where mmin and mmax are the minimum and maximum resistivity values
in the training dataset of S-RMD, and μ is the mean.
The target outputs, i.e. dB/dt,
are normalized by
dBndt=dB/dt-μdB/dt)σdB/dt,
where μ is the mean, and σ is the standard deviation of each
data point in the training dataset.
We use a simple DL method where a fully connected feed-forward neural
network is utilized with two hidden layers, each having 384 neurons. The
hyperbolic tangent function is used as an activation function between the
hidden layers, and the full-batch scaled conjugate algorithm is used for
backpropagation. The loss function for training is the sum of squared errors
with a regularization term consisting of the mean of sum of squares of the
network weights and biases. The network configuration used here is based on
our previous results (Asif et al., 2021b, 2022b). We also
apply an early-stopping criterion to ensure that the training stops when the
validation loss starts to increase. The validation set for the early-stopping criterion comprises of 70 000 models from S-RMD, which are excluded
from the training set. Once the network is trained, it can be used for
evaluation purposes. The evaluation metric for our baselines is the
percentage relative error, RLP, defined in Eq. (4), which effectively
deals with the large dynamic range and patterns of TEM data.
RLP=dB/dtDL-dB/dtNdB/dtN×100%,
where dB/dtDL is the output of the DL method, and dB/dtN is the numerically computed forward response.
Performance of the networks trained on S-RMD, von Kármán models,
and random resistivity models. (a) RLP distribution. (b) Cumulative
distribution of RLP.
Surrogate forward-modelling results
To test the performance of our DL method trained on S-RMD, we use 697
resistivity models inverted from field data from a survey conducted in
Søften, a region in Denmark. The data processing and inversion step of the
field data follows the method developed by Auken et al. (2018), which
covers averaging, anomaly detection, manual inspection, etc. on the data.
The minimum and maximum resistivity values in the test dataset are 3.9
and 127.1 Ωm, respectively. The forward responses of the
field-inverted resistivity models are calculated numerically to compare them
with the output of our DL method. Since the output of our DL algorithm is
the normalized forward response, it is de-normalized to raw data values by
manipulating Eq. (3). For a relative comparison, we train another DL network
with the same configuration using the initial von Kármán resistivity
models. The comparison to the initial von Kármán resistivity models
also allows us to examine the effect of the forward/inversion process, as described in
Sect. 2, in the generation of the DL-RMD. We also train an additional
network using the random resistivity models, similarly to several DL studies
(Colombo et al., 2021b; Moghadas, 2020; Moghadas et al., 2020; Noh et
al., 2020; Puzyrev and Swidinsky, 2021; Qin et al., 2019; Wu et al., 2021b)
as mentioned in Table 1. To have the same level of complexity, the number of
layers, depth discretization, and the number of random resistivity models are
kept the same as used to train the other two networks for a fair comparison, and
the resistivity of each layer is chosen randomly from a log-uniform
distribution to take into account the non-linearity of the forward responses
with the resistivity values. As such, a resistivity change from 1
to 10 Ωm would affect the forward data more than a change from 100
to 110 Ωm (Asif et al., 2021a).
Figure 3 shows the performance comparison of the
trained networks based on the evaluation metric in Eq. (3) against the
forward responses of 697 resistivity models from the Søften survey.
Figure 3a shows the distribution of RLP of
the DL network trained on S-RMD. We also show the accuracy performance of
the DL networks trained on von Kármán and the random resistivity
models. It is evident that the network trained on S-RMD results in lower
errors as compared to the network trained on von Kármán resistivity
models. On the other hand, the network trained on random resistivity models
results in a poor accuracy performance. In quantitative terms, 71 % of the
data points are evaluated to be within half a percent relative error for the
network trained on S-RMD. In comparison to S-RMD, the network trained on von
Kármán resistivity models results in 65 % of data points within
half a percent relative error. The network trained on random resistivity
models performs the worst, and only 34 % of the data points are calculated
to be within half a percent relative error.
We also show the cumulative distribution of RLP for the networks
trained on S-RMD, von Kármán models, and random models in
Fig. 3b. A maximum of 9 % improvement in
accuracy is achieved for the network trained on the S-RMD as compared to the
von Kármán models. In comparison to the network trained on random
resistivity models, an improvement of 43 % is achieved when S-RMD is used
for training. The increase in accuracy is achieved only by using an
appropriate dataset for training. The prediction accuracy can be improved
with different data pre-processing, network configurations, loss functions,
etc. while using the same training dataset to allow for consistency in
benchmarking of DL algorithms. It is also important that a balance between
the prediction performance and computational efficiency is maintained. As
such, the computational time for the forward pass of the proposed network
configuration can serve as a baseline for time comparison.
Figure 4 shows a visual comparison of a numerical
forward response against the forward response from the trained networks for
one of the resistivity models from the Søften survey. It is evident from
Fig. 4 that the forward response from the network
trained on S-RMD is the most accurate and has a maximum relative error of
1.4 % for the data point at ∼ 72 µs (see
Fig. 4a). The highest error for the forward
response from the network trained on von Kármán models is observed
to be 2.5 % for the data point at ∼ 160 µs as shown
in Fig. 4b. The forward response from the
network trained on random models results in the worst accuracy performance
and results in a maximum error of 22.3 % for the data point at 100 µs (see Fig. 4c).
Comparison of performance of the networks trained on S-RMD, von
Kármán models, and random resistivity models with a numerical forward
response from the test set. The forward responses are shown only within the
time range of tTEM data, and the inset shows the forward response from 16 to 20 µs (a) Numerical forward response vs. the forward
response from the network trained on S-RMD. (b) Numerical forward response
vs. the forward response from the network trained on von Kármán
models. (c) Numerical forward response vs. the forward response from the
network trained on random resistivity models.
Discussion
The network trained on random resistivity models results in a poor accuracy
performance as many of the resistivity models in the training dataset are
geologically unrealistic. The complex, unrealistic resistivity structures in
the randomly generated training models would result in forward responses
similar to the ones obtained from simpler resistivity models, which further
decreases the quality of the training dataset. The von Kármán models
may be considered pseudorandom resistivity models where the resistivity
structure of the models has a geologically realistic nature, as it considers
multiple correlation lengths with a stochastic nature resembling geological
processes. Due to the geological nature of the von Kármán models,
the network trained on such models results in a decent performance accuracy.
However, the network trained on von Kármán models has a lower
accuracy performance as compared to the network trained on S-RMD, where the
resolution capability of the EM method has been taken into account, resulting
in resistivity structures resolvable by the EM method.
The resolution capability and the depth of investigation for a given TEM
system strongly depend on the underlying resistivity model. Therefore,
stating a single depth of investigation value for a given TEM system is not
appropriate. A single exploration depth, depth of investigation, or a
similar value stated by the instrument manufacturers will often be an
optimistic one. For TEM systems with short transmitter current turn-off, the
early data points provide the near-surface resolution, while the late
data points strongly control the depth of investigation for a given
resistivity model. The transmitter moment and the background noise level
also influence the depth of investigation, but these factors are not
considered in our case, since we have assumed a uniform data uncertainty in
the forward and inversion process. The three DL-RMDs span different TEM
systems and resolutions. Therefore, for a particular TEM system, one should
pick the DL-RMD that has a similar resolution as the underlying generic TEM
system. This is best evaluated by matching the time interval of the data for
the particular TEM system to the data time interval (data time start/end in
Table 3) for the generic TEM system.
In Table 4, we list some examples of the
compatibility of our DL-RMD with some well-known TEM systems. Despite
I-RMD and D-RMD being compiled for a generic airborne system, I-RMD and
D-RMD are also appropriate for ground-based TEM systems since the simulated
flight altitude of 40 m does not lead to a drastic change in the vertical
resolution.
Examples of DL-RMD compatibility for some TEM systems.
SystemResistivity model database S-RMDI-RMDD-RMDEQUATOR (Karshakov et al., 2017)✓tTEM (Auken et al., 2018)✓MEGATEM (Smith et al., 2003)✓AEROTEM (Balch et al., 2003)✓SkyTEM (Sørensen and Auken, 2004)✓✓GEOTEM (Smith, 2010)✓✓SPECTREMPLUS (Leggatt et al., 2000)✓
Since FEM and TEM systems follow the same laws of physics, the DL-RMD is
also applicable for many FEM systems, despite the generic EM system in the
forward/inversion process mimicking the TEM systems. In general, the FEM
systems have a shallower depth of investigation than that of the TEM
systems, hence, the S-RMD is best suited for FEM systems. An alternative to
the DL-RMD is to generate the resistivity model realizations by following
the described methodology for the specific EM system by using the von
Kármán models provided (Asif et al., 2022a). This will
ensure a 100 % match between resolution, depth of investigation, etc. in
the model domain compared to sensitivity in the EM data domain.
Despite the initial von Kármán models with superimposed layering,
the resistivity models in the DL-RMD have a pronounced vertical smooth
behaviour due to the minimum structure (smooth) regularization scheme
(Viezzoli et al., 2008) used in the inversion phase. When applying another
regularization scheme in the inversion phase, e.g. the minimum support norm
(Vignoli et al., 2015), or when using a few-layer model discretization with no
vertical regularization, one could compile a resistivity model database with
different appearances. For our DL-RMD, we chose the minimum structure
regularization scheme, since it is commonly used for inverting airborne and
ground-based EM data. It is important to point out that a TEM data curve
itself does not hold information about whether subsurface boundaries are
smooth or sharp. As such, both smooth and sharp-layered models will
explain the recorded data equally well in most cases. With our approach of
compiling resistivity models, we have tried to avoid the inclusion of models
with different smooth/sharp behaviours that result in identical or close to
identical forward data responses (equivalent models).
The DL-RMD is generated in the resistivity range of 1–2000 Ωm, which
covers most of the geological settings, taking into account the EM mapping
capability in the high-resistivity range. The resistivity limit of 2000 Ωm was chosen since EM methods have no or very low sensitivity in
the high-resistivity range, since high-resistivity materials (granite,
basalt, glacier ice, etc.) produce an EM signal below the detection level.
Despite the 2000 Ωm limit, the resistivity distribution of the
models in the DL-RMD is slightly skewed towards lower resistivities due to
the limited sensitivity of the EM method to high-resistivity values. A
slight bias towards lower-resistivity values may affect the performance of a
DL method for highly resistive models. However, even if an actual subsurface
model is represented by a highly resistive model, it is expected that any TEM
method would have difficulty in resolving such a model. The RMD also has a
limitation in the low-resistivity end, e.g. in settings with seawater and
saltwater intrusion, which may result in subsurface materials with
resistivity values below 1 Ωm.
Since the 1-D models of the DL-RMD hold resistivity variations in one
dimension (vertical) only, they cannot be used for calculating 2-D or 3-D
EM responses. Examples of geological settings where a 1-D approach would be
inappropriate include steep-dipping geological structures, thin sheet
mineralization, mapping close to or on the shoreline, or areas with strong
topographical variations. However, one could apply the same methodology to
compile a 2-D or 3-D resistivity database. In this case, one would generate
the initial von Kármán models as a 2-D section or 3-D volumes and use
a 2-D or 3-D forward and inversion process, which of course would be much
more computationally expensive compared to the 1-D case. However, the DL-RMD
provided in this study opens up the possibility of exploring more deep learning
frameworks, which have reliability and consistency in performance comparisons
for 1-D models.
Code and data availability
The DL-RMD is freely available at 10.5281/zenodo.7260886
(Asif et al., 2022a), and a ready-to-run demo code in Python Jupyter
Notebook that uses the network trained on S-RMD and reproduces the results
of this paper is available at https://github.com/rizwanasif/DL-RMD (last access: 17 March 2023) (DOI: 10.5281/zenodo.7740243, Asif, 2023).
The EM modelling code “AarhusInv” used to
generate EM forward responses in this study is freely available to
researchers for non-commercial activities. The details are available at
https://hgg.au.dk/software/aarhusinv (Auken et al., 2015).
Conclusion
We have presented a methodology for compiling a geophysically constrained
subsurface resistivity model database for applications related to
electromagnetic data. We generated three 1-D resistivity databases,
discretized to depths of 120, 350, and 500 m in the resistivity range of
1–2000 Ωm, hence covering various ground-based and airborne
frequency-domain and time-domain electromagnetic systems and most of the
geological settings. The upper resistivity limit of the model database is
satisfactory as the electromagnetic methods have limitations for high
resistivity; however, the model database has limitations in the low
resistivity limit for subsurface materials below 1 Ωm that may occur in
some cases. Additionally, the database holds 1-D models and therefore
inherits the limitations of 1-D electromagnetic modelling.
An example is included using the proposed resistivity model database and deep
learning for surrogating TEM forward modelling, showing that high accuracy
can be obtained with our resistivity model database. Furthermore, the
example shows that the forward/inversion steps in the generation of the
database lead to a significantly increased performance in the forward
modelling.
Despite some limitations, the generated resistivity model database is a
well-organized database, which empowers the geoscience community to have
consistency and credibility in the development of deep learning methods for
many tasks including surrogating forward modelling, inverse modelling, data
de-noising, automatic data processing, etc. Therefore, we urge the
geophysical community to utilize the presented database to develop and
investigate different network configurations, data pre-processing
strategies, loss functions, etc. while using the presented model database to
allow for consistency in benchmarking deep learning algorithms. The resistivity
model database has already proven valuable in significantly improving the
accuracy of neural networks for the forward modelling of electromagnetic
data.
Author contributions
Conceptualization: MRA and TB.
Data curation, software, and visualization: MRA.
Formal analysis, methodology, and investigation: MRA, NF, TB, and AVC.
Funding acquisition, project administration, resources, and supervision: JJL and AVC.
Validation: NF and MRA.
Writing; original draft preparation, review and editing: MRA, NF, TB, JJL, and AVC.
Competing interests
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Special issue statement
This article is part of the special issue “Benchmark datasets and machine learning algorithms for Earth system science data (ESSD/GMD inter-journal SI)”. It is not associated with a conference.
Acknowledgements
The authors would like to thank the handling chief editor Kirsten Elger, topical editor Martin Schultz, and the two anonymous reviewers for their comments and feedback on this paper.
Financial support
This work has been supported by the Innovation Fund Denmark (IFD) under the
projects “MapField” (grant no. 8055-00025B) and “SuperTEM” (grant no.
0177-00085B).
Review statement
This paper was edited by Martin Schultz and reviewed by two anonymous referees.
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