Magnetotelluric 3-D inversion—a review of two successful workshops on forward and inversion code testing and comparison

Full text

Geophysical Journal International
Geophys. J. Int. (2013) 193, 1216–1238 doi: 10.1093/gji/ggt066
Advance Access publication 2013 March 13
GJI Geomagnetism, rock magnetism and palaeomagnetism
Magnetotelluric 3-D inversion—a review of two successful
workshops on forward and inversion code testing and comparison
Marion P. Miensopust,
1,2,∗
Pilar Queralt,
3
Alan G. Jones
1
and the 3D MT modellers
†
1
Dublin Institute for Advanced Studies, School of Cosmic Physics, Dublin, Ireland. E-mail: Marion.Miensopust@bgr.de
2
Institut f
¨
ur Geophysik, Westf
¨
alische Wilhelms-Universit
¨
at M
¨
unster, Germany
3
Deptartment Geodin
`
amica i Geof
´
ısica, Universitat de Barcelona, Spain
Accepted 2013 February 14. Received 2013 February 14; in original form 2012 May 27
SUMMARY
Over the last half decade the need for, and importance of, three-dimensional (3-D) modelling
of magnetotelluric (MT) data have increased dramatically and various 3-D forward and in-
version codes are in use and some have become commonly available. Comparison of forward
responses and inversion results is an important step for code testing and validation prior to
‘production’ use. The various codes use different mathematical approximations to the problem
(finite differences, finite elements or integral equations), various orientations of the coordinate
system, different sign conventions for the time dependence and various inversion strategies.
Additionally, the obtained results are dependent on data analysis, selection and correction as
well as on the chosen mesh, inversion parameters and regularization adopted, and therefore, a
careful and knowledge-based use of the codes is essential. In 2008 and 2011, during two work-
shops at the Dublin Institute for Advanced Studies over 40 people from academia (scientists
and students) and industry from around the world met to discuss 3-D MT inversion. These
workshops brought together a mix of code writers as well as code users to assess the current
status of 3-D modelling, to compare the results of different codes, and to discuss and think
about future improvements and new aims in 3-D modelling. To test the numerical forward
solutions, two 3-D models were designed to compare the responses obtained by different codes
and/or users. Furthermore, inversion results of these two data sets and two additional data sets
obtained from unknown models (secret models) were also compared. In this manuscript the
test models and data sets are described (supplementary files are available) and comparisons
of the results are shown. Details regarding the used data, forward and inversion parameters as
well as computational power are summarized for each case, and the main discussion points
of the workshops are reviewed. In general, the responses obtained from the various forward
models are comfortingly very similar, and discrepancies are mainly related to the adopted
mesh. For the inversions, the results show how the inversion outcome is affected by distortion
and the choice of errors, as well as by the completeness of the data set. We hope that these
compilations will become useful not only for those that were involved in the workshops, but for
the entire MT community and also the broader geoscience community who may be interested
in the resolution offered by MT.
Key words: Numerical solutions; Inverse theory; Magnetotelluric; Geomagnetic induction.
∗ Now at: Federal Institute for Geosciences and Natural Resources,
Hannover, Germany.
† Dmitry Avdeev, Anna Avdeeva, Ralph-Uwe B
¨
orner, David Bosch, Gary
Egbert, Colin Farquharson, Antje Franke-B
¨
orner, Xavier Garcia, Nuree
Han, Sophie Hautot, Elliot Holtham, Juliane H
¨
ubert, David Khoza, Duygu
Kiyan, Florian Le Pape, Juanjo Ledo, Tae Jong Lee, Randall Mackie, Anna
Mart
´
ı, Naser Meqbel, Greg Newman, Doug Oldenburg, Oriol Rosell, Yutaka
Sasaki, Weerachai Siripunvaraporn, Pascal Tarits and Jan Vozar.
1 INTRODUCTION
Testing and validation of new approaches and techniques are es-
sential, and therefore, measures to verify electromagnetic (EM)
forward and inversion algorithms are required. Within the geosci-
entific EM community, several prior workshops and projects dealt
with comparison of results obtained from test models or of data
sets designed by teams or individuals to address specific tasks
(i.e. 2-D forward modelling, 3-D f orward modelling). These results
were calculated by different users applying different algorithms and
1216
C

The Authors 2013. Published by Oxford University Press on behalf of The Royal Astronomical Society.
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

MT 3D forward and inversion comparison 1217
approaches to allow for comparison amongst each other. The suc-
cess of such workshops and projects lies not only in the value of
the models and data sets but also in the active par ticipation of sci-
entists attending and contributing to these comparisons. One of the
first and the most important ones was COMMEMI (Comparison Of
Modelling Methods for ElectroMagnetic Induction) that was pro-
posed and adopted at the VI EMI International Workshop (Victoria,
B.C., Canada, 1982) (Zhdanov et al. 1997). The project comprises
a set of six 2-D and two 3-D models covering a wide range of
resistivity contrasts and structures. It includes a simple model, for
which an analytical solution exists, as well as a highly complex
model that represents a structure very similar to a real geological
setting. The main contribution of this project was to assemble the
numerical forward solutions for these models, and some of them
have since become the standard models for testing new approaches
and algorithms.
During the 1990s, five Magnetotelluric Data Interpretation Work-
shops took place namely MT-DIW1 in Wellington, New Zealand,
1992; MT-DIW2 in Cambridge, England, 1994; MT-DIW3 in
Tsukuba, Japan, 1996; MT-DIW4 in Sinaia, Romania, 1998 and
MT-DIW5 in Cabo Frio, Brazil, 2000. Together with a compari-
son of the inversion models, the main contributions of these work-
shops were the data sets provided for the interpretation exercise.
For the first workshops, MT-DIW1 and MT-DIW2, two profiles
of real data—COPROD2 and BC87 (Jones 1993a,b)—were exam-
ined. There are a large number of publications using these data
to test 2-D inversion codes and different techniques for MT data
decomposition. The results of those workshops are published as
special sections in the Journal of Geomagnetism and Geoelectric-
ity, volume 45(9) (1993) and 49(6) (1997) with summaries by Jones
(1993c) and Jones & Schultz (1997) and a comparison of all 2-D
models presented at the MT-DIW1 workshop by Jones (1993a). For
the subsequent workshops (MT-DIW3, MT-DIW4 and MT-DIW5)
the Kayabe data set was provided consisting of 209 stations on a
rectangular grid. This data set allowed testing different approaches
of handling 3-D MT data as well as studying the errors intro-
duced by 2-D interpretation of 3-D bodies (i.e. Garcia et al. 1999).
Also, two synthetic data sets (COPROD2S1 and COPROD2S2)
were created for MT-DIW4 and MT-DIW5. COPROD2S1 is error-
free, whereas COPROD2S2 contains added noise and static shifts.
They are very valuable to compare inversion procedures (Varentsov
1998). All these data sets with full information can be downloaded
from: http://mtnet.dias.ie/workshops/mt-diw (last accessed 1 March
2013).
In parallel, the 3-D EM International Symposium was estab-
lished, not to compare models and results, but to stimulate sym-
biosis and gather information about new developments to improve
approaches and techniques used in 3-D: 3DEM-1 (Ridgefield, USA,
1995); 3DEM-2 (Salt Lake City, USA, 1999); 3DEM-3 (Adelaide,
Australia, 2003); 3DEM-4 (Freiberg, Germany, 2007) and 3DEM-
5 (soon in Sapporo, Japan, May 2013). These symposia covering
3-D EM modelling, inversion and application are a record of the
growing interest in 3-D EM modelling and inversion methods.
The EM Induction workshop reviews are also worth highlighting
as they offer a thorough summary of the state of the art for spe-
cific subjects in a format addressed to the entire community, not
only specialists. In 1988, at the EM Induction workshop in Sochi
(Russia),
ˇ
Cerv & Pek (1990) presented a review on some aspects
of modelling EM fields in 3-D inhomogeneous media consider-
ing various 3-D modelling techniques and their comparison. At the
same workshop, Oldenburg (1990) focused on new techniques of
EM data inversion especially related to the 2-D MT problem. In
2002, at the workshop in Santa Fe (USA), Ledo (2006) gave an
excellent review about 3-D effects on 2-D interpretations. At the
following EM Induction workshop (2004) in Hyderabad (India),
Avdeev (2005) presented a review concentrating on recent develop-
ments in the finite-difference (FD), finite element (FE) and integral
equation (IE) methods for 3-D forward modelling and inversion. He
also included an overview of certain optimization procedures. At
the 2008 workshop in Beijing (China), B
¨
orner (2010) gave a review
of numerical solutions of 3-D EM forward modelling, and he ad-
dressed several challenges in reducing the computing costs in terms
of code efficiency and speed. In Giza (Egypt) at the 2010 work-
shop, Siripunvaraporn (2012) reviewed 3-D magnetotelluric data
inversion techniques, summarizing work done related to various
optimization methods and provides a comparison of those. Finally,
at the most recent workshop (2012) held in Darwin (Australia), two
more modelling reviews followed: one on joint inversion by Haber
and the other one on parallel computing by Newman.
There is significantly increased interest and awareness in 3-D
modelling and inversion, and the efforts to develop 3-D codes have
increased and the number of projects and publications presenting 3-
D MT data is rising quickly (i.e. Chave & Jones 2012, and references
therein). Nevertheless, 3-D approaches and algorithms are not yet
standard tools for MT data modelling and inversion, and numerous
issues remain to be addressed to avoid pitfalls by ill-informed users.
These start with simple things, like a comparison of the forward
responses calculated by different codes and by different users, to
more applied issues, like the question of which strategies to use for
the inversion of a large data set, but also more complex problems
as, for example, how to handle distortion in 3-D environments (e.g.
Jones 2011). Combining the idea of COMMEMI and the MT Data
Interpretation Workshops, the first 3-D MT Inversion Workshop
was held on 2008 March 12–14 in Dublin, Ireland. It was such a
success that the 3-D MT Inversion Workshop II followed on 2011
March30toApril1(againinDublin).
The workshops are designed to be an opportunity to bring to-
gether a mix of MT 3-D inversion code writers and users and see
how well the codes perform as well as how good the understanding
of how to use them is. The main topics of the workshops were to
assess the state of the art, to compare the results of the codes, and
think about future improvements and new aims in 3-D modelling
and inversion. Prior to each workshop, first, a test model and sec-
ondly a test data set retrieved from a secret model were distributed,
so that forward modelling and inversion could be performed in
advance and the results were available for comparison during the
workshop. During both workshops, a total of 42 people from in-
dustry and academia participated, most attending both workshops,
and made the workshops such a success. Herein the main results
of both 3-D MT Inversion Workshops will be presented showing
some representative sites and model slices (see supplementary files
for more information available). The discussion summarizes issues
related to the presented test data sets and models as well as reflects
some of the points that were raised at the open discussions during
both workshop.
2 FORWARD MODELLING
The first task for the attendees of the two workshops was to forward
model a given 3-D electrical conductivity structure and obtain syn-
thetic data at given locations and periods. In the following we will
briefly describe the different techniques and codes used. Then, the
models
and instructions are represented and finally the results are
illustrated and compared.
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

1218 M. P. Miensopust et al.
2.1 Different algorithms and approaches
The participants of the workshops used different numerical algo-
rithms, based on FD, FE or IE methods, to calculate the 3-D forward
responses. To solve the problem, all of these codes use numerical ap-
proximations of the Maxwell’s equations in the frequency domain.
The most commonly employed method for our comparison was FD;
namely the codes by Mackie et al. (1994), Newman & Alumbaugh
(2000), Sasaki (2001), Farquharson et al. (2002), Siripunvaraporn
et al. (2002), Egbert & Kelbert (2012) and B
¨
orner (personal com-
munication). Using the FD approach, the electrical properties of
the Earth and the EM fields or their potentials are sampled over a
rectangular mesh and the associated differential equations are ap-
proximated by FDs resulting in a system of linear equations to solve.
However, there are important differences between the various codes;
for example, the solvers applied to the system of linear equations,
the type of grid and boundary conditions used. It is beyond the scope
of this paper to compare the technical aspects of the different codes
(the interested reader may consult the above references for more de-
tails). Despite these differences, the design of an appropriate mesh
is absolutely crucial and affects the derived results significantly.
Although the number of available MT forward codes based on
FE and IE approaches is limited, some attendees obtained responses
using such codes (i.e. FE: Franke et al. (2007), Nam et al. (2007),
Farquharson & Miensopust (2011); IE: Avdeev et al. (2002)). The
numerical implementations of FE and IE are more complex than FD,
and often require a highly detailed knowledge of the algorithm to
use it successfully. These two methods represent extreme situations:
IE is efficient for models with a localized anomalous conductivity
in a 1-D background given that only the anomalous region needs to
be discretized. In contrast, FE methods are powerful for modelling
complex media with the inclusion of detailed topography. The code
of Avdeev et al. (2002) is based on the IE approach to describe
Maxwell’s differential equations and applies the Green’s function
technique to obtain the scattering equation. In this way, only the
volume where the conductivity is different from the background
requires discretization and results in a linear equation system of
reduced size. The FE method is based on the approximation of the
EM fields, or their potentials, by a set of nodal- or edge-element
basis functions. Either Galerkin’s method or the FE analysis of the
weak form of the MT boundary value problem yields a discrete
approximation and results in a system of linear equations. As in the
FD case, there are similar differences between the FE codes used
for the comparison (e.g. the applied solver, the type of grid and
boundary conditions used). Additionally, the discretization can be
implemented in different ways, for example, as rectilinear mesh,
structured mesh of hexahedral elements (i.e. distorted bricks) and
unstructured grids of tetrahedral elements. Furthermore, the type of
basis function used (i.e. nodal or edge), as well as its order, varies
depending on the code. (For more details, the interested reader is
referred to the above references of the various codes.)
Note that all of the codes used do not mimic field acquisition
correctly, in that electric field calculations in the codes are made for
a point in space, whereas in the field the electric fields are derived
by measuring the potential differences between two points in the
ground, typically 100 m appart. See Jones (1988) and Poll et al.
(1989) for a discussion of this issue.
2.2 Dublin Test Model 1 (DTM1)
The design of the forward model for the first workshop was aimed at
a comparison of forward responses obtained from various codes and
Figure 1. Sketch of Dublin Test Model 1 (DTM1). On top two sections are
shown, one across body 3 (left) and one across body 2 (right). Additionally,
a plan view of all three bodies is shown. The background is a 100 m
homogeneous half-space and the dimensions and resistivities of the blocks
are listed in Table 1. The blue dashed lines represent the four profiles with
5-km site spacing (North, Centre, South: 16 sites each, Vertical: 11 sites).
Tabl e 1. Dimensions and resistivity values of the three blocks in the
Dublin Test Model 1 (DTM1). For x-, y-andz-direction, the extent of
the blocks is specified.
x (km) y (km) z (km) resistivity (m)
body 1 −20 to 20 −2.5to2.5 5to20 10
body 2 −15 to 0 −2.5to22.5 20to25 1
body 3 0 to 15 −22.5 to 2.5 20 to 50 10 000
different users, but also to see how they dealt with strong resistivity
contrasts. The 3-D forward model consists of three different blocks
embedded in a homogeneous, 100 m half-space. Fig. 1 shows dia-
grams of the structure in section and plan view. Table 1 summarizes
the dimensions and resistivity values of the three bodies. The origin
of the coordinate system is defined at the centre (in lateral direction)
of body 1, and is a right-handed system with z positive downwards.
The sites are located on four profiles (see Fig. 1). Three are parallel
to the y-direction at x =−15, 0, 15 km with 16 equally distributed
siteswith5kmspacingfrom−37.5 to 37.5 km. The fourth profile
is parallel to the x-direction at y = 0 km and has 11 sites from
−25 to 25 km (also 5-km site spacing). The task was to calculate
the forward responses at all s ites for the period range of 0.1 s to
10 000 s at four periods per decade.
Table 2 lists, for all forward response sets, the name of the
user(s), the code and information about the mesh and calcula-
tion times whereas the obtained response curves of an example
site (x = y = 0 km), which is considered as representative for
the discrepancies between different results, are shown in Fig. 2.
From top to bottom, each row of this figure shows the xx, xy, yx
and yy components (i.e. resistivity and phase curves). Each column
highlights a group of responses in colour, whereas the respective
other responses are shown in grey. The first column presents re-
sults from different people using the FD code of Farquharson et al.
(2002) (contoured colour symbols) and the FD code of Mackie et al.
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

MT 3D forward and inversion comparison 1219
Tabl e 2 . List of DTM1 forward results. The code and its type (finite-difference FD or finite-element FE) as well as the user are specified. Further more the used
mesh (i.e. number of cells in each direction and—if available—in parentheses the lateral width of the centre cells), required CPU time (and target tolerance TT
or achieved tolerance AT) and the specification of the used computer are listed.
Code User Mesh (# of cells) CPU time and tolerance AT/TT Computer used
FD (Egbert & Kelbert 2012) Naser Meqbel 42 × 42 × 56 — —
(plus 10 air layers)
FD (Farquharson et al. 2002) Colin Farquharson 72 × 72 × 72 (2.5 km) AT: 10
−8
to 10
−10
Mac PowerPC G5, 2 GHz
(serial) (low tolerance for long periods)
(frequency parallelization) Elliot Holtham & >1s:44× 52 × 43 (2.5 km) 49 h 37 min Dual AMD 244 Opteron CPU,
Doug Oldenburg ≤1 s: 104 × 114 × 46 (0.5 km) TT: 10
−9
4GB RAM
(not achieved at all periods)
(serial) Marion Miensopust 92 × 58 × 37 (0.5–1 km) — —
FD (Mackie et al. 1994) Randall Mackie 95 × 95 × 73 30 min 36 dual processor Xeons, 3.2 GHz
(parallelization with PETSc: (without air layers) TT: 10
−10
= rtol
distributes linear system/not (|r| < rtol ·|b| with r = b − Ax;
exactly domain decomposition) b = right hand side; all preconditioned)
(serial) Marion Miensopust 85 × 62 × 34 51 min Intel Core2 CPU E6300, 1.86 GHz,
(plus 10 air layers) TT: 10
−8
3.25GB RAM
FD (Newman & Alumbaugh 2000) Greg Newman 285 × 207 × 125 (0.3 km) 4 min - 38 min per period Parallel Linux cluster using 64 cores
(frequency parallelization & domain to 161 × 171 × 145 (1 km) (full impedance estimation)
decomposition; see reference) TT: 10
−12
FD (Sasaki 2001) Yutaka Sasaki 65 × 83 × 51 1 h 14 min Pentium 4 PC, 2GB RAM
FD (Siripunvaraporn et al. 2002) Marion Miensopust 39 × 40 × 18 (1-2.5 km) 1 h 18 min Intel Core2 CPU E6300, 1.86 GHz,
(serial) (plus 7 air layers) TT: 10
−7
3.25GB RAM
FE (Nam et al. 2007) Nuree Han & 48 × 47 × 31 57 h 16 min cluster: 256 nodes IBM x335, each
Tae Jong Lee 2 CPUs (Pentium IV Xeon DP 2.8 GHz)
IE (Avdeev et al. 2002) Dmitry Avdeev 120 × 135 × 23 4 h 11 min Laptop: T7200 2 GHz CPU, 2 GB RAM
(serial)
(1994) (filled colour symbols). The other four FD response sets ob-
tained from different codes (see Table 2) are shown in the middle
column. The FE and IE response-curves can be found in the right
column.
In general, all obtained response curves are in a good agreement
with each other. Although the methods are different, and although
the meshes differ substantially, the responses that Han & Lee ob-
tained using the FE code by Nam et al. (2007) are almost identical
with the IE responses Avdeev obtained with his code (Avdeev et al.
2002). Both results using the code by Mackie et al. (1994) agree
reasonably well, and we surmise that the small discrepancies are
due to the limited number of cells available in the WinGLink se-
rial version (used by Miensopust) compared to the parallel version
(used by Mackie). Sasaki’s responses using his code (Sasaki 2001),
the responses obtained by Meqbel using the code by Egbert & Kel-
bert (2012) and Newman’s data obtained from the finest mesh of
all using his code (Newman & Alumbaugh 2000), are also in a
good agreement with the FE and IE responses. The three responses
calculated using the code of Farquharson et al. (2002) show some
obvious differences. The data set calculated by Farquharson suffers
from the rather coarse mesh (2.5-km centre cells) and, therefore,
has problems to estimate the responses at the short periods correctly
(some data points are even outside the axes limits). Alternatively,
the responses from Miensopust differ slightly for the longer periods,
which is probably related to a mesh that suits the shorter periods
better than the longer ones. The best data set using this code was
obtained by Holtham & Oldenburg. They used two separate meshes,
one very fine mesh for short periods equal or less than 1 s and a
larger scale one to suit the long periods greater than 1 s. Merging
the data from both meshes, they obtained good responses over the
whole period range. Due to memory limitations, the mesh used by
Miensopust applying the code by Siripunvaraporn et al. (2002) is
by far the smallest. Therefore, the discrepancy is slightly larger than
for the other results. Although the differences might seem relatively
large, we would like to emphasize that is not the case. It is an illusion
caused by the extremely stretched axes for the xy and yx resistivi-
ties. On a normal scale, hardly any differences are visible amongst
the different results; the resistivity axes were enlarged to about one
decade (in contrast to five decades for the periods). Furthermore,
the logarithmic scale additionally emphasises differences for small
resistivity values. For example, the maximum xy resistivity value
obtained for the longest period is 54.7 m, whereas the smallest
is 45.4 m. On a normal logarithmic scale these differences would
hardly be recognizable, although they would be well outside normal
MT field data precision of typically 1 per cent in impedance, which
is 2 per cent in apparent resistivity (approx. ±1 m).
It is also obvious that the diagonal elements have relatively large
and well-defined resistivity values for long periods that are about 2
orders of magnitude smaller than those for the off-diagonals. Above
1 s, the diagonal values all match very well, but for shorter peri-
ods, when the resistivity values are below 10
−4
to 10
−5
m, the
phase values become more random (as expected). Therefore, for
this particular case we would suggest a threshold value of 10
−5
m
to distinguish if the diagonal element values contain structural in-
formation, and hence should be included in an inversion, or not.
For this synthetic case study, the threshold only needs to consider
numerical errors, and, therefore, it is orders of magnitude below
errors introduced by noise. In reality, a threshold value of 0.1 m
is more likely to be appropriate, given typical signal-to-noise con-
siderations.
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

1220 M. P. Miensopust et al.
Figure 2. Comparison of the obtained DTM1 responses at an example site (central site at x = y = 0 km). From top to bottom each row shows the xx, xy, yx
and yy components (i.e. apparent resistivity and phase curves). The left column shows in colour a comparison of responses from different users obtained using
the FD codes by Farquharson et al. (2002) and Mackie et al. (1994). In the middle responses obtained by various other FD codes: Newman used the code by
Newman & Alumbaugh (2000), Sasaki the one by Sasaki (2001), Miensopust the one by Siripunvaraporn et al. (2002) and Meqbel applied the code by Egbert
& Kelbert (2012). The right column shows the responses obtained with the FE code (Nam et al. 2007) by Han/Lee and the IE responses of Avdeev using the
code by Avdeev et al. (2002). In each column, the grey symbols show the respective other responses for comparison. More details can be found in Table 2 and
in the text. Note that the extremely stretched xy and yx resistivity and phase axes cause exaggeration and therefore overemphasise discrepancies which would
hardly be visible on a normal scale.
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

MT 3D forward and inversion comparison 1221
Figure 3. Sketch of Dublin Test Model 2 (DTM2). A plan view (left) and
a section (right) show the structure of DTM2. The radius R is 5 km and the
locations of the sites are given in Cartesian coordinates (see Table 3).
2.3 Dublin Test Model 2 (DTM2)
The motivation for the design of the forward model for the second
workshop was again to compare the solutions of various codes
amongst each other, but additionally to investigate how well galvanic
effects are dealt with by the codes. Therefore, the 3-D forward
model consists of a 10 m hemisphere of radius R = 5kmdirectly
beneath the surface of a homogeneous, 300 m half-space (after
Groom & Bailey 1991). The advantage of this structure is that there
are analytical solutions at the galvanic limit (see Groom & Bailey
1991). The centre of the hemisphere is defined as the origin of the
coordinate system, and Fig. 3 shows sketches of the section and
plan view. One site is located at the centre of the hemisphere, all
other sites are located on circles of different radii from the centre
inside and outside the hemisphere. The site locations are defined in
Cartesian coordinates and listed in Table 3. The task was to calculate
the forward responses at all sites for a period range of 0.01–10 000 s
(four periods per decade).
Table 4 lists, for all forward response sets, the name of the user(s),
the code and information about the mesh and calculation times.
Tabl e 3. Coordinates of the site locations for Dublin Test Model 2
(DTM2).
Site no. x (m) y (m) Site no. x (m) y (m)
0 0 0 25 5500 0
1 2500 0 26 3889 3889
2 1768 1768 27 0 5500
3 0 2500 28 −3889 3889
4 −1768 1768 29 −5500 0
5 −2500 0 30 −3889 −3889
6 −1768 −1768 31 0 −5500
70−2500 32 3889 −3889
8 1768 −1768 33 15 000 0
9 4500 0 34 10 607 10 607
10 3182 3182 35 0 15 000
11 0 4500 36 −10 607 10 607
12 −3182 3182 37 −15 000 0
13 −4500 0 38 −10 607 −10 607
14 −3182 −3182 39 0 −15 000
15 0 −4500 40 10 607 — 10 607
16 3182 −3182 41 25 000 0
17 5100 0 42 17 678 17 678
18 3606 3606 43 0 25 000
19 0 5100 44 −17 678 17 678
20 −3606 3606 45 −25 000 0
21 −5100 0 46 −17 678 −17 678
22 −3606 −3606 47 0 −25 000
23 0 −5100 48 17 607 −17 607
24 3606 −3606
Fig. 4 shows the obtained response curves for sites 10 and 18
(considered to be representative) in comparison to analytical solu-
tions. First, the analytic solutions consider only the galvanic effects
on the electric field (i.e. only the galvanic distortion tensor C is ap-
plied to the true impedance tensor Z
true
→ Z = CZ
true
, after Groom &
Bailey 1991) and are indicated by a grey dashed line, and secondly,
the full analytic solution of the galvanic scatterer was calculated by
David Bosch (after Groom & Bailey 1991) and is represented by
the solid black line. Note that the xx and yy components of CZ
true
and the full analytical solution are indeterminate for sites located
inside the hemisphere (e.g. site 10), and are therefore not plotted.
From top to bottom, each row of this figure shows the xx, xy,
yx and yy components (i.e. resistivity and phase curves)—on the
left for site 10 and on the right for site 18. Blue symbols repre-
sent the various FD response curves, green symbols are the FE
results and the IE responses are in red (more details about the dif-
ferent responses regarding the user, code, mesh etc. can be found
in Table 4). With respect to the hemisphere’s extent at the sur-
face, site 10 is located inside the hemisphere and 500 m away
from the edge of it, whereas site 18 is only 100 m away from
the edge but outside the hemisphere (assuming point electric field
measurements).
A structure like this one is a challenge for mesh and model design,
and of particular difficulty especially for FD codes or other codes
using rectilinear meshes. The blocky discretization of the hemi-
sphere subsurface causes a large variation in the response curves at
sites in its proximity (such as sites 10 and 18). For sites further away
from the hemisphere, the agreement between the various responses
is good, and also for sites inside the hemisphere, but with greater
distance to the edge the agreement is better but still influenced by
the variation in volume and the position of the bottom boundary
(both cases are not shown here). Due to the proximity of sites 10
and 18 to the edge of the hemisphere, and the tangent of the edge
being not parallel to the coordinate system, the diagonal elements
are significant for almost all periods.
For site 18 (i.e. outside the hemisphere) all four phases are in a
very good agreement for all computations (including the analytic
long period limits). Inside the hemisphere (i.e. site 10) the off-
diagonal phases agree well, but the diagonal phases show two groups
of responses for long periods. As responses calculated by different
people but the same code are present in both groups, we can exclude
the possibility that the grouping is algorithm related. It rather seems
to be a consequence of mesh design and the approximation of the
surface of the hemisphere.
The resistivity values of all components for both sites show dis-
crepancies (site 10 for longer periods but site 18 for all periods).
Considering the distance of site 18 to the edge of the hemisphere
(100 m) and the 300 m of the hemisphere, the edge of the hemi-
sphere should be visible for periods longer than approximately
0.0001 s and therefore f or the whole periods range as observed. Site
10 is 500 m away and, considering the 10 m of the hemisphere, pe-
riods longer than approximately 1 s should be affected. The reason
why the responses differ for the various results is the discrepancy
in the discretization of the hemisphere using different meshes. As
slightly different meshes and approximations cause a different dis-
tortion, each response is basically statically shifted differently for
site 18 (i.e. outside the hemisphere). This shift is strongest for the
responses obtained using the coarsest, rectilinear meshes (500-m
cell width compared to 250 m and less), that is, the response from
B
¨
orner, Farquharson and H
¨
ubert. (In rectilinear meshes, a step-like
realization of topography can introduce a similar static shift that
decreases with decreasing cell size.) In general, it seems that the
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

1222 M. P. Miensopust et al.
Tabl e 4 . List of DTM2 forward results. The code and its type (finite-difference FD or finite-element FE) as well as the user are specified. Further more the used
mesh (i.e. number of cells in each direction and—if available—in parentheses the lateral width of the centre cells), required CPU time (and target tolerance TT
or achieved tolerance AT) and the specification of the used computer are listed.
Code User Mesh (# of cells) CPU time & tolerance AT/TT Computer used
FD (B
¨
orner, pers. comm.) Ralph-Uwe B
¨
orner 129 600 (500 m) 45 min AMD Opteron6136 (2 CPUs each 4 cores,
(parallel/direct solver PARDISO) 2.4 GHz, 64 GB)
FD (Egbert & Kelbert 2012) Naser Meqbel 96 × 96 × 45 (125 m) 35 min parallel/51 processors
FD (Mackie et al. 1994) Duygu Kiyan & 68 × 69 × 40 (∼250 m) 2 h 15 min Dell 2 CPUs E5420@2.5 GHz, 3.25 GB
(serial) David Khoza (+ 1-D base model) TT: 10
−8
(serial) Florian Le Pape 66 × 64 × 42 (∼250 m) 2 h Dell 2 CPUs E5420@2.5 GHz, 3.25 GB
(+ 1-D base model) TT: 10
−8
(parallelization with PETSc: Randall Mackie 85 × 85 × 59 2 h Dell 690 workstation, Intel Xeon quad-core
distributes linear system/not TT: 10
−10
= rtol 5150@2.66 GHz, 4 GB
exactly domain decomposition) (|r| < rtol ·|b| with r = b − Ax;
b = right hand side; all preconditioned)
(serial) Marion Miensopust 72 × 72 × 30 (250 m) 2 h Dell 2 CPUs 6300@1.86 GHz, 4GB
(+ 1-D base model) TT: 10
−8
FD (Siripunvaraporn et al. 2002) Juliane H
¨
ubert 45 × 45 × 23 (500 m) 15 h desktop: 2.6GHz, 3.5GB
(serial)
FD (Siripunvaraporn & Egbert 2009) Jan Vozar 153 × 153 × 45 12 h 24 CPUs on ICHEC cluster Stokes
(frequency parallelization) TT: 10
−7
FE (Farquharson & Miensopust 2011) Colin Farquharson 70 × 70 × 43 (500 m) 2 h per period (full impedance estimation) 2 to 3 GB
(serial) AT: from 0.01 (short periods)
to 10
−15
(long periods)
FE (Franke et al. 2007) Antje Franke-B
¨
orner ∼300 000 elements 6 h AMD Opteron6136 (2 CPUs each 4 cores,
(parallel/direct solver PARDISO) 2.4 GHz, 64 GB)
IE (Avdeev et al. 2002) Dmitry Avdeev 249 × 249 × 32 (∼40 m) 56 h Intel Core i7-2630@2GHz,4GB (laptop)
(serial) (mesh finer than needed) TT: 0.0003 > ||b − Ax||/||b||
(system of linear equations: Ax = b)
(serial) Anna Avdeeva 120 × 120 × 28 (83.3 m) 10 h Adakus cluster (Intel Xeon
TT: 0.001 ≥||b − Ax||/||b|| X5570@2.93 GHz, 24 GB)
(system of linear equations: Ax = b)
shift for sites inside the hemisphere (on the conductive side; e.g.
site 10) tends to overestimation of the resistivity values, whereas
outside (on the resistive side; e.g. site 18) resistivities are, by trend,
underestimated.
3INVERSION
The inversion tasks were of two different types; first all participants
were asked to invert the data obtained from the forward model, that
is, a known structure, and secondly, to deal with a data set of an
unknown structure, as it would be the case for real field data. In the
following we will briefly describe the different inversion algorithms
that were used, and then present a comparison of the inversion results
for each of the models.
3.1 Different algorithms and approaches
As the forward solver is the driving engine of an inversion algo-
rithm, the inversion codes are also based on FD, FE and IE methods.
For the inversion task of both workshops only FD (Mackie et al.
2001; Siripunvaraporn et al. 2005; Hautot et al. 2007, 2011; Siripun-
varaporn & Egbert 2009; Egbert & Kelbert 2012) and IE (Avdeev
& Avdeeva 2009; Avdeeva et al. 2012) codes were used. The 3-
D magnetotelluric inverse problem is far from being solved, but
these codes using different inversion schemes showed that one can
recover the conductivity structure of simple, synthetic test models
reasonably well, to greater or less degrees depending on resolu-
tion and sensitivity. The major issues of making MT 3-D inversion
a r outinely applied procedure are (i) the long computation time
and (ii) the requirement of fast workstations with large memories,
preferably even huge clusters for parallel computation. The time-
wise most expensive part is the construction of the sensitivity ma-
trix. To avoid this time consuming procedure, diff erent schemes
were developed that use approximations of the sensitivity matrix
(e.g. approximate sensitivities from a homogeneous or 1-D/layered
half-space, quasi-linear approximation of Green’s functions). As
such approximate methods have their limitations (e.g. work best for
small resistivity contrasts, and the accuracy of the inversion is ques-
tionable), they are of value but cannot replace methods based on the
full solution of the EM induction equations. Methods such as the
conjugate gradient (CG) or non-linear CG (NLCG) (e.g. Mackie
et al. 2001; Hautot et al. 2007, 2011; Egbert & Kelbert 2012)
and Quasi-Newton (QN) methods (e.g. Avdeev & Avdeeva 2009;
Avdeeva et al. 2012) avoid both the explicit computation and storage
of a full sensitivity matrix. Only the gradient vector of the objective
functional is needed. Other codes are based on a Gauss-Newton-
type (GN) approach in model or data space (e.g. Siripunvaraporn
et al. 2005; Siripunvaraporn & Egbert 2009). Some of these codes
are written using OPENMP or the message passing interface (MPI)
protocol or similar to make them run on PC-clusters or massively
parallel systems in order to reduce the computation time. In the
review paper by Siripunvaraporn (2012), the different approaches
are described in more detail (also see the individual references of
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

MT 3D forward and inversion comparison 1223
Figure 4. Comparison of the obtained DTM2 responses. From top to bottom each row shows the xx, xy, yx and yy components (i.e. apparent resistivity and
phase curves). The left column shows the responses at site 10 whose location is indicated by the red dot in the small sketch on the top left (inside the hemisphere;
at surface 500 m away from the edge). Similarly, the red dot in the sketch on the top right indicates the location of site 18 (outside the hemisphere, at surface
100 m away from the edge), whose responses are shown in the right column. The grey dashed line represents CZ and the solid black line the analytic solution
calculated by David Bosch (after Groom & Bailey 1991). (Note that both are indeterminate for diagonal elements at sites inside the hemisphere.) Blue symbols
represent responses obtained by FD codes, green symbols those by FE codes and the IE responses are in red. More details can be found in Table 4 and in the
text.
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

1224 M. P. Miensopust et al.
each code) and a comparison of their advantages and disadvantages
is given.
One important part of inverse numerical modelling is to deter-
mine the measure of the misfit. The sum of the measures of data
misfit and model complexity is called an objective function (or a
penalty function). These measures are often defined by an L-norm,
but also non-standard norms and measures are used. The objective
function is the sum of these measures balanced, or ‘regularized’, by
a trade-off parameter and therefore often is called Tikhonov regu-
larized penalty function. It is this function that will be minimized
using CG, NLCG, GN or QN methods.
3.2 Dublin Test Model 1 (DTM1)
In Section 2.2 (see also Fig. 1 and Table 1), this model was described
in detail. To illustrate the range of data, Fig. 5 shows pseudo-sections
of resistivity and phase values of all four components along the
centre profile using the data set calculated by Miensopust utilising
the code by Mackie et al. (1994). The diagonal resistivity and phase
values are blanked in these pseudo-sections if the resistivity of the
component is less than a threshold value of 10
−5
m(seeabove).
Although the model is relatively simple, the pseudo-sections show
increasing resistivities for the diagonal elements that are only a
few orders of magnitude smaller than the off-diagonal resistivities.
Already the pseudo-sections indicate that there is a more conductive
structure a few kilometres below the surface and another one deeper
and shifted to the right (positive side of the y axis). Mackie inverted
the synthetic data using the FD code of Mackie et al. (2001). Table 5
lists the different settings and mesh parameters that were used for
the inversion, as well as information regarding r un time and final
misfit. As expected, his inversion result (see Fig. 6) recovers the
lateral extent and the top of the shallow conductor (body 1 in Fig. 1)
very well. The lower conductor (body 2 in Fig. 1) is recognizable
in section x =−7.5 km; the resistive structure is also indicated
in the model but not resolved as well as the conductors. It is a
general problem that MT data, as with all inductive EM techniques,
are more sensitive to conductive structures than to resistive bodies
and therefore, while the top of a conductor is well defined, its base
is usually smeared out. It is also nearly impossible to reproduce
the extreme resistivity contrasts of the true model using a smooth
inversion approach, as undertaken here. Rodi & Mackie (2012)
used this model to discuss 3-D MT inversion, and illustrate the
effect of the regularization parameter resulting in models that vary
from overly smooth to extraneous structures. Another challenge of
this inversion was the sparse distribution of sites mainly along the
edges of the bodies rather than across the structure.
Fig. 7 shows a comparison of Mackie’s forward (red) and in-
version (blue) responses. The overall fit is good and therefore in
agreement with the achieved final RMS misfit value of 1. Never-
theless, while RMS = 1 suggests that everything is well fit, looking
at the comparison of the response curves for this example site (at
x = y = 0 km) clearly illustrates that some periods and components
are fit better than others, and there is serial correlation in the resid-
uals. So, ideally one should not just rely on this single number but
more carefully examine t he misfit over all data space.
3.3 Dublin Test Model 2 (DTM2)
In Section 2.3 (see also Fig. 3 and Table 3) the model was already
described. Note that the inversion results obtained are based on data
that the person calculated her-/himself during the forward exercise,
Figure 5. Pseudo-section of DTM1 along the Centre profile (see Fig. 1)
using exemplary the data set calculated by Miensopust utilizing the code
by Mackie et al. (1994). Top panels show the resistivity pseudo-sections
for all four components and phase pseudo-sections are below. Note that the
diagonal element values are blanked for resistivity and phase if the resistivity
values is less than a threshold value of 10
−5
m.
and therefore the data sets inverted are not identical for the results
showninFig.8.WhileH
¨
ubert and Le Pape both started from
a homogeneous half-space of 100 m and used the FD code by
Siripunvaraporn et al. (2005) and Siripunvaraporn & Egbert (2009),
respectively, Avdeeva obtained the model using her IE inversion
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

MT 3D forward and inversion comparison 1225
Tabl e 5. List of DTM1 inversion results. Beside code and user, information about the used data (sub)set, mesh and inversion parameters are given as well as
the number of iterations, final RMS value and target tolerance (TT).
Code/User Info Data Mesh
FD (Mackie et al. 2001) 73 NLCG iterations all 59 sites 75 × 96 × 45 cells
Randall Mackie RMS 1 8 components + tipper 158 km × 178 km × 128 km
(parallelization with PETSc: 30 h on cluster 2 periods per decade → 11 1 km × 1km× 100 m centre cells
distributes linear system/not TT: 10
−10
= rtol 2 per cent error floors (0.003 on Tz) (vertical extent increasing with depth)
exactly domain decomposition) (|r| < rtol ·|b| with r = b − Ax; starting model: 100 m half-space
b = right hand side; all preconditioned)
Figure 6. Inversion result obtained for the Dublin Test Model 1 (DTM1).
The left column shows slices of the true model. On the right, the same slices
are plotted for the inversion model that Randall Mackie obtained using the
code by Mackie et al. (2001) with the settings specified in Table 5.
code (Avdeev & Avdeeva 2009; Avdeeva et al. 2012) starting from
the correct half-space resistivity of 300 m. More details about
the adopted meshes and data sets, as well as inversion settings,
can be found in Table 6. All three inversion results recover the
circular shape of the hemisphere well, and all models also indicate
the correct order of resistivity for the hemisphere. Similar to the
forward results in Section 2.3, rectangular meshing also limits for
the inversion the correct representation of the round structure. As
all inversion codes used are based on a smooth model approach, the
edges of the hemisphere are not represented by a sharp boundary, as
in the true model, but by a gradual change in resistivity resulting in
a more smeared out structure. Fig. 9 shows a comparison of forward
(red) and inversion (blue) response curves for site 10 and 18 (i.e.
those discussed in Section 2.2). The columns show a comparison for
the inversion data by Avdeeva, H
¨
ubert and Le Pape, respectively,
each with respect to their own forward data. Although all three
obtained an overall RMS misfit of about 1 (see Table 6), Fig. 9
clearly shows how different the fit of those responses curves are. In
general, it seems that for all three the data of site 18 (outside the
Figure 7. DTM1 comparison of forward versus inversion responses. Pre-
sented in red are the forward responses obtained by Randall Mackie at the
central site (x = y = 0 km) and in blue his inversion results. Note that not
all periods were used for the inversion and the error floor was set to 2 per
cent of the off-diagonal values.
hemisphere) is fit better than for site 10 (inside the hemisphere).
While Le Pape’s results fit the shor t periods well and have a bit of
a shortcoming on the longer periods, t he data obtained by Avdeeva
shows the opposite behaviour. H
¨
ubert only used short periods, but
even for those it is obvious that the fit is different for different
components and periods. Once again it becomes clear that a single
number RMS misfit is not a satisfying way to represent data fit.
3.4 Dublin Secret Model 1 (DSM1)
3.4.1 Model and data description
Inversion codes are commonly developed and tested by inver ting
synthetic data from a known structure, but later they will be used
to invert real data where the true structure is unknown. To investi-
gate how much the inversion results depend on knowledge of the
structure, and therefore, the specifically chosen mesh and parameter
settings, we provided a synthetic data set for inversion where the
original structure was not known to the participants. (Data s et avail-
able as supplementary file.) Additional to the data set, the following
information was given to them:
(1) 100 sites on 10 profiles covering a 50 km by 50 km area (equal
site spacing between sites and profiles of 5 km),
(2) data set covers period range from 0.56 to 10 000 s (18 periods),
(3) expected resistivity range of the structure is 1–1000 m,
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

1226 M. P. Miensopust et al.
Figure 8. Comparison of the inversion results obtained for the Dublin Test Model 2 (DTM2). From left to right, true model and the inversion results from Anna
Avdeeva, Juliane H
¨
ubert and Florian Le Pape are shown. The black dots represent the used sites. Table 6 lists the different settings and parameters chosen to
obtain the represented models. Note that Anna Avdeeva’s model covers a smaller area than the others resulting in the blank area around her model.
(4) impedance values are given in SI units (i.e. ) and are based
on the e
+iωt
convention,
(5) used coordinate system is x pointing North, y pointing East
and z positive downwards,
(6) how many and which sites and periods are used for the inver-
sion was up to individual choice.
The synthetic data were distortion- and noise-free and data errors
were not specified. The structure of the model was motivated by
finding a synthetic model that results in relatively large diagonal
element values, which is not the case for most synthetic data sets. In
addition, we were seeking a structure that is different to the block-
like structures of most other test models. The chosen structure is a
spiral like conductor of 1 m, which becomes thicker and wider
with depth (see Fig. 10). Its top is at 4 km depth and both of its ends
extend to infinity (i.e. the edge of the forward modelling mesh).
Fig. 11 shows resistivity and phase pseudo-sections for all four
components at y =−17.5 km (across the left part of the spiral).
It is obvious that the diagonal resistivities are relatively large—
sometimes even of the same order than the off-diagonal elements.
Therefore, the diagonal elements contain information and should
not be ignored or rejected during the inversion.
3.4.2 Results
Three 3-D inversion models (Avdeeva using her code (Avdeev &
Avdeeva 2009; Avdeeva et al. 2012), Hautot using her code (Hautot
et al. 2007, Hautot et al. 2011) and Miensopust using the code by
Siripunvaraporn et al. (2005)) and one model merged from 1-D
models of the determinant at each site (Queralt et al.), are shown in
Fig. 12. Note that Queralt (and colleagues) and Miensopust did not
apply any data analysis, decomposition nor dimensionality check
and used default inversion parameters to avoid any manipulation of
the inversion result as the true model was designed by (and therefore
known to) them. Table 7 lists the different inversion parameters and
information about the meshes and data sets used.
The 3-D inversion models recover the near-surface structure well.
The deeper part of the structure is more diffuse and laterally shifted.
Even the 1-D approach indicates the correct near surface structure,
although the conductor is more resistive. Similar to the hemisphere
model, the spiral also has a round shape that had to be approximated
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

MT 3D forward and inversion comparison 1227
Tabl e 6. List of DTM2 inversion results. Beside code and user, information about the used data (sub)set, mesh and inversion parameters are given as well as
the number of iterations, final RMS value and target tolerance (TT).
Code/User Info Data Mesh
FD (Siripunvaraporn et al. 2005) 2 iterations all 49 sites 45 × 45 × 23 cells (same as for forward)
Juliane H
¨
ubert RMS 0.9502 only periods < 56 s 80 km × 80 km × 20 km
(serial) 14 h error floor 5 per cent of main impedance∼500 m ×∼500 m × 500 m centre cells
(σ Z
xx
= σ Z
xy
, σ Z
yx
= σ Z
yy
) (vertical extent increasing with depth)
starting model: 100 m half-space
FD (Siripunvaraporn & Egbert 2009) 3 iterations all 49 sites 66 × 64 × 42 cells (same as for forward)
Florian Le Pape RMS 1.1594 24 periods 893 km × 893 km × 434 km
(frequency parallelization) 20 h 8 components ∼250 m ×∼250 m × 200 m centre cells
24 CPUs on ICHEC cluster Stokes error floor 5 per cent (vertical extent increasing with depth)
TT: 10
−7
(forward), 10
−4
(sensitivities) starting model: 100 m half-space
IE (Avdeev & Avdeeva 2009) RMS 1 33 sites (0 to 32) 54 × 54 × 16 cells
(Avdeeva et al. 2012) 77 h 0.1, 0.32, 1, 3.2, 10, 32, 100 s 13.5 km × 13.5 km × 10 km
Anna Avdeeva Abakus cluster ↓ 250 m × 250 m × 500 m cells
(serial) (Intel Xeon X5570@2.93 GHz, 24 GB) 0.032, 0.179, 1, 5.59, 32, 179 s (vertical extent increasing with depth)
filter parameters a
x
=a
y
= 3 → 8 → 3 error floor 10 per cent of |Z| starting model: 300 m half-space
regularization: gradient
(from 10
−3
to 10
−6
)
TT: 0.003 ≥||b − Ax||/||b||
(system of linear equations: Ax = b)
Figure 9. DTM2 comparison of forward versus inversion responses. For the example sites 10 (left) and 18 (right), the data of the inversion (red) obtained by
Anna Avdeeva, Juliane H
¨
ubert and Florian Le Pape, respectively, are shown in comparison to their forward responses (blue). Although all three obtained an
overall RMS misfit of about 1 for their inversion, it is obvious how different the fits are for the different results but also for different periods and components.
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

1228 M. P. Miensopust et al.
Figure 10. Sketch of Dublin Secret Model 1 (DSM1). The 1 m spiral (red)
is embedded in a homogeneous, 100 m half-space. Its top is 4 k m below
surface and its width and thickness increases with depth to a maximum
depth of 45.5 km. Both ends extend to infinity, i.e. to the edge of the forward
modelling mesh.
by rectangular meshes. It would be interesting to see an inversion
using an unstructured grid which probably could reproduce the
boundaries a bit better. Nevertheless, the obtained inversion results
are promising. Fig. 13 shows pseudo-sections (y =− 17.5 km) of
the response data obtained by the three 3-D inversions (i.e. Avdeeva,
Hautot and Miensopust). (Note as Queralt et al. used a 1-D Occam
inversion of the determinant only to s ee if such a 1-D approach can
give a first idea of the structure, no pseudo-sections are shown for
their result.) The three pseudo-sections are in a good agreement
with the true data (see Fig. 11), although none of the inversions
was able to recover the true structure well. All three inversions
resulted in similar low RMS misfit values (<2, some even around
1), which is an average of the fit of all data points (i.e. all periods
and sites). However, inspecting the pseudo-sections clearly shows
that differences in the models go back to differences in the fit
of each individual data point. As all modellers used all sites, the
lateral data distribution is identical for all three models (and also
pseudo-sections). While Hautot and Miensopust used (nearly) all
periods (17 and 18, respectively), Avdeeva only included half (9)
of the periods. For the pseudo-sections, as well as for the inversion
models, it seems that the inversion result did not improve with
increased number of periods. This might be an effect of the different
sensitivities due to diverse inversion algorithms and parameters
applied. Note that in this case a reduction in the number of sites
would most likely lead to loss of information, and therefore, a worse
result. It is obvious that the deeper part of the structure is shielded by
the shallower conductive structure. In this case, the resolution of the
deeper structure is not improved by using more periods per decade
and even a coarse and simple mesh (e.g. the one by Hautot) is able to
represent the general structure well. Therefore, a reduced number of
periods and/or a simple mesh can be efficient to obtain a sufficient
result in a short time and possibly even on a standard laptop.
3.5 Dublin Secret Model 2 (DSM2)
3.5.1 Model and data description
As for the first workshop, we provided again a synthetic data set of
unknown structure prior to the second workshop. (Data set available
as supplementary file.) The additional information given was:
Figure 11. Pseudo-section of DSM1 at y =−17.5 km. Top panels show
the resistivity pseudo-sections for all four components and phase pseudo-
sections are below. Note that for some sites and periods, the resistivity values
of the diagonal elements are of the same order as those of the off-diagonal
elements.
(1) 144 sites on 12 profiles covering approximately a
80 km by 80 km area (equal site spacing between sites and pro-
files of 7 km),
(2) data set covers period range from 0.016 to 10 000 s (30
periods),
(3) expected resistivity range of the structure is 0.1–100 m,
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

MT 3D forward and inversion comparison 1229
Figure 12. Comparison of the inversion results obtained for the Dublin Secret Model 1 (DSM1). From left to right, the true model and the inversion results
from Anna Avdeeva, Sophie Hautot, Marion Miensopust and Pilar Queralt (and colleagues) are shown. The black dots represent the used sites. Note that the
model by Queralt et al. is not a 3-D inversion model but projection of 1-D models at each site. More details on the used data and mesh as well as information
about the inversion parameters can be found in Table 7. White areas indicate where the model extent is less than the plotted area.
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

1230 M. P. Miensopust et al.
Tabl e 7. List of DSM1 inversion results. Beside code and user, information about the used data (sub)set, mesh and inversion parameters are given as well as
the number of iterations, final RMS value and target tolerance (TT).
Code/User Info Data Mesh
FD (Hautot et al. 2007, 2011) RMS 1.9 8 components 12 × 12 × 12 cells
Sophie Hautot 3–4 d all 100 sites 65 km × 65 × 178 km
standard laptop 17 periods (without shortest) 5 km × 5km× 500 m centre cells
2 per cent error floor for off-diagonal (vertical extent increasing with depth)
5 per cent error floor for diagonal starting model: 100 m half-space
(or 20 per cent if diagonal 
a
< 0.2)
no decomposition
FD (Siripunvaraporn et al. 2005) 3 iterations 8 components 44 × 44 × 29 cells
Marion Miensopust RMS 0.9703 all 100 sites 190 km × 190 km × 1130 km
(serial) approx. 18 d all 18 periods 1–2 km × 1–2 km × 100 m centre cells
AMD Opteron CPU 2.193GHz, 16 GB no decomposition (vertical extent increasing with depth)
default inversion parameter settings no dimensionality check starting model: 100 m half-space
TT: 10
−7
(forward), 10
−4
(sensitivities) no data analysis
IE (Avdeev & Avdeeva 2009) RMS 1 8 components 40 × 40 × 15 cells
(Avdeeva et al. 2012) 50 h all 100 sites 200 km × 200 km × 100 km
Anna Avdeeva Abakus cluster 2 periods per decade → 95km× 5km× 1 km cells
(serial) (Intel Xeon X5570@2.93 GHz, 24 GB) 5 per cent error floor (vertical extent increasing with depth)
filter parameters a
x
=a
y
= 2 starting model: 100 m half-space
regularization: gradient
(from 10
−5
to 10
−7
)
TT: 0.003 ≥||b − Ax||/||b||
(system of linear equations: Ax = b)
Occam 1-D (Constable et al. 1987) 1-D models determinant (
a
and φ) 3-D model constructed from 1 D results
Pilar Queralt, Juanjo Ledo 19 layers (from 0–60 km) all sites 10 × 10 cells (5 km × 5km)
& Anna Mart
´
ı <1 min (standard laptop) all 18 periods each cell represents the 1-D model of one site
1-D starting model: 100 m
Figure 13. Pseudo-section of the inversion results of DSM1 at y =−17.5 km. Only resistivity and phase pseudo-sections of the off-diagonal elements are
shown (compared with true data pseudo-section in Fig. 11) for the results of all three 3-D inversions (from left to right: Avdeeva, Hautot, Miensopust). As the
1-D approach by Queralt et al. uses the determinant of the tensor, their resulting data cannot be shown here.
(4) impedance values are given in field units (i.e.
mV
km
/nT )and
are based on the e
+iωt
convention,
(5) used coordinate system is x pointing North, y pointing East
and z positive downwards,
(6) how many and which sites and periods are used for the inver-
sion was up to individual choice.
The data set was calculated using a model on the basis of the
COMMEMI 3D-2A model (Zhdanov et al. 1997) but a 1-km thick,
near-surface cover layer of 50 m was introduced to avoid numer-
ical problems and effects due to the outcropping structures. While
the COMMEMI layered background was kept below the cover layer,
the lateral dimensions of the two blocks were modified ( see Fig. 14).
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

MT 3D forward and inversion comparison 1231
Figure 14. Sketch of Dublin Secret Model 2 (DSM2). A section (top) and
a plan view (bottom) show the structures of DSM2. Black crosses in the
plan view and triangles in the section view indicate the site locations. This
model is on the basis of the COMMEMI 3D-2A model by Zhdanov et al.
(1997) but with slightly modified lateral dimensions of the two blocks and
an additional cover layer.
Then random galvanic distortion was applied to the synthetic data
set. Therefore, the distor tion matrix C was calculated according to
the Groom–Bailey Decomposition (Groom & Bailey 1989) from
randomly generated values for the twist angle (within ±60
◦
), the
shear angle (within ±45
◦
) and the anisotropy ( within ±1). The gain
value was fixed to be equal to one at all locations. Finally, random
Gaussian noise of 5 per cent of the maximum impedance value was
applied to the distorted data set. The given errors were determined
as 5 per cent of the maximum values of the absolute impedance
values for each period. Fig. 15 shows resistivity and phase pseudo-
sections for all four components. The left column represents the
synthetic data at x = 3.5 km before and the right column after the
distortion and Gaussian noise have been applied. Applying distor-
tion and noise enhances the diagonal resistivity values by orders of
magnitude (note that the colour scale of undistorted and distorted
diagonal resistivities is different).
3.5.2 Results
Table 8 summaries the various inversion codes used, mesh and data
selection and inversion parameters of the models shown in Fig. 16.
One model was obtained using an IE code (modified version of
Avdeev & Avdeeva 2009; Avdeeva et al. 2012), all others were
Figure 15. Pseudo-section of DSM2 at x = 3.5 km. The left column shows
the resistivity and phase pseudo-sections for all four components based on
the data as they were obtained by forward modelling. The right column
represents the same pseudo-sections after distortion was applied to the data
and 5 per cent Gaussian noise was added.
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

1232 M. P. Miensopust et al.
Tabl e 8. List of DSM2 inversion results. Beside code and user, information about the used data (sub)set, mesh and inversion parameters are given as well as
the number of iterations, final RMS value and target tolerance (TT).
Code/user Info Data Mesh
FD (Egbert & Kelbert 2012) (run A/run B)(run A/run B)(run A/run B)
Naser Meqbel & Gary Egbert 262/74 iterations 137 sites 50 × 50 × 40/70 × 70 × 48 cells
RMS 1.04/2.5 15 periods 100/880 km × 100/880 km × 367/914 km
13/6 h on 31 cores data error = variance/standard deviation 2/3.5 km × 2/3.5 km × 50/5 m centre cells
(vertical extent increasing with depth)
starting model: 50 m half-space
FD (Hautot et al. 2007, 2011) RMS 4.4 all 144 sites 16 × 16 × 12 cells (2298 unknowns)
Sophie Hautot & Pascal Tarits approx. 3 d all periods (cells for layers 10 -12 grouped)
(desktop PC: 2.5 GHz, 3GB) no static shift correction 133 km × 133 km × 140 km
2 per cent error floor for off-diagonal 7 km × 7km× 60 m centre cells
5 per cent error floor for diagonal (vertical extent increasing with depth)
(or 20 per cent if diagonal 
a
< 0.2) starting model: 10 m half-space
no decomposition
FD (Mackie et al. 2001) (No statics/statics) 142 sites 96 × 96 × 67 cells
Randall Mackie RMS 1.352/1.408 8 components 178 km × 178 km × 107 km
(parallelization with PETSc: NLCG iterations: 62 (10 periods) ever y 2nd/3rd period → 10/15 1 km × 1km× 20 m centre cells
distributes linear system/not → 11 (15 periods) → (starting at 63.1 Hz) (vertical extent increasing with depth)
exactly domain decomposition) 20 (lower τ , 15 periods)/59 (10 periods) 5 per cent error floor on lnZxy & lnZyx starting model: 10 m half-space
36/14 h on cluster 20 per cent er ror floor on Zxx & Zyy
TT: 10
−10
= rtol
(|r| < rtol ·|b| with r = b − Ax;
b = right-hand side; all preconditioned)
FD (Siripunvaraporn et al. 2005) 2 iterations 8 components 44 × 44 × 33 cells
Xavier Garcia RMS 4.3104 50 sites 172 km × 172 km × 102 km
(serial) default inversion parameter settings 3 periods per decade → 16 2.33 km × 2.33 km × 50 m centre cells
TT: 10
−7
(forward), 10
−4
(sensitivities) (only periods ≤ 1s or > 10s) (vertical extent increasing with depth)
no decomposition starting model: 50 m half-space
no dimensionality check
no data analysis
FD (Siripunvaraporn & Egbert 2009) (run A/run B)(run A/run B)54× 54 × 30 cells
Marion Miensopust 4/6 iterations 8 components 260 km × 260 km × 326 km
(frequency parallelization) RMS 3.036/4.63 64 sites (shifted to cell centre) 2 km × 2km× 50 m centre cells
default inversion parameter settings every second period (≥0.063s) → 14 (vertical extent increasing with depth)
TT: 10
−7
(forward), 10
−4
(sensitivities) data error = variance/standard deviation starting model: 100 m half-space
no decomposition
no dimensionality check
no data analysis
FD (Siripunvaraporn et al. 2005) 13 iterations 4 components 20 × 20 × 45 cells
Oriol Rosell/Pilar Queralt RMS 3.1 all 144 sites 7 km × 7km× 20 m centre cells
(serial) default inversion parameter settings 2 periods per decade → 12 (vertical extent increasing with depth)
TT: 10
−7
(forward), 10
−4
(sensitivities) no decomposition starting model: 100 m half-space
no dimensionality check
no data analysis
FD (Siripunvaraporn & Egbert 2009) 1 iteration 4 components 62 × 62 × 33 cells
Weerachai Siripunvaraporn RMS 2.9925 all 144 sites 230 km × 230 km × 520 km
(frequency parallelization) 16 periods 1.63 km × 1.63 km × 10 m centre cells
error floor 10 per cent of |Z
xy
Z
yx
|
1
2
(vertical extent increasing with depth)
‘triage’ and removal of bad data starting model: 1 m half-space
(various models tested → smallest RMS selected)
FD (Siripunvaraporn & Egbert 2009) (run A/run B) 8 components 50 × 50 × 50 cells
Jan Vozar 3/4 iterations 72 sites (different ones for A & B) 2795 km × 2795 km × 2275 km
(frequency parallelization) RMS 3.743/3.8164 24 periods 3.5 km × 3.5 km × 50 m centre cells
16/13 h on cluster error floor 5 per cent on impedance (vertical extent increasing with depth)
(24 CPUs on Stokes/ICHEC) starting model: 100 m half-space
default smoothing parameters
TT: 10
−7
(forward), 10
−4
(sensitivities)
IE (Avdeev & Avdeeva 2009) RMS 0.7 8 components 43 × 43 × 14 cells
(Avdeeva et al. 2012) 152 h (Abakus cluster) all 144 sites 150 km × 150 km × 70 km
Anna Avdeeva filter parameters a
x
=a
y
= 3 3 per decade (≤251s) → 10 3.5 km × 3.5 km × 250 m cells
(modified to invert for regularization: gradient ↓ (vertical extent increasing with depth)
full distortion matrix) (from 10
−6
to 10
−8
) 3 per decade (≤3981s) → 13 starting model: 2 m half-space
(serial) TT: 0.003 ≥||b − Ax||/||b|| error floor 10 per cent of |Z|
(system of linear equations: Ax = b)
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

MT 3D forward and inversion comparison 1233
Figure 16. Comparison of the inversion results obtain for the Dublin Secret Model 2 (DSM2). Various inversion results are shown in comparison to the true
model. The black dots represent the used sites. More details on the used data and meshes as well as information about the inversion parameters can be found
in Table 8. Note that the inversion results by Anna Avdeeva and Randall Mackie (Statics) consider the distortion during the inversion as full distortion matrix
and static shift , respectively, while all other results are obtained by inverting for the resistivity structure only. A more detailed discussion of the difference can
be found in the text. (Figure continued opposite)
calculated using various FD codes (i.e. Mackie et al. 2001; Siripun-
varaporn et al. 2005; Hautot et al. 2007, 2011; Siripunvaraporn &
Egbert 2009; Egbert & Kelbert 2012). Although the data set pro-
vided also contained information about errors (standard deviation),
they were labelled as variance in the file in error, which led to not
identical errors used for the various inversions. Some participants
used the values as standard deviation (i.e. used the intended er rors),
others understood the values as variance (i.e. variance = (standard
deviation)
2
, and therefore got unrealistically large errors for long
periods) and others calculated errors themselves or used large error
floors for all periods. These differences in errors have, of course,
effects on the inversion results, especially whether the deeper struc-
tures are resolved or not. The runs A and B by Meqbel and Egber t
(slightly different mesh for the two runs) and Miensopust (mesh
and inversion settings identical for both runs) show the different
models obtained by using the values as variance (run A) or standard
deviation/correct er rors (run B). This comparison clearly shows
that the large errors (when using errors as variance) blank out any
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

1234 M. P. Miensopust et al.
Figure 16. (Continued.)
deep structure in favour of the shallower structure (see sections for
the shallow conductors). Therefore, an unresolved lower conductive
layer has nothing to do with the inversion algorithm nor the mesh
nor inversion parameters, but is caused by the choice of data errors.
Although Avdeeva used variance, she obtained the conducting layer
at the bottom, but only when overfitting the data (i.e. RMS < 1) as
the long periods have large error bars and therefore have a RMS
value below 1.
Table 8 also shows that the diagonal elements were treated dif-
ferently during the inversion. For example, Siripunvaraporn and
Rosell/Queralt ignored them and inverted the off-diagonals only;
others (e.g. Mackie) used a higher error floor for the diagonals
than on the off-diagonals and Hautot defined a threshold value for
the diagonal values below which she assigned an larger error than
above. To date, there are only very limited and contradictory ideas
about the importance of the diagonal elements for 3-D inversion.
Hence, there is also no conclusion yet how to treat them during
inversion. The comparison shown here cannot promote one or the
other approach, and therefore this topic remains a subject for further
investigation.
Another challenge of this data set is the distortion applied. Most
inversions ignore distortion effects, or, at most, bad data point
elimination was conducted or s ites rejected (compare ‘Data’ in
Table 8) but no systematic distortion analysis or removal was con-
sidered. Avdeeva used a modified version of her code that allows
inversion for structure as well as for the full distortion matrix.
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

MT 3D forward and inversion comparison 1235
Figure 17. Pseudo-sections of the inversion results of DSM2 at x = 3.5 km. For all inversion results shown in Fig. 16, the pseudo-sections of the off-diagonal
resistivities and phases (from top to bottom for each set of pseudo-sections: 
xy
, 
yx
, φ
xy
, φ
yx
) are shown in comparison to the true data pseudo-section (i.e. data
with distortion and added noise as it was provided). Note that those pseudo-sections are based on different numbers of sites and periods used for the inversion
(numbers are specified for each pseudo-section set).
Therefore, the objective function has an additional trade-off term as
constraint for the distortion and a modified data misfit term consid-
ering the distortion matrix. Mackie provided two inversion results:
one with a normal inversion for resistivity structure only (No statics)
and one where not the full distortion matrix but the static shift is con-
sidered during the inversion process (Statics). (However, the statics
correction method applied was incorrect, as discussed by Jones
2011). The two models provided by Vozar only differ by the sites
selected for inversion. Although both results show similar general
structures, it is obvious that the selected sites influence the obtained
model. It seems that for this resistivity structure a chessboard-like
selection of every second site (run B) resolves the layered back-
ground better than selecting every second profile (run A). The pres-
ence of distortion and overestimating the conductivity of the near
surface conductor and underestimating its thickness is the possible
cause of a too shallow lower conductive layer/half-space (see e.g.
Hautot, Meqbel/Egbert (run B), Rosell/Queralt, Siripunvaraporn,
Vozar (run A & B)).
Fig. 17 shows a comparison of the off-diagonal resistivity and
phase pseudo-sections for all inversion results shown in Fig. 16 and
of the true data (i.e. data with distortion and added noise as it was
provided). Note that those pseudo-sections are based on different
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

1236 M. P. Miensopust et al.
numbers of data points for each model (i.e. different numbers of sites
and periods). Although the RMS misfit values (see Table 8) are all,
more or less, of the same order, once again the pseudo-sections show
clearly the difference in fit of the various data points. Most inversion
results seem to fit the phases quite well whereas the resistivity
values show more discrepancies. Of course, those differences are
also subject to the adopted error values and the chosen error floor
as well as to the selected data subset (i.e. inversion with or without
diagonal elements, rejection of some sites or periods). This once
more highlights the need for a better measure of data fit/misfit than
just a single number.
Interesting to note is that the models by Mackie (‘No statics’ and
‘Statics’) are very similar, but the pseudo-sections are very different.
Probably not only the distortion but also structural infor mation in the
data has been considered as static shifts by the inversion algorithm,
and therefore, the pseudo-sections are overly smooth compared to
the ‘No statics’ inversion and the true data.
In general, the differences in results depend on many things—
even if the same inversion code is used, differences can be caused
by:
(1) not only the number but also the distribution of sites
(compare, e.g. Garcia versus Rosell/Queralt, Vozar run A versus
run B),
(2) if diagonal elements are ignored (see Rosell/Queralt and
Siripunvaraporn) or used (all others),
(3) selection of errors and error floor; especially in this case,
variance versus standard deviation (see run A versus run B of Meq-
bel/Egbert and Miensopust).
Due to the different usage of the specified er r or values, it is
not possible to judge if more periods improve the results. (Hautot
was the only one using all periods whereas everyone else used
approximately half of them, but her good result could simply be
related to chosen errors/error floor.) A comparison of the results by
Garcia and Rosell/Queralt (using the same code) suggest that one
cannot gain anything with a finer mesh if only a few data values are
used.
Despite all differences, it is obvious that not the numerical algo-
rithms are the weak point but the way they are used, how the data are
prepared and selected and how the errors are estimated and used. It
seems that all codes returned sufficient results for what they were
‘fed’ with.
4 DISCUSSION
Although we tried to address a number of issues and were successful
at advancing understanding, many still remain unsolved. The exer-
cises emphasized once more that forward and especially inversion
algorithms must not be used in a black-box-like manner. Detailed
knowledge of the code and the parameters to choose is as essen-
tial as triaging the data used. The comparison of the results from
DSM2 showed how significant differences in the resulting inversion
models can be depending on the used data error and selected error
floors. Errors allow emphasis of s ome model parts and blank out
others; therefore they have to be chosen carefully.
It also became clear that a simple RMS misfit is not a satisfying
measure of how well the model fits the data. A single number can
clearly not represent the data fit and model roughness—not only in
3-D but also in 2-D—in a meaningful way. One approach might be
data misfit maps for each impedance tensor element and each period.
But no matter how it will be realized, a more sophisticated presen-
tation is required. It would also be beneficial for the community if
a common format for model (and data) files could be established
and would be used for all available codes. That would allow users
to compare results of and move their data and models between dif-
ferent algorithms in a straightforward manner. Currently, numerous
conversion scripts have to be used to go back-and-forth between
the different formats. But such scripts are always another possible
source of errors and mistakes. Perhaps even defining a standard
colour map could ease comparison of models from different people
and codes, that is, assuming the same maximum and minimum re-
sistivity values for the scale, the same colour would represent the
same resistivity value.
As we tried to address specific issues with synthetic models and
data sets, it is clear that test data sets should become more realistic,
for example, using a more realistic site distribution as in the field
hardly ever a perfect 3-D array with equidistant sites is possible or
using geological–tectonical driven models. It also seems beneficial
to follow the EarthScope example on a more global scale and make
measured MT data available for scientific use and code testing.
Under www.earthscope.org (last accessed 1 March 2013) one can
access the data portal, and also it is intended that data acquired
using the instruments of the Geophysical Instrument Pool Potsdam
(GIPP) will be available to the public 3 yr after completion of the
experiment (www.gfz-potsdam.de/gipp_client last accessed 1 March
2013). We consider this a good start, but more institutes and science
foundations should follow these examples.
Applying 3-D inversion codes to real data still raises the ques-
tion how to process and analyse the data before inversion. For 2-D
inversion it is standard practice to analyse distortion parameter and
strike direction and apply programs such as strike (by McNeice &
Jones 2001) to obtain 2-D responses. There are no such procedures
established yet for 3-D inversion. Determination of strike direction
may not be relevant for 3-D structures but distortion still remains a
problem (Jones 2011) and should be taken care of either prior to or
during the inversion (e.g. Miensopust 2010). Avdeeva used a simi-
lar approach to invert for the resistivity structure and the distortion
parameter for the DSM2 data set. Her obtained inversion model is
the first showing the successful application of such a joint inversion.
It is also known that 2-D isotropic inversion applied to data from
an anisotropic Earth can cause significant artefacts in the obtained
inversion model (Miensopust & Jones 2011). However, so far it is
unknown if adding one dimension will cause the same problem, or
if the effects become negligible or are even enhanced.
Using synthetic data the diagonal elements are often very small,
but in real data, even though they are a few orders of magnitude
smaller than the off-diagonal elements, they are far larger than for
the synthetic case. The cause of relatively large diagonal elements in
real data is presumably a consequence of imperfect data acquisition,
such as instrument noise and inaccuracy, external noise (e.g. wind
noise, artificial noise) and distortion. This raises another issue of
how diagonal elements should be treated in 3-D inversion. Different
ideas were discussed, from not using them at all to always including
them all. Probably the most sensible approach is to define a threshold
value below which they are rejected (i.e. considered to be below
experimental error) and above included in the inversion. But that
raises the next question of finding the optimal threshold value and if
it is data set dependent. Therefore, the issue of the diagonal elements
remains a subject to further investigation.
The models and data sets presented here all focused on the
impedance tensor information only and do not include geomag-
netic transfer function (also known as tipper) data or interstation
magnetic transfer function data. An increasing number of forward
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

MT 3D forward and inversion comparison 1237
modelling and inversion codes are now including the geomagnetic
transfer function data, which can provide valuable additional in-
formation about the subsurface structures. Also, other inversion
approaches, such as inverting for the phase tensor or invariants (the
latter taking all impedance tensor elements into account), are not
considered in the presented comparison but might provide a good
complement or even an alternative to the presented approaches.
5CONCLUSION
While Zhdanov et al. (1997) concluded in their report that the ma-
jority of 3-D forward algorithms belong to the class of IE methods
and that the differential equation methods revealed shortcomings
in theoretical and practical development, 15 yr later the situation
has clearly changed. These days the majority of 3-D codes in use
is based on the FD method and an increasing number on the FE
method. Only one IE code was presented. Clearly the increased
computational power (i.e. speed and memory) has promoted FD
codes. Although all three methods have their advantages and disad-
vantages, we found reasonable qualitative agreement between the
results. Also, the number of different forward codes presented in
comparison to the few 3-D inversion codes reflect clearly that devel-
opment in this field is still ongoing. The compilation of the forward
and inversion comparison illustrated that a number of issues in mod-
elling and inversion can be dealt with and solved sufficiently, but
there are also still various challenges to tackle. It is necessary to
continue the comparative calculations for 3-D modelling and in-
version. This comprehensive collection of test data sets and test
models in combination with all the responses and inversion models
obtained by various users and codes is a great opportunity to do so.
The COMMEMI models by Zhdanov et al. (1997)—still used by
many to test their codes—showed how valuable such tools are for
the entire community.
ACKNOWLEDGEMENTS
We thank all the other participants for making these two workshops
so successful: Masoud Ansari, Sebastien de la Kethulle de Ry-
hove, Hao Dong, Jimmy Elsenbeck, Mark Hamilton, Jelena Koldan,
Markus L
¨
ower, Jan Petter Morten, Laust Pedersen, Estelle Roux,
Stewart Sandberg, Jan Schmoldt, Toshihiro Uchida and Wenke Wil-
helms. The workshops were kindly financially supported by DIAS
(both workshops), EMGS (both workshops), WesternGeco (both
workshops) and Zonge (2nd workshop).
MPM was initially supported by a Science Foundation Ireland
grant to AGJ (05/RFP/GEO001) and subsequently by DIAS. PQ ac-
knowledges financial support from DIUE (Generalitat de Catalunya)
(2007BE-100200) and from MEC (Spanish Government) (PR2007-
0436/PR2011-0469) for her several visits at DIAS.
Last, but not least, we thank Weerachai Siripunvaraporn and one
anonymous reviewers for their helpful comments on the original
version of this manuscript that improved it.
REFERENCES
Avdeev, D.B., 2005. Three-dimensional electromagnetic modelling and in-
version from theory to application, Surv. Geophys., 26, 767–799.
Avdeev, D.B. & Avdeeva, A., 2009. 3D magnetotelluric inversion using
a limited-memory quasi-Newton optimization, Geophysics, 74(3), F45–
F57.
Avdeev, D.B., Kuvshinov, A.V. & Pankratov, O.V., 2002. Three-dimensional
induction logging problems, part I: an integral equation solution and
model comparisons, Geophysics, 67(2), 413–426.
Avdeeva, A.D., Avdeev, D.B. & Jegen, M., 2012. Detecting a salt dome
overhang with magnetotellurics: 3D inversion methodology and synthetic
model studies, Geophysics, 77, E251–E263.
B
¨
orner, R.-U., 2010. Numerical Modelling in Geo-Electromagnetics: Ad-
vances and Challenges, Surv. Geophys., 31, 225–245.
Chave, A.D. & Jones, A.G., 2012. The Magnetotelluric Method: Theory and
Practice, Cambridge University Press, ISBN: 9780521819275.
Constable, S.C., Parker, R.L. & Constable, C.G., 1987. Occam’s inversion—
a practical algorithm for generating smooth models from electromagnetic
sounding data, Geophysics, 52, 289–300.
Egbert, G.D. & Kelbert, A., 2012. Computational recipes for electromagnetic
inverse problems, Geophys. J. Int., 189, 251–267.
Farquharson, C.G. & Miensopust, M.P., 2011. Three-dimensional finite-
element modelling of magnetotelluric data with a divergence correction,
J. appl. Geophys., 75, 699–710.
Farquharson, C.G., Oldenburg, D.W., Haber, E. & Shekhtman, R., 2002. An
algorithm for the three-dimensional inversion of magnetotelluric data, in
Proceedings of the 72nd Annual Meeting of the Society of Exploration
Geophysicists, Salt Lake City, Utah.
Franke, A., B
¨
orner, R.-U. & Spitzer, K., 2007. 3D finite element simulation
of magnetotelluric fields using unstructured grids, in Proceedings of the
22nd Colloquium of Electromagnetic Depth Research, pp. 27–33, eds
Ritter, O. & Brasse, H., Decin, Czech Republic, ISSN 0946–7467.
Garcia, X., Ledo, J. & Queralt, P., 1999. 2D inversion of 3D magnetotelluric
data: the Kayabe dataset, Earth Planets Space, 51, 1135–1143.
Groom, R.W. & Bailey, R.C., 1989. Decomposition of magnetotelluric
impedance tensors in the presence of local three-dimensional galvanic
distortion, J. geophys. Res. (Solid Earth), 94(B2), 1913–1925.
Groom, R.W. & Bailey, R.C., 1991. Analytic investigations of the effects of
near-surface three-dimensional glavanic scatterers on MT tensor decom-
positions, Geophysics, 56(4), 496–518.
Hautot, S., Singel, R., Waston, J., Harrop, N., Jerram, D.A., Tarits, P., Whaler,
K. & Dawes, G., 2007. 3D magnetotelluric inversion and model validation
with g r avity data for the investigation of flood basalts and associated
volcanic rifted margins, Geophys. J. Int., 170, 1418–1430.
Hautot, S., Goldak, D., Tarits, P. & Kosteniuk, P., 2011. Three-dimensional
magnetotelluric inversion of large data sets: case study of Pasfield Lake
Saskatchewan for mineral exploration, GEM Beijing 2011, Beijing,
China.
Jones, A.G., 1988. Static shift of magnetotelluric data and its removal in a
sedimentary basin environment, Geophysics, 53, 967–978.
Jones, A.G., 1993. The COPROD2 dataset: tectonic setting, recorded MT
data, and comparison of models, J. Geomagnet. Geoelectr., 45, 933–
955.
Jones, A.G., 1993. The BC87 dataset: tectonic settings, previous EM results
and recorded MT data, J. Geomagnet. Geoelectr., 45, 1089–1105.
Jones, A.G., 1993. Introduction to MT-DIW1 special section, J. Geomagnet.
Geoelectr., 45,
931–932.
Jones, A.G., 2011. Three-dimensional galvanic distortion of three-
dimensional regional conductivity structures: comments on “Three-
dimensional joint inversion for magnetotelluric resistivity and static shift
distributions in complex media” by Y. Sasaki and M. A. Meju (2006), J.
geophys. Res.-Solid Earth.
Jones, A.G. & Schultz, A., 1997. Introduction to MT-DIW2 special issue, J.
Geomagnet. Geoelectr., 49, 727–737.
Ledo, J., 2006. 2-D versus 3-D magnetotelluric data interpretation, Surv.
Geophys., 26, 511–543.
Mackie, R.L., Rodi, W. & Watts, M.D., 2001. 3-d magnetotelluric inversion
for resource exploration, SEG Int’l Exposition and Annual Meeting, San
Antonio, Texas.
Mackie, R.L., Smith, J.T. & Madden, T.R., 1994. Three-dimensional electro-
magnetic modeling using finite difference equations: the magnetotelluric
example, Radio Sci., 29, 923–935.
McNeice, G.W. & Jones, A.G., 2001. Multisite, multifrequency tensor de-
composition of magnetotelluric data, Geophysics, 66(1), 158–173.
Miensopust, M.P., 2010. Multidimensional Magnetotellurics: a 2D case
study and a 3D approach to simultaneously invert for resistivity structure
and distortion parameters, PhD thesis, Faculty of Science, Department
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from

1238 M. P. Miensopust et al.
of Earth and Ocean Sciences, National University of Ireland, Galway,
Ireland.
Miensopust, M.P. & Jones, A.G., 2011. Artefacts of isotropic inversion
applied to magnetotelluric data from an anisotropic earth, Geophys. J.
Int., 187, 677–689.
Nam, M.J., Kim, H.J., Song, Y., Lee, T.J., Son, J.-S. & Suh, J.H., 2007.
3D magnetotelluric modelling including surface topography, Geophys.
Prospect., 55, 277–287.
Newman, G.A. & Alumbaugh, D.L., 2000. Three-dimensional magnetotel-
luric inversion using non-linear conjugate gradients, Geophys. J. Int.,
140(2), 410–424.
Oldenburg, D., 1990. Inversion of electromagnetic data: an overview of new
techniques, Surv. Geophys., 11, 231–270.
Poll, H., Weaver, J.T. & Jones, A.G., 1989. Calculations of voltages for
magnetotelluric modelling of a region with near-surface inhomogeneities,
Phys. Earth planet. Inter., 53, 287–297.
Rodi, W.L. & Mackie, R.L., 2012. The Inverse Problem, in The Magne-
totelluric Method—Theory and Practice, eds Chave, A., Jones, A.G.,
Mackie, R.L. & Rodi, W.L., chapter 8, Cambridge University Press, ISBN:
9780521819275.
Sasaki, Y., 2001. Full 3-D inversion of electromagnetic data on PC, J. appl.
Geophys., 46, 45–54.
Siripunvaraporn, W., 2012. Three-dimensional magnetotelluric inversion:
a introductory guide for developers and users, Surveys in Geophysics,
33(1), 5–27.
Siripunvaraporn, W. & Egbert, G., 2009. WSINV3DMT: vertical magnetic
field transfer function inversion and parallel implementation, Phys. Earth
planet. Inter., 173(3–4), 317–329.
Siripunvaraporn, W., Egbert, G. & Lenbury, Y., 2002. Numerical accuracy
of magnetotelluric modeling: a comparison of finite difference approxi-
mations, Earth Planets Space, 54, 721–725.
Siripunvaraporn, W., Egbert, G., Lenbury, Y. & Uyeshima, M., 2005. Three-
dimensional magnetotellurics: data space method, Phys. Earth planet.
Inter., 150(1–3), 3–14.
Varentsov, I.M., 1998. 2D synthetic data sets (COPROD-2S) to study
MT inversion techniques, Presented at the 14th Workshop on Electro-
magnetic Induction in the Earth. Available from: http://mtnet.dias.ie/
data/coprod2s.html
ˇ
Cerv, V. & Pek, J., 1990. Modelling and analysis of electromagnetic fields
in 3D inhomogenous media, Surv. Geophys., 11, 205–229.
Zhdanov, M.S., Varentsov, I.M., Weaver, J.T., Golubev, N.G. & Krylov,
V.A., 1997. Methods for modelling electromagnetic fields: results from
COMMEMI—the international project on the comparison of modelling
methods for electromagnetic induction, J. appl. Geophys., 37, 133–271.
SUPPORTING INFORMATION
Additional Supporting Information may be found in the online ver-
sion of this article:
DSM1.txt Data set of the secret model (DSM1).
DSM2.txt Data set of the secret model (DSM2).
DTM1.txt Response sets of the test model (DTM1).
DTM2.txt Response sets of the test model (DTM2) (http://gji.
oxfordjournals.org/lookup/suppl/doi:10.1093/gji/ggt066/-/DC1)
Please note: Oxford University Press are not responsible for the
content or functionality of any supporting materials supplied by
the authors. Any queries (other than missing material) should be
directed to the corresponding author for the article.
at DIAS on May 28, 2013http://gji.oxfordjournals.org/Downloaded from