The quantitative integration of geophysical measurements
with data and information from other disciplines is becoming increasingly
important in answering the challenges of undercover imaging and of the
modelling of complex areas. We propose a review of the different techniques
for the utilisation of structural, petrophysical, and geological information
in single physics and joint inversion as implemented in the Tomofast-x
open-source inversion platform. We detail the range of constraints that can
be applied to the inversion of potential field data. The inversion examples
we show illustrate a selection of scenarios using a realistic synthetic
data set inspired by real-world geological measurements and petrophysical
data from the Hamersley region (Western Australia). Using Tomofast-x's
flexibility, we investigate inversions combining the utilisation of
petrophysical, structural, and/or geological constraints while illustrating
the utilisation of the L-curve principle to determine regularisation
weights. Our results suggest that the utilisation of geological information
to derive disjoint interval bound constraints is the most effective method
to recover the true model. It is followed by model smoothness and smallness
conditioned by geological uncertainty and cross-gradient minimisation.
Introduction
Geophysical data provide detailed information about the structure and
composition of the Earth's interior otherwise not accessible by direct
observation methods, and thus these data play a central role in every major Earth
imaging initiative. Applications of geophysical modelling range from deep
Earth imaging to study the crust and the mantle to shallow investigations of
the subsurface for the exploration of natural resources. Recent integration
of different geophysical methods has been recognised as a means to reduce
interpretation ambiguity and uncertainty. Further developments introduce
uncertainty estimates from other geoscientific disciplines such as geology
and petrophysics to produce more reliable and plausible models. Various
techniques integrating different geophysical techniques have been developed
with the aim to produce more geologically meaningful models, as reviewed by
Parsekian et al. (2015), Lelièvre and Farquharson (2016), Moorkamp et al. (2016), Ren and
Kalscheuer (2019), and Meju and Gallardo (2016), and several
kinds of optimisation for such problems exist (Bijani et
al., 2017). In the natural resource exploration sector, the calls of
Wegener (1923), Eckhardt (1940), and
Nettleton (1949) for the development of comprehensive,
thorough multi-disciplinary and multi-physical integrated modelling have
been acknowledged by the scientific community, and data integration is now
an area of active research, to quote André Revil's preface of the
compilation of reviews proposed by Moorkamp et al. (2016):
“The joint inversion of geophysical data with different sensitivities
[...] is also a new frontier”. The integration of multiple
physical fields (both geophysical and geological) is particularly relevant
for techniques relying on potential field gravity and magnetic data, as
these constitute the most commonly acquired and widely available geophysical
data types worldwide. The need for integrated techniques is partly due to
the interpretation ambiguity of geophysical data and resulting effects of
non-uniqueness on inversion. Therefore, effective inversion of potential
field data necessitates the utilisation of constraints derived from prior
information extracted from geological and petrophysical measurements or
other geophysical techniques whenever available.
A number of methods for the introduction of geological and petrophysical
prior information into potential field inversion have been developed. For
example, when limited geological information is available, the assumption is
that spatial variation of density and magnetic susceptibility are
co-located. This can be enforced through simple structural constraints
encouraging structural correlation between the two models using Gramian
constraints (Zhdanov et al., 2012) or the cross-gradient
technique introduced in Gallardo and Meju (2003). When
petrophysical information is available, petrophysical constraints can be
applied during inversion to obtain inverted properties that match certain
statistics (see techniques introduced by
Paasche and Tronicke, 2007; De Stefano et al., 2011; Sun and Li, 2011,
2015, 2016; Lelièvre et al., 2012; Carter-McAuslan et al., 2015; Zhang and
Revil, 2015; Giraud et al., 2016, 2017, 2019c; Heincke et al., 2017).
Furthermore, when geological data are available, geological models can be
derived, and their statistics can be used to derive a candidate model for
forward modelling
(Guillen et al., 2008; Lindsay et al., 2013; de La Varga et al., 2019), to derive
statistical petrophysical constraints for inversion
(Giraud et al., 2017, 2019c, d), and to
restrict the range of accepted values using spatially varying disjoint bound
constraints (Ogarko et al., 2021a) or
multinary transformation (Zhdanov and Lin, 2017).
In this paper, we present a versatile inversion platform designed to
integrate geological and petrophysical constraints to the inversion of
gravity and magnetic data at different scales. We present Tomofast-x (“x”
for “extendable”) as an open-source inversion platform capable of dealing
with varying amounts and qualities of input data. Tomofast-x is designed to
conduct constrained single physics and joint physics inversions. The need for
reproducible research (Peng, 2011) is facilitated by
open-source code (Gil et al., 2016), and thus we
introduce and detail the different constraints implemented in Tomofast-x
before providing a realistic synthetic application example using selected
functionalities. We illustrate the use of Tomofast-x by performing a
realistic synthetic study investigating several modelling scenarios
typically encountered by practitioners and provide information to get free
access to the source code and to run it using the synthetic data shown in
this paper. We perform single physics inversion of gravity data and study
the influence of prior information using several amounts and types of
constraints, and run joint inversion of gravity and magnetic data. The
flexibility of Tomofast-x is exploited to test the effect of structural
constraints combined with petrophysical and geological prior information
that are yet to be demonstrated in the published literature. A challenging
geological setting is used to examine the capability of cross-gradient
constraints within the joint inversion method. The mathematical formulations
of geophysical problems and solutions are detailed throughout the paper, and
sufficient information is provided to allow the reproducibility of this work
using Tomofast-x.
The remainder of the contribution revolves around two main aspects. We
first review the theory behind the inversion algorithm and the different
techniques used, with an emphasis on the mathematical formulation of the
problem. We then present a synthetic example inspired from a geological
model in the Hamersley province (Western Australia), where we investigate
two case scenarios. In the first case, we apply structural constraints to an
area where geology contradicts the assumption of co-located and correlated
density and magnetic susceptibility variations. In the second case, we
investigate a novel combination of petrophysical and structural information
to constrain a single physics inversion. Finally, we place Tomofast-x in the
general context of research in geophysical inverse modelling and conclude
this article.
Inverse modelling platform Tomofast-xPurpose of Tomofast-x
Tomofast-x can be used in a wide range of geoscientific scenarios as it can
integrate multiple forms of prior information to constrain inversion and
follow appropriate inversion strategies. Constraints can be applied through
Tikhonov-style regularisation of the inverse problem
(Tikhonov and Arsenin, 1977, 1978). In single physics
inversion, these comprise model smallness (also called “model damping”,
minimising the norm of the model; see Hoerl and Kennard,
1970) and model smoothness (also called “gradient damping”, minimising the
norm of the spatial gradient of the model; see Li and Oldenburg,
1996). For more detailed imaging, petrophysical constraints using Gaussian
mixture models (Giraud et al., 2019c), as well as structural
constraints (Giraud et al., 2019d;
Martin et al., 2020) and multiple interval bound constraints
(Ogarko et al., 2021a), can be used
depending on the requirements of the study and the information available. In
the case of single physics inversion with structural constraints, structural
similarity between a selected reference model and the inverted models can be
maximised using structural constraints based on cross-gradients
(Gallardo and Meju, 2003) and locally weighted gradients in the
same philosophy as
Brown
et al. (2012), Wiik et al. (2015), Yan et al. (2017), and Giraud et al. (2019d).
Generally speaking, in the joint inversion case, the two models inverted for
are linked using the structural constraints just mentioned or petrophysical
clustering constraints in the same spirit as
Carter-McAuslan et al.
(2015), Kamm et al. (2015), Sun and Li (2015, 2017), Zhang and Revil (2015), and
Bijani et al. (2017). In addition to the underlying assumptions defining the
relationship between the properties that have been jointly inverted, prior information from
previous modelling or geological information can be incorporated in
inversion using model and structural covariance matrices by assigning
weights that vary spatially. In such cases, Tomofast-x allows for utilising prior
information extensively. Furthermore, Tomofast-x allows the use of an
arbitrary number of prior and starting models, enabling the investigation of
the subsurface in a detailed and stochastically oriented fashion. Tomofast-x was
initially developed for application to regional or crustal studies (areas
covering hundreds of kilometres) and retains this capability. The current
version of Tomofast-x is now more versatile as development is now directed
toward use for exploration targeting and the monitoring of natural resources
(kilometric scale).
Lastly, in addition to inversion, Tomofast-x offers the possibility to
assess uncertainty in the recovered models. The uncertainty assessments
include statistical measures gathered from the petrophysical constraints,
posterior least-squares variance matrices of the recovered model (in the least
squares with QR-factorisation algorithm – LSQR – sensu
Paige and Saunders, 1982; see
Sect. 2.5), and the degree of structural similarity
between the models (for joint inversion or structurally constrained
inversion). From a practical point of view, associated with the inversion
algorithm is a user manual covering most functionalities and a reduced 2D
Python notebook illustrating concepts (see Sect. 7
for more information) that can be used for testing or educational purposes.
A summary of the inverse modelling workflow of Tomofast-x is shown in
Fig. 1.
Modelling workflow summary (modified from Jérémie Giraud, Loop Workshop, March 2020).
General design
The implementation we present extends the original inversion platform
“Tomofast” (Martin et al., 2013, 2018).
Tomofast-x is an extended implementation proposed and modified by
Martin et al. (2018), Giraud et al. (2019c, d), Martin et al. (2020), and Ogarko et al. (2021a). Tomofast-x follows the object-oriented Fortran 2008 standard and utilises
classes designed to account for the mathematics of the problem. This
introduces enhanced modularity based on the implementation of specific
modules that can be called depending on the type of inversion required. The
utilisation of classes in Tomofast-x also eases the addition of new
functionalities and permits the reduction of software complexity while maintaining
flexibility. Our implementation uses an indexed hexahedral solid body mesh,
giving the possibility to adapt the problem geometry, allowing the regularisation of
the problem in the same fashion as Wiik et al. (2015) or to perform overburden stripping. By default, the sensitivity
matrix to geophysical measurements is stored in a sparse format (using the
compressed sparse row format) to reduce memory consumption and for fast
matrix vector multiplication.
Attention has also been given to computational aspects. The only dependency
of Tomofast-x is the message passing interface (MPI) libraries, which eases
installation and usage. This allows optimal usage of multi-CPU systems
regardless of the number of CPUs. Parallelisation is made on the model cells
using a domain decomposition approach in space; i.e. the model is
divided into nearly equal, non-overlapping contiguous parts distributed
among the CPUs, hence enforcing minimum load imbalance. Consequently, the
code is fully scalable as the maximum number of CPUs is not limited by the
number of receivers or measured data points. For large 3D models, Tomofast-x
can run on hundreds of CPUs for a typical problem with 105–106
model cells and 103–104 data points. Parallel efficiency tests
reveal excellent scalability and speed performance provided that the
portions of the model sent to the CPUs are of sufficient size. In the
current implementation, the optimum number of elements per CPU is 512.
Interested readers can refer to Appendix D for more information.
Cost functionGeneral formulation
Tomofast-x inversions rely on optimisation of a least-squares cost function
and are optimised iteratively. The choice of a least-squares framework was
motivated by flexibility in the number of constraints and forms of prior
information used in the optimisation process.
The objective function θ is derived from the log likelihood of a
probabilistic density function Θ (see Tarantola, 2005, chaps. 1 and 3, for details). In the case of geophysical inversion, it is representative of the “degree of knowledge that we have about the values of the parameters of our system” (Tarantola and Valette, 1982), as summarised
below. Let us first define Θ as follows:
Θ(d,m)=Θd(d,m)∏i∈constraintsΘi(m),
where Θd(d,m) is the density function over the geophysical data d that model m represents, and Θi(m) is the density function for the ith
type of prior information available (the “constraints” set).
On the premise that Gaussian probability densities approximate the problem
appropriately, Θd(d,m) can be
expressed as follows:
Θd(d,m)=Aexp(-‖Wd(d-g(m))‖22),A∈R+\{0},
where g(m) is the forward data set
calculated by the forward operator g and the matrix
Wd weights the data points. Similarly, we formulate
the different Θi∈constraints as follows:
Θi(m)=Ciexp(-αi2‖Wif(m)‖22),Ci∈R+\{0},
where f is a function of the model and prior information. Wi
is a covariance matrix weighting of f(m),
and αi contains positive scalars that are introduced to
adjust the relative importance of the ith constraint term.
Wi and αi are derived from prior
information or set according to study objectives.
From Eq. (3), it is clear that maximising Θ(d,m) is equivalent to minimising its negative logarithm, θ(d,m), defined as follows:
θ(d,m)=-logΘ(d,m)=αd2‖Wd(d-g(m))‖L224+∑i∈constraintsαi2‖Wif(m)‖L22,
which corresponds to the general formulation of a cost function as
formalised in the least-squares framework; αd is a
weight controlling the importance of the corresponding data term (i.e.
gravity or magnetic) in the overall cost function.
Definition of regularisation constraints
Adapting the formulation of the second term of Eq. (4) to the different
types of prior information that we can accommodate leads to the following
aggregate cost function:
θ(d,m)=‖Wd(d-g(m))‖22+αm2‖Wm(m-mpr)‖p2+αg2‖Wg∇m‖22+αx2‖Wx(∇m(1)×∇m(2))‖22+αpe2‖WpeP(m)‖22+αa2‖Wa(m-z+u)‖22,
where the different terms following the data misfit term ‖Wd(d-g(m))‖22 constitute constraints
for the inversion of geophysical data acting as regularisation in the
fashion of Tikhonov regularisation (Tikhonov and Arsenin, 1977).
In Eqs. (2)–(5), Wd represents geophysical data
weighting. Generally, Wd should be the data
covariance. It is calculated by Tomofast-x as follows:
Wd=∑i=1ndata(di)2-1Iw,
where Iw is a diagonal matrix equal to the identity matrix in
single domain inversion or giving the weight of one data misfit term (i.e.
gravity data) against the other (i.e. magnetic data) in joint inversion;
di is the ith datum. By convention, we fix Iw to the
identity matrix I for gravity inversion and use
Iw=Iαmag in joint inversion. In
such cases, αmag is a strictly positive scalar.
The main terms of the cost function are defined below. The other individual
terms are defined in the next subsection and summarised in
Appendix B:
mpr refers to the prior model;
‖Wm(m-mpr)‖p2
represents the smallness term (detailed in Sect. 2.4.1);
subscript p refers to the Lp norm (here
taken such that 1<p≤2);
∇ is the operator calculating the spatial gradient of the model;
‖Wg∇m‖22 represents the smoothness
constraint on the model (detailed in Sect. 2.4.2);
‖Wx(∇m(1)×∇m(2))‖22
represents cross-gradient constraints between the models
m(1) and m(2) (detailed in Sect. 2.4.3);
‖WpeP(m)‖22 represents petrophysics term (clustering
constraint), into which P(m) represents
petrophysical distributions used to impose petrophysical constraints
(detailed in Sect. 2.4.4);
‖Wa(m-z+u)‖22 is the formulation of the multiple bound
constraints using the alternating direction of multipliers method (ADMM,
detailed in Sect. 2.4.5).
In the case of joint inversion, the vectors defined above are concatenated
and the matrices are expanded as follows:
g(m)=gG(mG)gM(mM),m=m(1)m(2)=mGmM,d=dGdM,αi∈[d,m,g,x,pe,a]=αi∈[d,m,g,x,pe,a]G00αi∈[d,m,g,x,pe,a]M,Wi∈[d,m,g,x,pe,a]=Wi∈[d,m,g,x,pe,a]G00Wi∈[d,m,g,x,pe,a]M,
where T denotes the transpose operator. For the more illustrative purposes of
the joint inversion, here we take the gravity and magnetic joint inversion
example, where G and M refer to gravity and magnetic problems, respectively.
In the case of single domain inversion, m(1) is the model inverted for and is equal to mG or
mM depending on the type of geophysical data
inverted, and m(2) is a reference model that
can be used to constrain inversion from a structural point of view (see
Sect. 2.4.3, and
4.3 for the theory and an
example of utilisation, respectively).
In Eqs. (5)–(7), subscript m, g, x, pe, and a refer to model, gradient, cross-gradient, petrophysics, and ADMM bound constraints, respectively. The different α terms
are trade-off parameters that control the importance given to the different
terms during the inversion. These terms therefore play an important role in
inversion and need to be determined carefully (see Sect. 4.1 and 4.2 for more
details).
As mentioned above, the cross-gradient constraints can be applied either to
joint or single domain inversion. In the case of ADMM constraints, single or
multiple bounds can be applied to define bounds for inverted model values.
Such bound constraints can vary in space and be made of an arbitrary number
of intervals, regardless of whether they are disjointed or not (see Sects. 2.4.5 and 4.5).
Qualitatively, the case with multiple disjoint intervals can be interpreted
as applying a dynamic smallness constraint term.
In Tomofast-x, we introduce prior information in the diagonal variance
matrices Wi such that they are no longer homogenous and can
vary in space. Note that in the implementation of gravity and magnetic
inversion, g(m)=Sm, with
S the sensitivity matrix relating to measured geophysical data
and corresponding recovered physical property (see
Appendix C for details about their calculation).
Introducing the sensitivity matrix SG and
SM for gravity and magnetic data, respectively, we
obtain the following equation:
g(m)=Sm=SG00SMmGmM=gG(mG)gM(mM)
For reference, the terms defined or used here are summarised in
Appendix B. Tomofast-x uses the least squares with
QR-factorisation (LSQR) algorithm (Paige
and Saunders, 1982) to solve the least-squares problem. The full matrix
formulation of the problem and the related system of equations are provided
in Appendix E.
Generally, not all of the terms in Eq. (5) are used during a single
inversion. The activation of selected terms from the cost function (setting
αi>0 and non-null Wi) depends on
the information available or on the requirements of the modelling to be
performed. For example, a term not used during inversion has the
corresponding weighting simply set to 0 (the corresponding matrix
Wi is set to 0). Conversely, setting a specific
weight to a relatively large value leads to the corresponding constraint to
dominate the other terms. Such practice is typically used in sensitivity
analysis to examine the effect of incorrectly assigned extreme weighting
values on the inversion by providing an example to aid detection of this
unintended situation.
In the following subsection, we detail the implementation of the different
terms. The terms are introduced and detailed following the order they appear
in Eq. (5).
Detailed formulation of constraints for inversion
In this section, we introduce the mathematical formulation of constraints
applied during inversion. Throughout this paper, “geological information” relates to information
extracted from probabilistic geological structural modelling. Petrophysical
information relates to the statistics of the values inverted for (density
contrast and magnetic susceptibility).
Smallness term
We repeat the smallness term as follows:
‖Wm(m-mpr)‖p2.
The smallness term corresponds to the ridge regression constraint, i.e. the
smallness term of Hoerl and Kennard (1970). To simplify
the problem, the covariance matrix Wm is assumed to
be a diagonal matrix. In Tomofast-x, it is used to adjust the strength of
the constraint either globally (i.e Wm=I) or locally
(i.e. the elements of Wm may vary from one cell to
another). In the second case, Wm can be determined
using prior information such as uncertainty from geological modelling or
models and structural or statistical information derived from other
geophysical techniques (e.g. seismic attributes and probabilistic results from
magnetotellurics).
Smoothness term
The smoothness model term (Li and Oldenburg, 1996) is a total
variation (TV)-like regularisation term based on an original idea of
Rudin et al. (1992). It constrains the degree of
structural complexity allowed in the inverted model. We repeat the term as follows:
‖Wg∇m‖22.
The covariance matrix Wg modulates the importance of the term
by assigning the weights to each cell. For the sake of simplicity, the
matrix Wg is commonly assumed to be a diagonal matrix. It is
commonly set as the identity matrix (Wg=I), but several works
vary the values in space accordingly with prior information. For instance,
Brown et al.
(2012) and Yan et al. (2017) use seismic models to calculate such weights for
the inversion of electromagnetic data, and Giraud
et al. (2019a), who present an application case using Tomofast-x, invert
gravity data using geological uncertainty information to calculate
Wg. In Tomofast-x, it can be set either globally (i.e.
Wg=I) or locally (i.e. the elements of
Wg may vary from one cell to another).
Cross-gradient
The cross-gradient constraints were introduced as a means to link two models
that are inverted jointly by encouraging structural correlation between them
(Gallardo and Meju, 2004). We refer the reader to
Meju and Gallardo (2016) for a review of applications using
this technique. We repeat the term below:
‖Wx(∇m(1)×∇m(2))‖22.
The matrix Wx modulates the importance of the term by
assigning the weights to each cell. In previous works, it is always (to the
best our knowledge) set as the identity matrix (Wx=I), with the
exception of Rashidifard et al. (2020), who
define such weights using seismic reflectivity and apply this approach to
single physics inversion of gravity data constrained by fixed seismic
velocity. In Tomofast-x, three finite-difference numerical schemes can be
chosen to calculate the cross-gradient derivatives: forward, centred, and
mixed. In what follows, we use a “mixed” finite-difference scheme, where
inversion iterations with an odd number use a forward scheme and those with even numbers use a
backward scheme (e.g. iteration 3 will use a forward scheme and iteration 4
a backward scheme). This scheme was chosen as it reduces the influence of
the border effects of both the forward and backward schemes on the
inverted model.
Statistical petrophysical constraints
One strategy to enforce the petrophysical constraints using statistics from
petrophysical measurements is performed by encouraging the statistics of the
recovered model to match that of a statistical model derived either from
measurements made from the study area or literature values. In the current
implementation of Tomofast-x, a mixture model representing the expected
statistics of the modelled rock units is used. We use a Gaussian mixture
model to approximate the petrophysical properties of the lithologies in the
studied area. In the mixture model, the weight of each Gaussian can be set
in the input. We suggest using the probability of the corresponding rock
unit when this information is available. The mismatch between the statistics
of the recovered model and the mixture model is minimised in the
optimisation framework following the same procedure described in
Giraud et al. (2018, 2019b).
In the ith model cell, the likelihood term P(mi) of model cell mi is calculated as follows for the kth
lithology:
12Nk=ωkN(mi|μk,σk)13P(mi)=-log∑k=1nfNk+log(maxNk=1…nf),
where
ωk=1nfeverywhere in the absence of spatially-varying prior information (a)ωk=ψk,iin the ith cell using prior
information (b),N symbolises the normal distribution, and nf is the total number of rock formations observed in the modelled area.
In practice, an expectation maximisation algorithm (McLachlan and Peel, 2000) can be used to estimate the mean μk and standard deviation σk of the petrophysical measurements.
In Eqs. (12)–(14), ωk is the weight assigned to the Gaussian
distribution representative of the petrophysics of the
kth lithology in the mixture. In Eq. (14a), the weight ωk assigned to each lithology is constant across
the model, while in Eq. (14b) the weight ψk,i is derived from
information derived from another modelling technique (geology, seismic,
electromagnetic methods, etc.) and varies spatially.
We note that a small number of Gaussian distributions might not be suitable
to approximate certain types of distributions like bimodal (magnetic
susceptibility) or lognormal (electrical resistivity) distributions.
However, we point out that increasing the quality of such an approximation
depends on the number of Gaussian distributions used for approximation
(McLachlan and Peel, 2000).
Disjoint bound constraints using the ADMM algorithm
The objective of the disjoint bound constraints is to optimise Eq. (5) while
ensuring that in every model cell m1≤i≤nm the inverted value
lies within the prescribed bounds such that mi∈Bi,
defined as follows (Ogarko et al., 2021a):
Bi=⋃l=1Li[ai,l,bi,l],withbi,l>ai,l,∀l∈[1,Li],
where ai,l and bi,l are the lower and upper
bounds for the ith model cell and l is the lithology index;
Li≤nf is the total number of bounds allowed for the
considered cell, corresponding to the number of distinct rock units allowed
by such constraints. During inversion, such multiple bound constraints on the
physical property values inverted for are gradually enforced using the ADMM
algorithm. Implementation details are beyond the scope of this paper, but
more information is provided in Appendix E, and we
refer the reader to Ogarko et al.
(2021a) for details. Details about the general mathematical formulation of
the ADMM algorithm can be found in Dykstra (1983), chap. 7 in Bertsekas (2016), and in Boyd et al. (2011).
Note that the application of the ADMM bound constraints can be interpreted
as being analogous to clustering constraints where (taking the example of the
kth model-cell) the following conditions are met:
the centre values depend on both the current model m at any given
integration and petrophysical information defining a and b;
the weight assigned to each centre value changes from one iteration to the
next as a function of the distance between mk and the closest bound, and
the number of iterations mk has remained outside Bk.
Depth weighting and data weighting
To balance the decreasing sensitivity of potential field data with the depth
of the considered model cell, Tomofast-x offers the possibility for the
calculation of the depth-weighting operator. The first one, which we use in
this paper, utilises the integrated sensitivities technique following
Portniaguine and Zhdanov (2002). For each model cell i, a
weight Dii is introduced:
Dii=∑k=1ndata(Ski)214.
The second option relies on the application of an inverse depth-weighting
power law function following Li and Oldenburg
(1998) and Li and Chouteau (1998) for gravity and
Li and Oldenburg (1996) for magnetic data:
Dii=1zi+εβ,
where zi is the depth of the ith model cell and
ε is introduced to ensure numerical stability, such that
z≫ε; the value of β
depends on the type of data considered (gravity or magnetic). For more
details about the use of depth weighting and selection of values of β, the reader is referred to the references provided in this subsection. The
application of depth weighting as a preconditioner to the matrix system of
equation solved for during inversion is shown in
Appendix E.
Posterior uncertainty metrics
Uncertainty information is an important building block of modelling and a
critical aspect of decision making (Scheidt et al., 2018).
When available, uncertainty information can be communicated and used in
subsequent modelling or for decision making (see examples of
Ogarko et al., 2021a, who use uncertainty
information in the model recovered by another method as input to their
modelling using Tomofast-x). Tomofast-x allows the calculation of metrics
reflecting the degree of uncertainty in the models before and after
inversion. It allows monitoring the evolution of the different terms of the
cost function during inversion. Tomofast-x also calculates uncertainty
metrics that are specific to the kind of constraints used in inversion:
the posterior covariance matrix of model m as estimated by the LSQR algorithm (Paige and Saunders, 1982, p. 53, for details and Sect. A1.1 for a brief introduction),
the value of the cross-gradient in each cell,
the individual Nk values (Eq. 12) of the different Gaussians making up the Gaussian mixture used to define the petrophysical constraints.
The implementation of this series of indicators was performed with the
intent to provide metrics for posterior analysis in detailed case studies.
More information about these posterior uncertainty indicators is provided in
Appendix A, which details functionalities of
Tomofast-x not explored here.
Synthetic model and data
In this section, we introduce how the data used for synthetic modelling were
derived, and we present examples of using prior information derived from
geological modelling. The process of simulating a realistic field case study
is described with the design of the numerical experiment.
Geological framework
The original geological model is based on a region in the Hamersley province
(Western Australia). It was built using the map2loop algorithm
(Jessell et al., 2021b) to parse the raw data and the
Geomodeller® implicit modelling engine for geological
interpolation (Calcagno
et al., 2008; Guillen et al., 2008) to model the contacts, stratigraphy, and
orientation data measurement in the area (see the geographical location in
Fig. 2). Data used to generate the model comprise
the 2016 1:500000 Interpreted Bedrock Geology map of Western Australia
(https://catalogue.data.wa.gov.au/dataset/1-500-000-state-interpreted-bedrock-geology-dmirs-016,
last access: 2 December 2020) and the WAROX outcrop database
(https://catalogue.nla.gov.au/Record/7429427, last access: 2
December 2020).
Geological modelling was assisted by interpretation of the magnetic anomaly
grid compilations at 80 m and the 400 m gravity anomaly grid from
the Geological Survey of Western Australia (https://www.dmp.wa.gov.au/Geological-Survey/Regional-geophysical-suvey-data-1392.aspx,
last access: 2 December 2020). More information about data availability is
provided in Sect. 7.
True model used for geologic modelling and geophysical inversion.
The top panel shows the map view. The black line represents the surface
coordinates of the 2D profile considered here for illustration of
Tomofast-x's utilisation. The values given on either side of the colour bar
indicate the density contrast and magnetic susceptibility attached to each
rock unit. Note that several rock units present similar density contrast or
magnetic susceptibilities, making them undistinguishable using either
gravity or magnetic inversion.
In what follows, we use an adapted version of the original structural
geological framework of the selected region by increasing the vertical
dimensions of the model cells and assuming a flat topography. The resulting
reference geological model used to generate the physical properties for
geophysical modelling measurements is shown in Fig. 2 in terms of its geological units.
In addition to the modification of the structural model, we make adjustments
to the original density values derived from field petrophysical measurements
by reducing the differences between the density contrasts of different rock
units. By doing so, we increase the interpretation ambiguity of inversion
results and decrease the differentiability of the different rock units. The
same procedure is applied to magnetic susceptibility to make accurate
imaging using inversion more challenging. The three-dimensional (3D) density
contrast and magnetic susceptibility models used to generate the gravity and
magnetic data are shown in Fig. 3.
True synthetic density contrast (a) and magnetic susceptibility
(b) model used for the simulation of geophysical data. The black line
represents the surface coordinates of the 2D profile considered here. The
voxels represent lithologies 10 through 15, colour-coded with their
respective density contrast and magnetic susceptibility values.
Geophysical simulations and model discretisation
The core 3D model is discretised into Nx×Ny×Nz=103×113×33 cells of dimensions equal to 999×996×745m3. For both gravity and magnetics, we generate one
geophysical measurement per cell along the horizontal axis, leading to
Nx×Ny=11639 data points for each method, and add 10 padding
cells in each horizontal direction to limit numerical border effects in the
forward calculation, leading to a total of 123×133×33=539847
model cells. We simulate a ground gravity survey by locating the
measurements 1 m above ground level, and aeromagnetic data acquired by a
fixed-wing aircraft flying 100 m above surface. To test the robustness
of our inversion code to noise content in the data, the geophysical data
inverted here are contaminated by noise.
The noise component was generated as follows. For each gravity measurement,
we first add a perturbation value randomly sampled from the standard normal
distribution of the whole data set multiplied by 9 % of the measurement's
amplitude. We then add a second perturbation value randomly sampled from the
standard normal distribution with an amplitude of 3 mGal (2 % of the
dynamical range). These values were derived manually to obtain a realistic
noise contamination. To simulate small-scale spatial coherence in the noise
generated in this fashion, we then apply a two-dimensional Gaussian filter
to the 2D noise map obtained from the previous step. We then apply a
two-dimensional median filter to the resulting noise-contaminated gravity
data to simulate de-noising. For magnetic data, we apply the same procedure,
using 12.5 % of the measurement's amplitude for the first step and 15 nT
(1 % of the dynamical range) for the second step. Similarly to gravity
data, these values were derived manually; no levelling noise was simulated.
For comparison, the noise-free and contaminated synthetic measurements are
shown in Fig. 4. The resulting noise standard
deviation σnoise for gravity and magnetic data are equal to 1.2 mGal and 8.5 nT, respectively.
Noise (a, d) added to the data calculated from
the true model (b, e) and resulting noisy data (c, f)
for the gravity (a–c) and magnetic (d–f) data sets. The contour lines
shown correspond to the ticks shown in the palette's colour bar. The
black line represents the location profile we use for the inversions
performed here.
The gravity data modelled here correspond to the complete Bouguer anomaly.
Magnetic data are simulated using the magnetic strength of the Hamersley
province (53 011 nT, which approximates the International Geomagnetic
Reference Field in the area) reduced to the pole.
To complete the 3D modelling procedure, a series of 100 structural
geological models are generated using Monte Carlo perturbations of the
geological measurements (foliation and contact points between geological
units) constraining the geological structures. This was performed using the
Monte Carlo Uncertainty Estimator (MCUE) technique of
Pakyuz-Charrier et al.
(2018, 2019). The result is an ensemble of models that all fit the
geological measurements within a given set of prior uncertainty parameters.
The ensemble is thus assumed to represent the geological model space, rather
than just a single “best-guess” model. Probabilities for the occurrence of
different rock units can be calculated from the ensemble and used to
constrain geophysical inversion (see examples of
Giraud et al., 2017,
2019a; Ogarko et al., 2021a). More specifically, MCUE is useful to obtain
the probability ψi,l of occurrence of the different lithologies l
for every ith model cell and to calculate the related uncertainty
indicators (Sect. 3.3). Detailing the probabilistic
geological modelling procedure and analysing the results in 3D is beyond the
scope of this paper and interested readers are referred to
Lindsay
et al. (2012), Pakyuz-Charrier et al. (2018), Wellmann and Caumon (2018), and
references therein.
In this contribution, we simulate a case study where modelling is carried
out along the 2D profile materialised by the black line in
Figs. 3 and 4,
extracted from the 3D modelling framework as detailed in the next
subsection.
2D model simulation in a 3D world
As mentioned above, we perform the inversion of geophysical data along a 2D
profile for simplicity and to simulate the challenging case of 2D data
acquired in a 3D geological setting, in a part of the model where
subhorizontal or gently dipping features can be observed. The philosophy of
the numerical study presented here is summarised in
Fig. 5.
Summary of experimental protocol for synthetic study and testing
of different functionalities of Tomofast-x.
The 2D geological Sect. considered is shown in Fig. 6 (the black line marked in Figs. 3 and 4). Geological certainty is estimated using a
measure of the dispersion away from the perfectly uninformed case where the
all rock units are equiprobable. In each model cell, this measure, which we
write as σψ′, is calculated as a function
of the standard deviation σψ of the
probability ψ of observing the different rock units as
follows:
σψ′=1/card-σψ,
where card is the geological cardinality of the model. It is equal to the number of possible rock units observed in one location
across the entire ensemble. From Eq. (18), σψ′ is maximum where rock units are well constrained
and minimum where the model is the most uncertain. This geological certainty
metric is shown in Fig. 6 for the 2D section
considered in this example.
Two-dimensional slice extracted from the 3D model along the
profile: geological reference model (a) and the σψ′ metric (b). Here, the geological
uncertainty metric considered is the standard deviation of the probability
of the different lithologies as per Eq. (18).
The probabilities of observation of the different lithologies are shown in
Fig. 7. Note that for the purpose of the tests we
perform on gravity data inversion, we reduce the set of probabilities by
grouping rock units into fictitious units with the same density contrasts as
single rock units. This reduces the number of rock units to six units that
can be distinguished by gravity inversion, as several units may be assigned
the same density contrast.
Observation probabilities for the rock units that present
differencing density contrasts. Units 3 and 4 are nearly absent from the
section (see maximum probability area marked by the arrow).
The geophysical data we use for inversion are extracted along the line
marked in Figs. 3 and 4. The geophysical
data and reference (true) petrophysical model extracted in this fashion are
shown in Fig. 8. Care was taken not to use the
same mesh for both generating the synthetic data set and its inverse
modelling.
Two-dimensional slice extracted from the 3D model: gravity data
and density contrast (a) and magnetic data and magnetic susceptibility (b).
To inverse model the data shown in Fig. 8, we
generate a mesh centred on the profile (oriented along the y direction) and
add padding to either side and the northern and southern extremities. The
resulting model comprises nx×ny×nz=13×133×33 cells of dimensions equal to 2998×996×745 m3. Note that we
increased the model cell size in the direction perpendicular to the profile.
The gravity and magnetic data sets along the profile each comprise 113 data
points evenly distributed along the line.
While we focus on a 2D section extracted from a 3D model presented here (see
location in Figs. 3 and 4), the 3D model and the associated gravity
and magnetic data sets shown here are made publicly available (see Sect. 7).
Application example: sensitivity analysis to constraints and prior
information
In the examples shown below, we first perform single domain and joint
(multiple domain) inversion (using the cross-gradient constraint) assuming
identity matrices for Wm, Wg,
and Wx. We then investigate the influence of prior
information on single domain inversion by combining structural and
petrophysical information in the case of gravity inversion. The combination
of petrophysical and structural constraints derived from geology is tested.
The intention is to address knowledge gaps in the literature that describe
the effects of parameterisation of such constraints.
Experimental protocol
It is necessary to determine the appropriate weights α assigned to
the terms defining the constraints applied during inversion to optimise the
cost function in Eq. (5). The α values that define the weights of
the different terms in the cost function constitute hyperparameters of the
inverse problem. Appropriate estimation of these hyperparameters is
necessary to approximate the optimum value of the global misfit function. To
this end, we use the L-curve principle
(Hansen and O'Leary, 1993;
Hansen and Johnston, 2001; Santos and Bassrei, 2007) for each of the cases
presented below. We perform series of inversions, sampling α values
spanning the plausible range of potential choices using a heuristic
approach.
When two constraint terms are used in inversion (i.e. with α>0), we extend the L-curve approach to the two-parameter cases. In such
cases, the optimum values for the α weights are determined by
applying the L-curve criterion using L-surfaces (or elbow surface) instead
of L-curves (Belge et al., 2002) (we note that
the L-curves as plotted here can also be referred to as “Tikhonov curves” in
the case where data misfit is plotted as a function of regularisation
value). The optimum value for the α weight of the two constraint
terms is therefore obtained by identification of the inflection point of the
surface made up of the variations of the data misfit as a function of the
weights under consideration. We chose this approach for its simplicity and
note that there exist other techniques that use an automated process, such
as the generalised cross-validation technique (Craven and Wahba,
1978). We refer the reader to Farquharson and
Oldenburg (2004) for a general introduction and
Giraud et al. (2019b) and Martin et al.
(2020) for an application of this principle to inversions using Tomofast-x.
The role the L-surface analysis plays in the synthetic case presented here
is reminded in the workflow shown in Fig. 5. In
our analysis, we set the objective value for the data misfit
‖Wd(d-g(m))‖22 to be equal to the
objective data misfit Θdobj(d,m), defined as follows:
Θdobj(d,m)≥ndataσnoise∑i=1ndata(di)2,
so that the data is reproduced with a level of error superior or equal to
the estimated noise level of the data. Here, this leads to Θdobj=5.01×10-4 for gravity
inversion and to Θdobj=1.55×10-4 for magnetic data inversion.
For the sake of consistency in our study of the influence of constraints
on inversion, we set mpr=0kgm-3 for gravity
data inversion and mpr=0 SI for magnetic data
inversion.
Homogenously constrained potential field inversions
We first perform single physics inversion following the common strategy of
constraining the model using smallness and smoothness constraints. Obtaining
a good approximation of the optimum values of these parameters gives
insights into the numerical structure of the problem. It constitutes
valuable knowledge when using other kinds of constraints, and we consider it
good practice to run such inversion prior to using more advanced
constraints. Here, the first α parameters to determine are
αm and αg, for both gravity and
magnetic data inversion, assuming identity Wm and
Wg matrices so that the constraints are applied homogenously
over the entire model.
We generate grids in the (αm,αg) plane using
αm∈[10-8,10-6] and
αg∈[10-6,10-3] for gravity inversion, and
αm∈[103,105] and αg∈[103,108] for magnetic data inversion. These ranges were
determined empirically and assumed to comprise the optimums. In this
subsection, all matrices W in Eq. (5) are set as the
identity matrix.
For accurate estimation, the (αm, αg) values are sampled more finely closer to the estimated optimum
values. The resulting L-surfaces are shown in Fig. 9, where the vicinity of the optimum value of (αm,
αg) is shown with a green dot. From these values, we
estimate the optimum values of (αm, αg) reported in Table 1.
Elbow surfaces for gravity and magnetic inversions (top row)
and a plot of the data misfit term as a function of the α
weights (bottom row). Each plot uses a total of 1260 points sampling the (αm,αg) plane. The black lines show the contour values corresponding to the ticks shown in the palette's colour bar, which shows the value of the data misfit term. The red line materialises the contour value of Θdobj, guiding the selection of the optimum (αm, αg) values, and the green dot materialises the vicinity of the curve's inflection point.
Optimum values of (αm, αg) estimated from L-surface analysis.
The models corresponding to values in Table 1 are
shown in Fig. 10.
Results from separate inversions using smallness and smoothness
constraints. The starting and prior models are equal to zero
everywhere, and the smoothness constraint is applied homogeneously.
The values of αm and αg obtained for such constraints
can be used as a starting point in subsequent inversions to understand the
influence of prior information when varying amounts and types of prior
information are available about the structure of the subsurface or its
petrophysics. For instance, in what follows we will investigate the
utilisation of geological information to define Wm and
Wg (Eqs. 9 and 10, respectively) and see how it can combined
with petrophysical data to define B (Eq. 15) (see the following subsection
where we use global and structural and/or petrophysical information). In the
case of structural constraints relying on the spatial derivatives of model
values (cross-gradient values or local smoothness), the value of αm may be kept constant and while the other α
parameters (αg or αx) are adjusted.
Conversely, αg may be kept and αm set to 0 for the utilisation of petrophysical constraints
acting on the model values themselves instead of the spatial derivatives
(ADMM or statistical petrophysical constraints). Here, we restrict our
analysis to two α values being strictly superior to zero,
thereby accounting for prior information in up to two constraints terms in
the definition of the regularisation term in Eq. (5).
Joint inversion using the cross-gradient constraint
We start from the previous step to perform joint inversion using the
cross-gradient constraint. Keeping the αm
weight constant and equal to the values determined from single domain
inversion, it remains necessary to estimate the optimum values of the
cross-gradient constraint weight, αx, and the
relative importance given to the gravity and magnetic data misfit terms
(setting αG=1, it remains to determine αM). We therefore investigate values in the (αx,αM) plane, which we sample in the same
fashion as in the previous subsection. The resulting surfaces are shown in
Fig. 11.
Determination of the optimum (αx,αM) parameters in the case of the joint inversion using the cross-gradient
constraint. Top view of the elbow surfaces for gravity (a) and magnetic (b)
inversions (top row) and a plot of the data misfit term as a function the
α weights (bottom row). The solid lines show the contour values
of the data misfit; values are given by the colour bar on the side. The solid red line shows the contour level of Θdobj for the corresponding data set (gravity or
magnetic), while the dashed line shows the same quantity for the other
data set. The green dot marks their intersection, indicating the optimum
(αx,αM) values.
In contrast to the single physics inversion shown in
Fig. 9, it appears from
Fig. 11 that the two hyperparameters to be
determined here, αx and αM, influence the
inversion differently. While the contour levels of the magnetic data misfit
show a linear trend in the (αx,αM) plane, it is clearly non-linear in the case of gravity data misfit.
This difference might be explained by the fact that the cross-gradient is a
second-order regularisation (product of two spatial derivatives of model
values) linking two models that are otherwise decoupled. In addition, this
suggests that in cases differing from this one, the hyperparameter selection
may be non-unique. Nevertheless, the value of the optimum value is
unambiguous in our case and can be determined easily. From
Fig. 11 we obtain (αx,αmag)=(1.995×104, 2.57×10-5). The corresponding inversion results are shown in
Fig. 12.
Joint inversion results obtained from utilisation of the
cross-gradient constraint.
Compared to Fig. 12, we observe that the
application of the cross-gradient constraint leads to adjustments of the
model ensuring more structural consistency between density contrast and
magnetic susceptibility, illustrating the applicability of the approach
presented here. Also note that the model is also visually closer to the true
model from approximately 7520 km northing and above. However, despite the
increased structural consistency between the density contrast and magnetic
susceptibility models, some of the structures of the model are not recovered
accurately. For instance, the basin-shape structure around 7500 km northing
mirrors the actual geological structure (see Fig. 8) and is an effect of non-uniqueness on inversion. In this case, this
illustrates the need for prior information in our inversion. While joint
inversion of gravity and magnetic data using the cross-gradient constraint
improves imaging comparatively with an inversion constrained only using
smallness and smoothness constraints, prior geological information or
petrophysical information may be necessary to alleviate the remaining
uncertainty.
Smallness and smoothness constraints using geological information
In this subsection, a sensitivity analysis to prior information in inversion
is performed through a series of scenarios where geological structural
information is introduced to adjust the smallness and smoothness constraints
through Wm and Wg, respectively. In what
follows, we apply this approach to gravity inversion.
The influence of geological information in defining the smallness and
smoothness terms (detailed in Sects. 2.4.1 and
2.4.2) is analysed by investigating three additional
scenarios allowed by the utilisation of either homogenous or
geologically derived Wm and Wg matrices. In each
case, we start from the (αm, αg)
weights estimated in Sect. 4.2 from the analysis of
the L surface, which we adjust to obtain the geophysical misfit sufficiently
close to objective values. We restate that αm and
αg weight the overall contribution of the model
smallness and smoothness, respectively, in the cost function (Eq. 5).
In the first scenario we investigate (scenario b in
Table 2), geological uncertainty information is used
to define Wm while keeping Wg homogenous. This
allows us to test the influence of geological prior information on the
smallness term. The values of the diagonal variance matrix Wm
are calculated using the geological certainty metric σψ′ (Eq. 18, shown in Fig. 6b for the
2D section modelled here) and keep Wg homogenous.
Contrary to the previous tests (see Sect. 4.2) where
Wm=Inm, we find the following result for the kth model cell:
(Wm)kk=(σψ′)k.
Because 0≤(σψ′)k≤1∀k, we have the following result:
tr(Wm)=∑i=1nm(σψ′)i≤tr(Inm)=nm.
Consequently, setting Wm in this fashion and keeping
αm constant translates to a lower overall relative
importance of the smoothness term in the least-squares cost function (Eq. 5), thereby moving away from the trade-off inferred from the L-curve
principle (Sect. 4.2). To mitigate this, we adjust
αm to a value αm′ such that
αm′=αmnm∑i=1nm(σψ′)i,
which equates (αm′)2tr(Wm) with (αm)2nm so that the overall weight assigned to the smallness term
remains the same with and without geological structural information
(left-hand side and right-hand side of inequality in Eq. 20,
respectively). Because the values of ‖Wmm‖ depend on both Wm and m, which
vary in space and also depend on the other terms of the cost function,
minor adjustments of the value of αm′ are
necessary to reach the objective value of data misfit. In this example, this
leads to tune the suggested αm′=8.4×10-7 to
αm′=8.85×10-7 (keeping αg constant). The
corresponding inverted model is shown in Fig. 13b.
The corresponding α weights are repeated in
Table 2.
Results from gravity inversion constrained by: (a) homogenous smallness and homogenous smoothness constraints, (b) geologically-derived smallness and homogenous smoothness constraints (c) homogenous smallness and
geologically-derived smoothness constraints, (d) geologically-derived smallness and geologically-derived smoothness constraints.
In the second scenario we test (scenario c in Table 2), geological uncertainty information is then used to define
Wg while keeping Wm homogenous. This allows us
to test the influence of geological prior information on the smoothness
term. Following the same procedure as for the smallness term, we adjust the
suggested αg′=3.6×10-4 to αg′=4.1×10-4. The
corresponding inverted model is shown in Fig. 13c.
Finally, we test the case where both Wm and Wg
are defined using geological information in the form of σψ′. Starting from values of αm′ and αg′ used in the
previous tests, minor tuning is performed, leading to αm′=6.1×10-7 and
αg′=6.0×10-4 inversion results in the model shown in
Fig. 13d.
The α values derived for simultaneous usage of local
and global smallness and smoothness constraints. Scenario (a) is a reminder
of the values obtained in Sect. 4.2 when only global
constraints are used.
αmαgGlobal smoothness, global smallness constraints (a)2.1×10-71.8×10-4Global smoothness, local smallness constraints (b)8.85×10-71.8×10-4Local smoothness, global smallness constraints (c)2.1×10-74.1×10-4Local smoothness, local smallness constraints (d)6.1×10-76.0×10-4
As can be seen in Fig. 13 by comparing
Fig. 13a–b with Fig. 13c–d, the utilisation of geological structural information to adjust the
smoothness regularisation strength spatially has more impact on inversion
than adjusting the smallness term. While incorporating prior geological
information in Wm constrains the model to a certain extent,
using Cv to derive Wg has more
influence on the inverted model than for Wm, with resulting
models that are closer to the reference model.
Comparing Fig. 13c and d indicates that the use of geological
uncertainty information to adjust the smallness regularisation strength
spatially (through Wm) in addition to the model smoothness
term (through Wg) modifies inversion results further towards
the reference model. Figure 13d, which results from
inversion using prior information in both constraint terms, provides the
model closest to the reference. While most interfaces are well-recovered
when using geological information to define both Wg and
Wm, the recovered density contrasts remain affected by the
ambiguity inherent to gravity data in the presence of subhorizontal
geological units (around 7460 km northing). This suggests that in this
example, more prior information might be useful in recovering the causative
model more truthfully, especially in cases where potential field inversion
is ambiguous (e.g. subhorizontal interfaces for gravity inversion). This is
the object of the next subsection, which describes a new single physics
inversion scenario where petrophysical constraints are combined with
structural constraints and geological information.
Structural and petrophysical constraints
In addition to the definition of matrices Wm and
Wg, geological information can be combined with
petrophysical knowledge to define the range of density values allowed in
inversion. This is achieved with spatially varying bound constraints on the
property inverted for density contrast in this case (see Sect. 2.4.5). Here, such bounds are defined using multiple
intervals, each one corresponding to the range of density contrast values
expected for a geological unit. Such bounds can be defined globally
(homogenously) where all intervals are allowed everywhere in the model or
locally when prior information about the presence of the rock units is
available. In this work, we use the probability of occurrence of the
different rock units to derive bounds that vary in space accordingly with
the probability of observation of each of the rock units. In a given cell,
only the bound values corresponding to rock units with a probability Ψ>0 are considered. Starting from Eq. (15), such spatially varying bounds
Bk of the kth model cell are obtained as follows:
Bk=⋃l=1nfΨk,l>0[ak,l,bk,l],
where a and b correspond to lower and upper bounds. We consider narrow
bounds such that bk,l=ak,l+ε, with
ε≪ak,l, to encourage inversion to use density
contrasts that closely resemble values defined a priori. Equation (23)
corresponds to the application of a Boolean operator to the probabilities
Ψk=1…nf in every cell to divide the studied area into
domains defined by rock units with a probability Ψ>Ψth. In such
cases, the ADMM bound constraints act as a proxy for a prior models that have been
dynamically constrained by petrophysical information.
Four additional scenarios are tested to determine the influence of prior
information on inversion to accommodate the addition of both the damping
gradient and ADMM bound constraint term. The use of prior information is
illustrated in Fig. 14.
Tested combinations for the utilisation of prior information into
inversion. Bold frames indicate the utilisation of geological information to
define the constraints.
At a given iteration, the ADMM bound constraint encourages inverted values
to evolve inside one of the prescribed intervals depending on the current
model m. As mentioned above, we can then make the analogy with a smallness
term that is dynamically updated. For this reason, we treat the ADMM bound
constraints in the same fashion as the smallness term, which we apply
simultaneously to the model smoothness term.
Following the same protocol as Sect. 4.1 to
determine αg and αa, we first
perform inversion without the use of geological information in the form of
the probabilities for the occurrence of different rock units or metrics that
can be derived from them (Fig. 14a, i.e. with
Wg and Wa equal to the identity matrix and the
corresponding regularisation term weighted by αg and
αa, respectively). It is therefore necessary to
determine the value of αg and αa
(Eq. 5). We perform an L-surface analysis and sample values in the (αg,αa) plane to
estimate the optimum values for these hyperparameters (see
Fig. 15). Values of αg vary
from 1.585×10-7 and
1.585×10-5, and values of
αa vary from 2.484×10-5 and 2.484×10-7. The resulting L-surface is shown in
Fig. 15.
Elbow surfaces for gravity inversions. A total of 840 points
sampling the (αg,αa) plane were used. The black lines show the contour
values corresponding to the ticks shown in the palette's colour bar, which
shows the value of the data misfit term. The red line indicates the contour
value of Θdobj=5.008×10-4, guiding the selection of the optimum
(αg,αg)
values, and the green point indicates the curve inflection point.
From Fig. 15, we estimate the hyper-parameters
(αg,αa) to be (αg,αa)=(2.2×10-5,1.3×10-7) in the case no geological information is used, meaning that both
constraints are applied homogenously across the model.
From there, we follow the same procedure as described above (Sect. 4.1 and 4.2) to obtain an
estimate for the values of αa and αg
in the different configurations shown in Fig. 14b–d. The resulting inverted models are shown in
Fig. 16, and the estimates of (αg,αa) are provided in Table 3.
The α values derived for simultaneous usage of global
and local smoothness and ADMM bound constraints. Cases (a) through (d)
correspond to cases (a) through (d) in Fig. 14.
αgαaGlobal constraints (a)2.2×10-51.3×10-7Global ADMM, local gradients (b)3.3×10-42.6×10-7Local ADMM, global gradients (c)1.1×10-43.6×10-7Local ADMM, local gradients (d)3.1×10-43.25×10-7
Results from gravity inversion using (a) global ADMM clustering
and homogenous smoothness constraints, (b) global ADMM clustering and
geologically derived smoothness constraints, (c) ADMM clustering and
geologically derived smoothness constraints, and (d) geologically derived
ADMM clustering and geologically derived smoothness constraints. For visual
comparison, the true model is repeated at the bottom.
Figure 16 shows that the use of ADMM facilitates
recovery of better-defined interfaces between rock units than in previous
inversions (Figs. 10, 12, and 13), and decreases the misfit with
the causative model (shown in Fig. 6).
Unsurprisingly, without the use of geological information
(Fig. 16a) inversion results remain inconsistent
with geology in several parts of the model, especially around the position
7500 km northing. The inconsistent results can be partly mitigated by using
geologically derived smoothness constraints (Fig. 16b). In comparison, however, Fig. 16c shows that
use of geological information to determine the bounds recovers features much
closer to the causative model.
While Fig. 16d shows the more robust results
overall, Fig. 16c and d present generally similar features. This
indicates that in this case geological uncertainty information in
structural constraints only allows the refining of features largely controlled by
the utilisation of the ADMM constraints. This statement is supported by
Fig. 16, where the comparison cases (a and b) and (c and d)
reveal that the effect of using geological information to define bounds
dominates over the effect of using uncertainty to define structural
constraints.
The comparison of cases (a and b) and (c and d) in Fig. 16
can be extrapolated to Figs. 13 and 16 to compare constraints more broadly. This
is discussed in Sect. 5.1, which presents a short
comparative analysis of all gravity inversion results.
DiscussionSensitivity analysis summary: comparison of constrained inversions
Tomofast-x was developed with the intent of providing practitioners with an
inversion platform accounting for various forms of prior information and
geophysical data sets. We have tested a series of constraints involving joint
inversion and geological and petrophysical information. The inverted density
contrast models for inversion using global smallness and smoothness
constraints, joint inversion using the cross-gradient technique,
geologically derived smallness and smoothness constraints, and ADMM bound
constraints (both global and using geological information) are shown in
Fig. 17. We remind that all models shown here
produce a similar data misfit Θdobj accordingly with
Eq. (19).
General comparison of all inversion results obtained from gravity
inversion. The legend identifies the different types of
inversion shown. We repeat the true model at the bottom. The
model misfit indicated on each panel is calculated as the root mean square
of the difference between the inverted and true models.
Firstly, it appears from Fig. 17 that regardless
of the type of constraints considered, the utilisation of geological
information (cases b, d, e, g–i) to derive spatially varying constraints
for the W matrix of both terms used provides the models that are
visually closest to the true model. In this category, the utilisation of
petrophysical information in the ADMM bound constraints provides (cases h–i)
models that are closest to the true model (the lowest model misfit values are
indicated in the titles of the panels in Fig. 17). Secondly, the
comparison of cases (a) and (d), (b) and (e), (f) and (g), and (h) and (i)
indicates that while it has a less significant influence on the results,
incorporating geological information in the definition of the smoothness
term also influences inversion results significantly. Lastly, comparison of
cases (a) and (b) and (d) and (e) suggests that the utilisation of
geological information to adjust the smallness strength spatially has an
effect on inversion that is the
lowest with the cross-gradient constraints (where
structural information is passed on from another geophysical data set).
From the results shown in Sect. 4.2 through
4.5 and compared in Fig. 17, it is possible to make a qualitative ranking of the constraints
according to their influence on the resulting model (from the most
important influence to the least important influence): ADMM bound constraints > smoothness constraints > smallness
constraints > cross-gradient constraints.
This observation is also corroborated by the values of the root-mean-square
misfit between the true and inverted model. We note that this ranking
remains speculative as it might apply only to models sharing similarities
with the case we investigated.
From these observations we also deduce that when geological uncertainty
information is added to the definition of constraints (i.e. σψ′ for defining Wm and
Wg and probabilities for defining B), the term of the cost
function with the highest influence on the process will determine the main
features of the model, which will be adjusted by the other term.
Tomofast-x was developed with the intent of providing practitioners with an
inversion platform allowing various forms of prior information and
geophysical data. Constraints that represent uncertainty and our level of
epistemic knowledge provide useful constraint to inversion. This is
encouraging as the Tomofast-x platform addresses a gap in inversion schemes
that rely on a single model, with the model being as similar as possible to
the target region, an often impossible requirement to meet. Thus, Tomofast-x
opens additional research avenues to the community that are widely
acknowledged but remain largely unaddressed. Conceptual uncertainty
relating the prior assumptions made about tectonic event history of the
region, and thus the structure under study, can be analysed. Different event
histories and topologies can be considered, giving a wider scope to the
model space, and allowing the geophysics to invalidate implausible
histories, while giving us pause to consider other histories that may be less likely
but that are nonetheless possible.
Outlook for future developments
Another research avenue under consideration is the integration of results
from probabilistic modelling of seismic and electrical data into Tomofast-x.
As stated in the introduction, one of the goals born in mind during the design of Tomofast-x is interoperability. Current work involves the integration of Tomofast-x into the Loop
https://www.loop3d.org/ (last access: 7 September 2021)
open-source 3D probabilistic geological and geophysical modelling platform
(Ailleres et al., 2019), in an effort to unify geological
and geophysical modelling at a more fundamental level than the more common
cooperative approaches. Ongoing developments include the possibility to
adjust weight assigned to the ADMM bound constraints accordingly with uncertainty
levels in prior information used to define spatially varying intervals.
Future research includes the utilisation of implicit geological modelling
(in the sense of Calcagno et al., 2008)
with Tomofast-x to define geological structures and rules that inversion
will be encouraged to follow. It also comprises the incorporation of
topological laws previously used a posteriori (Giraud et al.,
2020) directly into inversion. The electrical capacitance tomography
component of Tomofast-x (Martin et al.,
2018), which we have not detailed here, can be extended to acoustic (seismic)
or electromagnetic data inversions that rely on the resolution of similar
non-linear inversion problems. It opens the door to more versatility in the
code and can be applied to joint inversion in similar ways but on more than
two physical domains.
In addition, future developments comprise the collaborative and joint
inversion of seismic and potential field data. It is planned to develop an
interface between Tomofast-x and Unisolver (not yet released as open source by
its authors), which is an extension of Seimic_Cpml code
(Komatitsch
and Martin, 2007; Martin et al., 2010, 2019), where integrated seismic imaging
solvers are implemented. Unisolver is a multi-purpose 2D and 3D seismic imaging
platform based on high-order finite-difference and finite-volume
discretisation and non-linear seismic data inversion procedures. Such an
interface would allow performing collaborative or joint inversion of seismic
and gravity or magnetic data and could obtain the resulting models on the same
mesh while benefitting from Tomofast-x's various functionalities. This will
be an easy way to provide Tomofast-x with separate seismic information on the fly like
sensitivity kernels as another physical domain.
In the implementation presented here, only the truncation of the matrix
system based on maximum distance thresholding was discussed. It is planned
to reduce memory requirements using the wavelet compression of the matrix
system of the inverse problem in the same fashion as
Martin et al. (2013).
We have shown a number of tests using a selected set of functionalities of
Tomofast-x. However, more or different tests could be done. For instance,
an interesting research avenue is to exploit Tomofast-x's capability to read
an arbitrary number of prior and starting models to test the geological
archetypes that can be identified by clustering of the set of geological
models probabilistic geological modelling can produce
(Pakyuz-Charrier et al., 2019). Additional features of
Tomofast-x, the testing of which lies beyond the scope of this paper, are Jacobian
matrix truncating and different kinds of depth weighting and their effects
on the different types of inversion. Finally, we have not used posterior
uncertainty indicators listed in Sects. 2.5 and
A1 as the paper focusses on the inversion
capabilities of Tomofast. However, the output results of Tomofast-x allow us
to study uncertainty in the same fashion as Giraud et
al. (2017, 2019c) where some of them are used.
Results obtained using the cross-gradient technique for joint inversion of
gravity and magnetic data showed that it can improve imaging of geological
structures. However, our study also revealed some of the limitations of this
method. In the synthetic example, structurally coherent features of the
resulting model contradict the geology of the true model. In addition, our
L-curve (or L-surface) analysis suggests that the determination of the
optimum α weights of the cost function using the cross-gradient
technique may be affected by non-uniqueness and that multiple sets of
weights could equally satisfy the L-curve criterion. One interpretation is
that this method remains affected by uncertainty and could be producing
several families of models fitting geophysical data equally well. This
observation differs from similar analysis performed in the case of joint
inversion using petrophysical constraints, where such potential ambiguity
was not suggested by the L-surfaces (Giraud et al., 2019c).
These impressions, however, require a more detailed investigation and
constitute a new research avenue.
In our sensitivity analysis, we have produced a series of models that can be
considered geophysically equivalent because they fit the geophysical data
equally well. These models are the result of deterministic inversion, where
prior information guides inversion towards one of the modes of the
probability density function describing the problem (Eq. 1) or
modifies them. It is therefore safe to assume that each mode is
representative of an archetype of models from the geophysical data's
null space. This highlights the interest of using “null-space shuttles”,
allowing navigation of the null space
(Deal and Nolet, 1996;
Muñoz and Rath, 2006; Vasco, 2007; Fichtner and Zunino, 2019) to explore
the space of possible models without extensive sampling and to assess the
robustness of the result. In addition, the plots of the L-curves
corresponding to the problem we presented suggest the presence of multiple
optima in the hyperparameter space (weights α), which it might be
interesting to investigate in future research, especially in the joint
inversion case.
Conclusions
We have introduced the open-source joint inversion platform Tomofast-x and
demonstrated its capabilities with a realistic data set taken from the
Hamersley region in central Western Australia. The geophysical theoretical
background of Tomofast-x was explained in depth to guide users in
understanding and using the modelling approach implemented in the source
code.
We leveraged the modularity of Tomofast-x to study the sensitivity of
inversion to prior structural, geological, and petrophysical information;
joint inversion; and the code's scalability. We tested a new combination of
constraints incorporating geological structural information in the
smoothness term and dynamic prior model definition using petrophysical
knowledge (ADMM bound constraints), a feature usually not available to most
inversion software. Our sensitivity analyses on prior information and
different constraints reveal that constraints using petrophysics (ADMM bound constraints) dominate over gradient-based constraints
(smoothness and cross-gradient constraints), which in turn exert more
influence on inversion than smallness constraints. This shows the
importance of prior information in inversion and illustrates the need to
study the space of geophysically equivalent models.
The examples described here were designed to replicate a typical, rigorous
approach to the development of a geoscientific model and be relevant to
real-world application. The aim to ensure rigour and reproducibility of the
result presented is facilitated by the release of the source code, data sets,
and a reduced modified Python version of the algorithm that accompany this
paper.
Other functionalities of Tomofast-xPosterior uncertainty indicatorsPosterior LSQR variance matrix
At the first and last iteration of the inversion, the diagonal elements of
the posterior covariance matrix of the recovered model are calculated in
Tomofast-x (see Sect. 2.5). The
variances are part of the outputs of Tomofast-x for further analysis by the
user, such as the estimation of uncertainty in the recovered property model.
Jacobian of the cost function
Tomofast-x offers the possibility to examine the Jacobian matrix of the
cost function (Eq. 5), which encapsulates the contribution of several
constraint terms (see for example Eq. 5), by calculating its derivative with
respect to the model m, ∂θ(d,m)/∂m. This feature takes advantage of the LSQR solver. In
the LSQR algorithm, ∂θ(d,m)/∂m is calculated at the beginning of each iteration when
approximating a solution to the system of equations
(Appendix E). The value of ∂θ(d,m)/∂m can then be calculated before or
after application of the depth-weighting operator. It is computed as the
product of the transpose of the matrix representing the left-hand side of
the system of equations to be solved by the vector of
the corresponding quantities to minimise data misfit, cross-gradients,
damping terms, etc. (constituting the right-hand side of the corresponding
equation, as shown in Appendix E). Importantly, its
dimension is equal to the number of model parameters. It is therefore
possible to store it on disk to provide a measure of the sensitivity of the
data and the different terms of the misfit function to model variations at
depth or in any part of the computational domain. By computing ‖∂θ(d,m)/∂m‖, it
is possible to study the convergence of the algorithm, with small values
indicating convergence. In addition, it is a metric that measures the
stability of the algorithm and which is useful to determine whether the
system of equations is well conditioned.
Identification of rock units
Membership analysis of the inverted model can be performed when statistical
petrophysical constraints were applied to inversion from the values of
Nk reached after inversion converged. Membership values can be used to
assess inverted models by reconstructing a rock unit model from the
recovered inverted physical properties
(Doetsch et al., 2010; Sun et
al., 2012; Giraud et al., 2019c). Rock units labels can also be assigned to
model cells when the ADMM bound constraints have converged. It allows
attaching a petrophysical property interval to each model cell, allowing
direct identification of rock types.
Cross-product of gradients
The cross-product of model gradients in 3D can be stored after inversion and
its L2 norm is given after each inversion cycle. It allows us to
assess the degree of structural similarity between the models and to
delineate areas showing specific structural similarities or dissimilarities.
Jacobian matrix truncation
Tomofast-x offers the possibility to use a moving sensitivity domain
approach (Čuma et
al., 2012; Čuma and Zhdanov 2014), limiting the sensitivity domain to a
cylinder, the radius of which is chosen by the user, to reduce computational
requirements (the option for a sphere is also present in the source code
but commented in the current version). The underlying assumption is that
cells beyond a given distance exert a negligible influence on the
measurement. Generally speaking, this radius is provided by the users and
should be chosen carefully.
Lp norm
Tomofast-x also offers the possibility of performing data inversion using a
Lp norm (1<p≤2) to define the smallness term, as it has
been proposed for electrical capacitance tomography (ECT) in
Martin et al. (2018) in the framework
of Tomofast-x. The Lp norm inverse problem is non-linear and can be
solved iteratively using L2 minimisation. In the Lp norm case, the
regularisation parameter can be approximated by a p-power law of the model
at each point of the computational domain and must also be recomputed at
each new inversion cycle. When the Lp norm is introduced for p<2, this procedure allows us to obtain sharper models with better
interface definition and determine stronger contrasts for the specific
cases under study. If p=2, the smallness term is reduced to L2 norm
minimisation (the commonly used Tykhonov-like regularisation) as used in
this work. The choice of other values such that 1<p≤2 is at
the discretion of the user or depending on prior information.
Electrical capacitance tomography
Detailing electrical capacitance tomography (ECT) is beyond the scope of
this paper, but we can apply to joint gravity and magnetic inversion the
functionalities that have been introduced to solve the ECT inverse problem
based on L2 data misfit norm and Lp (1<p≤2) damping
term minimisation (see Sect. A3). In Tomofast-x,
ECT is based on the finite-volume method for the forward problem and on a
non-linear and iterative LSQR method to solve the inverse problem. As in
propagative and diffusive geophysical inverse problems in frequency domain,
the sensitivity matrix and the damping term depend on the current model and
must be recomputed at each new iteration. We refer the reader to Martin et
al. (2018) for more details on this technique. Note that algorithms
developed in relation to this method can easily be extended to propagative
and diffusive geophysical inverse problems.
Summary of the notation and terms used in the paperSymbolDefinitionSubscripts and superscripts drefers to “data”, i.e. “mag” or “grav”mmodelprrefers to “prior”ggradientxcross-gradientperefers to “petrophysics”aADMMGgravityMmagneticsModel and physical quantities mproperty model inverted forzADMM variableuADMM variableωmembership value in Gaussian mixtureψmembership value in Gaussian mixture (from geology)μmean value of petrophysical propertiesσstandard deviation of petrophysical propertiesεpositive threshold real number such that z≫εndatanumber of geophysical data pointsnfnumber of rock formationsdgeophysical dataβexponent for depth-weighted power lawMathematical operators or notations g(.)geophysical forward operatorP(⋅)petrophysical distribution operator∇⋅gradient operatorLpLp normL2L2 norm (sum-of-squares)Sgeophysical data sensitivity matrixdiagonal matrices WWddiagonal matrix where all elements are equal to the inverse of the sum of squares of the dataWmsmallness term covariance matrixWgsmoothness term covariance matrixWxcross-gradient term covariance matrixWpepetrophysical term covariance matrixWaADMM term covariance matrixDdepth-weighting operatorweighting terms ααmmodelαggradientαxcross-gradientαpepetrophysicsαamultiple bound constraintsαGweight assigned to the gravity inverse problem (used only in joint inversion)αMweight assigned to the magnetic inverse problem (used only in joint inversion)Forward gravity and magnetic data calculation
In this Appendix we summarise the forward calculation of gravity and
magnetic data as performed in Tomofast-x. In practice, Tomofast-x calculates
forward data using input data expressed in units from Système International (SI), expressed in
kilograms, metres, and seconds. The gravity field f of a
distribution of density anomalies Δρ over a volume of rock V
at a location r′=[x′,y′,z′] can
be expressed as follows:
f(r)=G∭VΔρ(r)r-r′|r-r′|3dV,
where r=[x,y,z] defines
the location of mass density anomaly Δρ(r) and G is the universal gravity constant.
While Tomofast-x is implemented in such a way that the three spatial
components of f can be obtained, we consider only the vertical
direction here, which we simply write as f for sake of clarity (note that
here, when taken for the whole model, f=gG).
In our implementation, the volume V is discretised in Nm rectangular
prisms (model cells) of constant density. Discretised,
Eq. (C3) then
rewrites as follows:
f(x,y,z)=G∑k=1NmΔρk×zk-z′((x-xk)2+(y-yk)2+(z-zk)2)32C2×ΔxkΔykΔzk,
where Δxk, Δyk, and Δzk define the
dimensions of the kth rectangular prism. In our discretisation, we
assume a model constituted of nx×ny×nz cells, with
nx, ny, and nz representing the number of cells in each
direction. This leads to the computation of f using the following
formulation of Eq. (C1):
f(x,y,z)=∑i=1nx∑j=1ny∑k=1nzΔρi,j,kSi,j,k,
where, using the formulation of Blakely (1995), the elements of
the sensitivity matrix S are given as follows:
Si,j,k=G∑m=12∑q=12∑t=12(-1)m+q+t×[ζtatan(ξmηqζtRm,q,t)C4-ξmlog(Rm,q,t+ηq)-ηqlog(Rm,q,t+ξm)],
where ξm, ηq, and ζt are the coordinates of the
vertices of the prism and
Rm,q,t=[(x-ξm)2+(y-ηq)2+(z-ζt)2]12.
In Tomofast-x, the total magnetic field anomaly is calculated by summing the
responses of the prisms making up the model, following
Bhattacharyya (1964, 1980). The regional magnetic
field is denoted F=(Fx,Fy,Fz), and the magnetisation is denoted M=(Mx,My,Mz). We write F=‖F‖ and M=‖M‖. Note that
remnant magnetisation is not accounted for.
Using the formalism of Blakely (1995), we denote ΔT the
magnitude of the total magnetic field anomaly generated by a prism oriented
parallel to the x, y, and z axes of the mesh similarly to the gravity
case. We have the following equation:
ΔT(x,y,z)=∑i=1nx∑j=1ny∑k=1nzχi,j,kSi,j,k,
where χ is the magnetic susceptibility. The sensitivity S is given
as follows:
Si,j,k=μ0F∑m=12∑q=12∑t=12(-1)t+1×[αyz2log(Rm,q,t-ξmRm,q,t+ξm)+αxz2log(Rm,q,t-ηqRm,q,t+ηqm)-αxylog(Rm,q,t+ζt)-MxFxatan(ξmηqξm2+Rm,q,tζt+ζt2)-MyFyatan(ξmηqRm,q,t2+Rm,q,tζt-ξm2)C7+MzFzatan(ξmηqRm,q,tζt)],
where, ξm=1,2, ηq=1,2, and ζt=1,2 are the
coordinates of the vertices of the prism along the x, y, and z
directions, respectively. The other terms of Eq. (C7) are defined below:
C8αxz=FxMz+FzMx,C9αxy=FxMy+FyMx,C10αyz=FyMz+FzMy.
Scaling tests
Although Tomofast-x can run on personal computers in a few seconds or
minutes for 2D inversions and small 3D volumes (typically a few minutes on a
laptop for models smaller than approx. 100 000 model cells), it necessitates
a supercomputer for realistically sized 3D case studies (e.g. models exceeding
500 000 model cells and 10 000 geophysical data points).
We assess Tomofast-x's parallel efficiency using the strong scaling as an
indicator. The strong scaling curve is given by plotting the number of CPUs
as a function of user time. It is complemented by the relative speedup curve
t1/tncpu, where t1 and
tncpu are the user times to
complete inversion using the number of CPUs
ncpu=1 and a given number of CPU
ncpu, respectively. We performed the scaling tests on
the EOS machine from the CALMIP supercomputing centre
(https://www.top500.org/site/50539/,
https://www.calmip.univ-toulouse.fr/spip.php?article388 – the latter
being in French only, both last accessed on 10 November 2020).
The full-sized model we used is made of
NxNyNz=128×128×32=524288 cells (i.e. 219
cells), which we reduce by a factor of 2 by reducing the physical domain
incrementally to
NxNyNz=32×32×32=32768 cells (i.e. 215
cells) to be able to use it on a single CPU for the purpose of
parallelisation efficiency analysis. In the configuration we use, the number
of data points modelled is equal to
Ndata=NxNy.
Strong scaling (a), relative speedup (b), and number of elements
per CPU plots for a number of CPUs equal to 1, 2, 4, 8, 16, 32, 64, 128,
256, and 512. The line marked “ideal law” indicates perfect scalability for the tests that were performed.
Figure D1a shows the parallelisation efficiency. It
reveals that the scaling is nearly perfect for up to 16 CPUs, that it is very good for
32 CPUs, and that it deteriorates above 64 CPUs. This corresponds to relative
speedups (ratio between run time for a single CPU and a given number of CPUs) of
about 14.5, 25, and 40 (Fig. D1b), respectively.
For the cases using 64, 128, and 256 CPUs, speedup increases from 40 to 65,
indicating that overhead inter-processor communication time for
ncpu≥64 increasingly impacts the total
computation time for this (small) problem size. This is noticeable in
Fig. D1a and b,
as both curves seem to adopt an asymptotic behaviour for the largest numbers
of CPUs. This illustrates the deterioration of performances due to
inter-processor communications (Kumar et al., 1994). The
deterioration of performances due to inter-processor communications is due
to the number of elements (or model cells) processed by each CPU becoming
smaller, while the number of elements involved in communications increases;
ultimately, the time spent in pure computations in each core becomes smaller
than the time spent in inter-processor communications.
The efficiency curves (e.g. Fig. D1a and b) allow us to determine the minimum
number of elements per CPU that run efficiently (Hammond and Lichtner, 2010). The case of ncpu=64 marks an inflection point in Fig. D1b, corresponding to point of diminishing return equal to a number of element per CPUs of 512. This indicates that for this particular configuration, it is preferable to run inversions with ncpu≤64 to
maximise parallel efficiency. For better understanding and interpretation of scaling and speedup, we repeat that the number of elements
nel per CPU as a function of ncpu is as follows:
nel=NxNyNzncpu.
As a general rule, we recommend respecting the condition of
nel≥512. For a smaller number of
elements, the allocated resources are used in a suboptimum manner. Note that
the memory requirements vary proportionally with
NxNyNzNdata=nmnd, and thus no Jacobian matrix compression is used.
Matrix formulation of least-squares problem and resolution of the
inverse problem
This Appendix introduces the matrix formulation of Eq. (5) and its
resolution.
We can write the system of equations to be solved in the least-squares sense
as follows.
SmαmWmmαgWg∇mαpeWpeP(m)αxWx(∇m(1)×∇m(2))αaWam=dαmWmmpr0αpeWpePmax0αaWa(z-u)
At iteration k, the system of the equation is linearised around the current
model. It is solved for the optimum update of the current model
mk model update as described below. Models
mk(1) and mk(2)
are set accordingly with the type of inversion considered.
In Tomofast-x, depth-weighting D is applied as a sensitivity
matrix preconditioner. The resulting system is solved using the LSQR
algorithm (Paige and Saunders, 1982) as follows.
D-1SαmD-1WmαgD-1Wg∂∇mk∂mαpeD-1WpeP′(mk)αxD-1Wx∂∂m(∇mk(1)×∇mk(2))αaD-1WaΔm‾k+1=-g(mk)-dαmWm(mk-mpr)αgWg∇mkαpeWpe(P(mk)-Pmax)αxWx(∇mk(1)×∇mk(2))αaWa(mk-zk+uk)
At each kth inversion cycle, we solve this system of equations and
calculate the model update the model as follows:
Δmk+1=D-1Δm‾k+1.
The model mk can then be updated to obtain mk+1.
mk+1=mk+Δmk+1
Following Ogarko et al. (2021a), u0=0 and
z0=0. The updated ADMM variables zk+1 and uk+1 are calculated using the ADMM algorithm introduced by Boyd (2010):
E5zk+1=πB(mk+1+uk),E6uk+1=uk+mk+1-zk+1,
where πB is a projection onto the bounds B such that
E7πB(x)=[πB1(x1),πB2(x2),…,πBn(xn)], withE8πBi(xi)=argminy∈Bi‖xi-y‖2,
The value of the starting model m0 is provided by the
user.
Code and data availability
The source code for Tomofast-x as used in this paper can be found in Ogarko et al. (2021b, 10.5281/zenodo.4454220). The latest version of Tomofast-x is available at https://github.com/TOMOFAST/Tomofast-x (last access: 7 September 2021).
The geological model, a description of the input data, and the geophysical
models are given in Jessell et al. (2021a, 10.5281/zenodo.4431796). It also contains a data set using the same model projected onto a finer mesh of
approximately 4.2 M cells and 80 000 geophysical data. The data sets are
licensed under the Attribution-ShareAlike 4.0 International (CC BY-SA 4.0)
license (see https://creativecommons.org/licenses/by-sa/4.0/legalcode, last access: 7 September 2021, for details).
Tomofast-x's source code is licensed under the MIT License
(https://opensource.org/licenses/MIT, last access: 7 September 2021).
Author contributions
JG designed the geophysical study and ran all inversions shown in the paper.
He adjusted the geological model presented here, which was initially built
by MJ and ML. JG performed posterior analysis and interpretation of results.
JG is the main contributor to the writing of this article and preparation of
the assets. JG carried out the scaling tests shown in the Appendices with the
collaboration of VO. VO wrote the user manual of Tomofast-x that can be found in the GitHub repository.
VO and JG worked together on the implementation of the gravity and magnetic
inversion methodologies in Tomofast-x. VO is the main developer of this
version of Tomofast-x, the development of which was carried in collaboration
with JG and RM. The main contributors to the code are VO, JG, and RM. JG
performed the initial testing of the different data integration techniques
presented, with the exception of the cross-gradient technique, which was
implemented independently by VO.
RM participated in the redaction of this paper. RM initiated the Tomofast
project and implemented the initial LSQR solver for gravity inversion, which
was subsequently used as a starting point for the current version of the
code. RM and VO added the possibility of performing data inversion using an
Lp norm (1<p≤2) smallness term. RM and VO developed and tested
the electrical capacitance tomography component of Tomofast.
JG and RM explored the functionalities of Tomofast-x and performed extensive
testing for robustness and validation of techniques implemented, especially
the cross-gradient technique.
ML and MJ produced the reference geological model from field measurements
and carried out probabilistic geological modelling. ML and MJ were involved
in the redaction of the manuscript and participated in the supervision of
the project. MJ and ML were involved in the validation of the
methodology at the initial development stage and supervised the overall
progress of the presented work.
All authors participated in the revisions of the manuscript.
Competing interests
The authors declare that they have no conflict of interest.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
Appreciation is expressed to the CALMIP supercomputing centre (Toulouse,
France). Roland Martin acknowledges the was supported from the CNES (French National Space Agency). Jérémie Giraud, Mark Lindsay, and Mark Jessell
acknowledge the support from the Australian Government's Cooperative Research Centre Program. This is MinEx CRC Document
2021/3. Mark Lindsay acknowledges funding from the ARC and DECRA.
The authors acknowledge the contribution of Clement Barriere to the reduced 2D Python version of Tomofast-x (called “Tomoslow”).
The authors appreciated discussions about geophysics–geology integration
with Laurent Aillères and the rest of the Loop consortium's researchers. The
authors thank Guillaume Pirot for his comments on the manuscript. The
authors are also thankful to Mahtab Rashidifard, Nuwan Suriyaarachchi,
Damien Ciolczyk, and Marina Zarate-Jeronimo for providing interesting discussions.
Financial support
This research has been supported by the Department of Industry, Science, Energy and Resources of the Australian Government (grant no. GA22270); the CALMIP supercomputing centre (Toulouse, France) via their support through Roland Martin's supercomputing projects
(no. P1138_2018, no. p1138_2019) and the
computing time provided on the Olympe machine; and the CNES (French National Space Agency) through the projects 2017 and 2018-TOSCA/GET-GRAVI-GOCE (no. AG68ANGS) grants. Jérémie Giraud, Mark Lindsay, and Mark Jessell
are supported in part by Loop – 3D Enabling Stochastic 3D Geological
Modelling (LP170100985) and the Mineral Exploration Cooperative Research
Centre (MinEx CRC), whose activities are funded by the Australian
Government's Cooperative Research Centre Program. This is MinEx CRC Document
2021/3. Mark Lindsay received funding from the ARC and DECRA (DE190100431).
Review statement
This paper was edited by Thomas Poulet and reviewed by Mehrdad Bastani and one anonymous referee.
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