Geophysical Journal InternationalGeophys. J. Int. (2013) 192, 58–66 doi: 10.1093/gji/ggs014

Three-dimensional inversion of magnetotelluric phase tensor data

Prasanta K. Patro,1 Makoto Uyeshima2 and Weerachai Siripunvaraporn3,4

1CSIR- National Geophysical Research Institute, Uppal Road, Hyderabad, India. E-mail: patrobpk@ngri.res.in2Earthquake Research Institute, the University of Tokyo, Tokyo, Japan3Department of Physics, Faculty of Science, Mahidol University, Bangkok, Thailand4ThEP Center, Commission on Higher Education, Si Ayutthaya Road, Rachatawee, Bangkok, Thailand

Accepted 2012 October 7. Received 2012 October 7; in original form 2010 May 17

S U M M A R YRecent increase in the application of 3-D inversion of magnetotelluric (MT) data is facilitatedby the availability of several 3-D inversion codes, which led to improved interpretation ofMT data. However, still the galvanic effects continue to pose problems in interpretation of theMT data. We have addressed this problem using the phase tensor (PT) concept and developeda scheme based on a modification of the sensitivity matrix in the 3-D inversion code ofSiripunvaraporn et al., which enables us to directly invert the phase tensor elements. We haveused this modified code for PT inversion of MT data and evaluated its efficacy in reducing thegalvanic effects through a few examples of inversion of synthetic data and its application onreal data. The synthetic model study suggests that the prior model (mo) setting is importantin retrieving the true model. Comparison of results obtained from conventional impedanceinversion and the proposed PT inversion method suggests that, even in the presence of thegalvanic distortion (due to near-surface checkerboard anomalies in our case), the new inversionalgorithm reliably retrieves the regional conductivity structure when the prior model or regionalresistivity value level can be estimated with sufficient accuracy. Data errors were propagatedto the PT elements using delta method while inverting the real data from USArray. The PTresults compare very well with those from full tensor inversion published earlier, signifyingthe efficacy of this new inversion scheme.

Key words: Inverse theory; Electrical properties; Electromagnetic theory; Magnetotelluric;Subduction zone processes.

I N T RO D U C T I O N

3-D inversion of the magnetotelluric (MT) data (Tuncer et al. 2006;Hiese et al. 2008, 2010; Patro & Egbert 2008) has become a com-mon practice among the MT community due to free availability ofthe WSINV3DMT code (Siripunvaraporn et al. 2005) to the aca-demic community. 3-D inversion codes have opened a world of 3-Dinterpretation of MT data and remove difficulties in interpretationfrom limitations of the 2-D assumptions. This enhances reliabilityof the model resistivity structures, though the galvanic effects re-main to be effectively taken care of to obtain realistic subsurfaceresistivity models (e.g. Meju 2002). Sasaki & Meju (2006) proposeda way to alleviate the difficulties from the galvanic effects, wherein both the resistivity values and static shift parameters are treatedas model parameters in their inversion schemes. However, sincethey expressed static shifts as gain factors multiplied to respectiveimpedance tensor elements, that is, phase of respective elements re-main the same regardless of existence of the galvanic effects. Theirformulation is therefore not consistent with the general representa-tion of the galvanic effects, where, an observed impedance tensoris expressed as a product of a galvanic real tensor and a regional

complex impedance tensor (e.g. Bahr 1988; Groom & Bailey 1989;Chave & Smith 1994; Berdichevsky & Dmitriev 2002). Alternateway in dealing with static shift is during data pre-processing stepssuch as tensor decomposition (e.g. Heise et al. 2008, 2010) or usingTransient Electromagnetic (TEM) data (e.g. Arnason et al. 2010) orwith different assumptions such as that of Zhdanov et al. (2010). Todeal with galvanic distortion of MT data in more general cases, Cald-well et al. (2004) introduced the novel concept of a ‘phase tensor’(PT). They demonstrated how the regional phase information can berecovered directly from the observed impedance tensor, where in,both the near-surface inhomogeneity and the regional conductivitystructures can be 3-D. Heise et al. (2008) modelled a complicated3-D resistivity structure with the use of the PT approach by perform-ing a series of forward modelling tests. Their contribution suggestedthe potential of the PT method in obtaining a dependable regionalstructure.

Stimulated by these developments on the PT approach, we at-tempted to modify the WSINV3DMT code to directly invert thePT elements. In this study, we present the PT inversion method andillustrate its effectiveness through application on a few examples ofsynthetic as well as real data.

58 C© The Authors 2012. Published by Oxford University Press on behalf of The Royal Astronomical Society.

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3-D inversion of MT phase tensor data 59

M E T H O D O L O G Y O F 3 - D P T I N V E R S I O N

To realize a 3-D PT inversion, we modified the WSINV3DMT 3-DMT inversion code that is based on a data space 3-D MT OCCAM’sinversion scheme (Siripunvaraporn et al. 2005). The theoreticalbackground of the WSINV3DMT is discussed in greater detail inthe earlier papers (Siripunvaraporn & Egbert 2000; Siripunvara-porn et al. 2005). Hence, here we briefly mention the modificationscarried out to the code.

The principal modification to achieve the PT inversion was car-ried out in the sensitivity matrix computations of the WSINV3DMT.The sensitivity of the PT at a specific site and frequency i for a modelconductivity j is expressed as:

∂φi/∂σ j = (∂(�Zi)−1(�Zi))/∂σ j

= −(�Zi)−1 ∂�Zi

∂σ j(�Zi)−1 �Zi + (�Zi)−1 ∂�Zi

∂σ j, (1)

where φ denotes the PT = (�Z )−1 �Z (Caldwell et al. 2004), and�Z and �Z denote real and imaginary part of the impedance ten-sor Z, respectively. Since the complex sensitivity (∂ Z

/∂σ ) for the

MT impedance tensor is readily computed in the WSINV3DMTbased on the reciprocity approach (Rodi 1976; Siripunvarapornet al. 2005), the sensitivity for the PT expressed in eq. (1) can alsobe estimated by using the conventional MT impedance sensitivity.

After estimating the sensitivities for multiple sites and frequen-cies, the model renewal equation for each iteration step can be ob-tained by minimization of the penalized objective function, which iscomposed using the data misfit and the model variance terms. Thedata misfit and the model variance terms are, respectively, normal-

ized by the PT errors and the model covariance matrix. In the newlydeveloped PT inversion, PT errors are estimated by error propaga-tion expressions from estimates of errors for respective impedancetensor elements (see the Appendix), and model covariance is fixedthroughout the inversion process. The way how the model covari-ance is considered in the data space approach, was described inprevious papers in detail (Egbert et al. 1994; Siripunvaraporn &Egbert 2000; Siripunvaraporn et al. 2005). The trade-off parame-ter between the data misfit and the model variance is determinedthrough OCCAM’s inversion scheme; the smoothest model (or theleast variance model) with plausible data misfit level is selected asthe best model (Constable et al. 1987).

S Y N T H E T I C T E S T

We have tested the PT inversion scheme on a few synthetic models.In the first test, the synthetic model is constructed following a 3-Dmodel described in Caldwell et al. (2004). The model was createdwith Nx = 37, Ny = 36 and Nz = 26 layers (plus seven air layers) anda vertical factor of 1.20 with top layer thickness being 100 m. Thedimensions of the total model domain were 26 × 26 × 57 km3. Itconsists of a small conductive cube (10 Ohm m, ‘A’ in Fig. 1a) withits top at a depth of 100 m below the surface extending verticallyupto 350 m and a large rectangular body of 1 Ohm m (‘B’ in Fig. 1a)with its top buried at 0.5 km below the surface and extending downto 6 km deep. The conductive cube is positioned near the cornerof the rectangular body. Both of these bodies are embedded in a100 Ohm m half-space. The synthetic model is shown in Fig. 1(b).Inversion was done for the synthetic responses at 28 sites and 12

Figure 1. (a) Map view of the synthetic model used to generate synthetic data. The conductive body A is placed below 100 m of the surface, extends upto350 m. Another conductive body B starts from 500 m below the surface and extends upto 6000 m deep. (b) Slice plot of the synthetic model. (c) Slice plot ofthe phase tensor (PT) inversion solution resulted from 10 Ohm m half-space prior model. (d) Slice plot of the PT inversion solution resulted from 100 Ohm mhalf-space prior model. (e) Slice plot of the PT inversion solution resulted from 1000 Ohm m half-space prior model.

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60 P.K. Patro, M. Uyeshima and W. Siripunvaraporn

Figure 2. Difference between the predicted model and true model for 10 Ohm m prior model (A), 100 Ohm m prior model (B) and 1000 Ohm m prior model(C).

Table 1. Mean deviation between the true model and inverted model. Meandeviation is a comparison in terms of model values.

Inversion type Different prior model Model rms Mean deviation

Phase tensor inversion10 Ohm m prior 1.044 1.941100 Ohm m prior 0.965 0.8341000 Ohm m prior 1.740 2.356

Full tensor inversion10 Ohm m prior 1.039 2.042100 Ohm m prior 1.015 1.1651000 Ohm m prior 1.558 2.584

periods from 0.215 to 1000 s, which were equally distributed on alogarithmic scale. The inversion was carried out for different priormodels (mo) with the half-space having diverse resistivities, that is,10, 100 and 1000 Ohm m. For each test run the initial model andthe respective prior model remained the same. All these inversionswere performed after perturbing the synthetic data by a value of 5per cent of

√|φxx × φ

yy |. One can also consider taking the arithmetic

mean of main diagonal PT components as errors to avoid zeros thatmight exist in 90◦ phases. The inversions were run for 10 iterations.The inversion solution resulted from the 100 Ohm m prior model(see Fig. 1d) has the minimum rms of 0.965 compared to other twocases (see Figs 1c and e; rms = 1.044, 1.740 for 10 Ohm m and1000 Ohm m prior models, respectively). Differences between themodel parameters corresponding to the predicted and true modelsare plotted in Fig. 2. We also computed mean deviation betweeninverted and true model (see Table 1) for PT and full tensor (FT)

inversions applying the formula:√

[∑N

i=1 (RT i − RPi )2 /N ], where

N = total number of blocks in the model, RT = true model resistivityand RP = inverted model resistivity. Note that, from the Fig. 2 andthe Table 1, it is evident that the model derived from the 100 Ohm mprior model is closest to the synthetic model. In the other two cases,however, resultant resistivity values are on average more conductiveor more resistive depending on the prior model resistivity values.Moreover, thickness of the reproduced conductive rectangular bodyis thicker or thinner than that of the synthetic model for conductiveor resistive prior model, respectively. Misfit between the syntheticand predicted responses (for period = 2 s) in the case of 10,100 and1000 Ohm m prior models are presented in Fig. 3. Note that, the

synthetic PT elements, which are used as the observations, are wellreproduced from the inverted models by all the three prior models.

Further, using the synthetic data generated from the same model,we compared the results from the present PT inversion and con-ventional FT impedance (Zxx, Zxy, Zyx and Zyy) inversion. The priormodel was 100 Ohm m half-space and the errors used in the inver-sion were 5 per cent of

√Zxx × Z yy . Final model is obtained with an

rms of 1.015, is presented in Fig. 4. In both the resultant models theconductor B is recovered well. From the model deviation numberspresented in Table 1, it can be seen that both the inversion resultsare comparable.

In the second synthetic model test, we used the above-discussedmodel and introduced distortion by a checkerboard pattern of resis-tivites of 10 and 1000 Ohm m in the top layer (100 m thick). Thedimension of the checkerboard is set to 1 km × 1 km. The syn-thetic model is illustrated in Figs 5(a) and (d). A series of inversiontests were carried out for the synthetic full impedance tensors aswell as for the synthetic PTs, at 28 sites and seven periods from10 to 1000 s, which are equally distributed in the logarithmic scale.One important point in this demonstration is that the inductionscale length (or the skin depth) of the shortest period (10 s) usedis set to much longer than the thickness and size of the surfacecheckerboard. Thus the PTs are devoid of the surface checkerboardanomaly, while, magnitudes of the impedance are clearly dependenton site locations and checkerboard pattern due to the galvanic ef-fects. Results for the full impedance tensor inversion are presentedin Figs 5(b) and (e) and for the PT inversion in Figs 5(c) and (f).It is evident from these results that the synthetic regional modelshown in Fig. 5(d) is reproduced better in the PT inversion resultsshown in Fig. 5(f). However, as can be seen from a comparison ofFig. 5(d) with 5(e), the conventional impedance tensor inversion,where the magnitudes of the impedances are also considered, hasmoderately reproduced the synthetic regional model. The conven-tional impedance tensor inversion on the other hand tends to yieldspurious fine-scale anomalies in the surface layers to adjust the mag-nitude of the impedance, as shown in Figs 5(b) and (e). In contrast,the PT inversion does not produce any such spurious features inthe surface layers as is evident from Fig. 5(c). It may also be notedthat the feature ‘A’ is reproduced with higher amplitude in both FT(Figs 5b and e) and PT (Figs 5c and f) inversion results. In theseinversion tests, a 100 Ohm m prior model was assumed. As was thecase with the first example, however, the synthetic regional model

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3-D inversion of MT phase tensor data 61

Figure 3. Misfit between the synthetic and predicted phase tensor element (dimensionless) responses (�xx, �xy, �yx, �yy) of the inversion carried out with 10(left-hand side) 100 (centre) and 1000 (right-hand side) Ohm m half-space prior model.

Figure 4. Full tensor inversion result of the Caldwell model with 100 Ohm m prior model.

cannot be reproduced solely by the PT inversion, if the prior modelis set to be much different from the true background resistivity. Asimilar observation was made in a recent study on the 3-D inversionof vertical magnetic transfer functions (Siripunvaraporn & Egbert2009), suggesting the importance of correct estimation for the hostresistivity in the prior model.

A P P L I C AT I O N T O T H E R E A L DATA

We have also evaluated the usefulness of the PT inversion by apply-ing it to USArray MT data acquired in the Pacific Northwest, UnitedStates of America under EarthScope programme (Fig. 6). The MTstudy in USArray was aimed to cover the continental United Statesat an approximate spacing of 75 km in selected areas. During 2006–2007 long-period magnetotelluric (LMT) data were collected at 110sites covering Oregon, Washington State and a part of Idaho. LMT

data were acquired by a commercial contractor in a series of over-lapping arrays using Fluxgate magnetometers (Narod Geophysics).MT transfer functions were derived by applying robust remote ref-erence processing approach (Egbert 1997). Initial 3-D inversion ofthese data sets revealed many interesting features such as exten-sive areas of high conductivity in the lower crust below SE Oregoncovering Basin and Range, Blue Mountains and High Lava Plains;variation in the upper-mantle conductivity between the subductingocean and North American continent (Patro & Egbert 2008).

In this paper, we inverted the PT elements (�xx, �xy, �yx, �yy)of the above-mentioned 109 sites at four periods (100–8000 s).Because we want to compare with the results from FT inversioncarried out earlier (Patro & Egbert 2008), we used the same modelgrid for PT inversion. The model domain consists of Nx = 80,Ny = 78 grid cells with a grid spacing of 12 km in the centralpart of the grid and Nz = 34 layers. Pacific Ocean (conductivity of

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62 P.K. Patro, M. Uyeshima and W. Siripunvaraporn

Figure 5. (a) Distortion is simulated with a checkerboard pattern of resistivity bocks (1 km × 1 km) of 10 and 1000 Ohm m in the top layer (100 m thick). (b)Top layer slice of the inversion solutions resulted from the full tensor (FT) impedances. (c) Top layer slice of the inversion solutions resulted from the phasetensor (PT). (d) Depth section of the synthetic model. (e) Depth section of the inversion solutions resulted from the FT impedances. (f) Depth section of theinversion solutions resulted from the PT.

Figure 6. Location of USArray MT sites occupied in the NW Pacific during 2006 (blue star) and 2007 (red star) under EarthScope programme (Patro & Egbert2008).

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Figure 7. 3-D resistivity image derived from the inversion of the USArray MT data from (a) full tensor (Zxxr, Zxxi, Zxyr, Zxyi Zyxr, Zyxi Zyyr, Zyyi) inversion ateight periods (b) phase tensor (�xx, �xy, �yx, �yy) inversion at four periods. The subducting Juan de Fuca Plate is shown as grey feature in the figure.

Figure 8. Phase tensor ellipse plot for observed (a) and computed (b) responses at 1092 s.

3.33 S m–1) was included in the model. Though for FT eight periodswere considered, due to computational limitations only four peri-ods were used for PT inversion. The prior model was 100 Ohm mhalf-space with fixed ocean. The smoothing length scales were in-creased to four times the default value in the x and y directions tomaintain consistency with the earlier FT inversion (Patro & Egbert2008). The errors of PT elements (see the Appendix) were esti-mated directly from impedance tensor errors using delta method(Efron 1982). An error floor was assigned as 10 per cent of

√[|

�xx × �yy |] to stabilize the inversion. The final rms obtained was2.35. Figs 7(a) and (b) show the solution from FT and PT inversion.It may be noted that the conductive features (C1–C3) shown in redto yellow colour delineated from both FT and PT inversion. Themagnitude of the conductor C1 is reduced in the PT inversion whencompared to FT inversion. It may be noted that in the case of PTinversion we have considered only four periods apart from that theprior model was 100 Ohm m half-space with ocean fixed. The elon-gated conductor C2 where it coincides with Southwest WashingtonCascade Conductor is also recovered in the PT inversion. The PT

inversion results also brought out the conductor C3 in the NE partof the model, which seems to extend eastwards. Resistive oceanicmantle (R1) subducting beneath the North American continent isalso observed from PT inversion.

A comparison of the PT ellipses for observed data and computedresponses of the inverted model at 1092 s are plotted for all thesites. One can note the coast effect (A in Fig. 8) on the data, whichis clearly reflected in both observed and computed responses. Thereis a good amount of agreement between Figs 8(a) and (b) except atfew stations, for example, in the northeastern part of the array (B inthe Fig. 8).

D I S C U S S I O N A N D C O N C LU S I O N

We have modified the WSINV3DMT code to directly invert thePT elements of MT data. The main modification was done to thesensitivity matrix computations. We did a few synthetic tests withthe new code. In the first case, we generated synthetic PT data from amodel similar to that presented in Caldwell et al. (2004) and inverted

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64 P.K. Patro, M. Uyeshima and W. Siripunvaraporn

the same. The inversion was run for different prior models. The truemodel was recovered when the value of prior model was close tothat of the synthetic model. In the case of another synthetic testmodel, we could simulate galvanic distortion related to the modelby introducing fine-scale checkerboard pattern of conductors andresistors on the surface of the 3-D model. These data with galvanicdistortion effects were inverted using both the conventional 3-D FTimpedance inversion as well as the PT inversion. The results showthat, in the presence of the galvanic distortion, the PT inversionrecovers the regional structure more reliably compared to that fromimpedance inversion. All the results from the present studies providestrong evidence on the usefulness of the PT inversion but point tothe importance of selection of a reliable prior model to retrievethe true model. The reliable prior model for PT inversion could beobtained from the TEM, the Network-MT or other static shift-freeEM-responses (e.g. Uyeshima 2007; Heise et al. 2008). However,this issue needs further study. Application of the 3-D PT inversion tothe real data from USArray brought out the conductive and resistivefeatures described in Patro & Egbert (2008). This result furthersignifies the usefulness of the PT inversion to the real data.

A C K N OW L E D G M E N T S

This research was carried out with the research grant (No: 20.08024)from JSPS and MEXT to PKP and MU. It is also partly supportedby the research grant (No. 19540442) from MEXT to MU. Thisresearch has been supported by the Thailand Research Fund (TRF:RMU5380018) to WS. Thanks to IRIS for providing USArray MTdata. Most of the inversions were run in the EIC at the EarthquakeResearch Institute, Japan. PKP thanks the Director, NGRI, for grant-ing deputation to Tokyo. Thoughtful comments by the editor, OliverRitter has helped to improve the manuscript considerably. The au-thors are thankful to Maxim Smirnov and the anonymous reviewersfor their constructive suggestions that have enhanced the clarity ofthe manuscript.

R E F E R E N C E S

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Bahr, K., 1988. Interpretation of the magnetotelluric impedance tensor:regional induction and local telluric distortion, J. Geophys., 62, 119–127.

Berdichevsky, M.N. & Dmitriev, V.I., 2002. Magnetotellurics in the Contextof Theory of Ill-Posed Problems, Society of Exploration Geophysicists,Tulsa, OK.

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Constable, C.S., Parker, R.L. & Constable, C.G., 1987. Occam’s inversion:a practical algorithm for generating smooth models from electromagneticsounding data, Geophysics, 52, 289–300.

Efron, B., 1982. The Jackknife, the Bootstrap and Other Resampling Plans.CBMS-NSF Regional conference series in applied mathematics, Societyfor industrial and applied mathematics, Philadelphia, Pennsylvania

Egbert, G.D., 1997. Robust multiple-station mgntotelluric data processing,Geophys. J. Int., 130, 475–496.

Egbert, G.D., Bennett, A.F. & Foreman, M.G., 1994. TOPEX/POSEIDONtides estimated using a global inverse model, J. geophys. Res., 99, 24 821–24 852.

Groom, R.W. & Bailey, R.C., 1989. Decomposition of magnetotelluricsimpedance tensors in the presence of local three-dimensional galvanicdistortions, J. geophys. Res., 94, 1913–1925.

Heise, W., Caldwell, T.G., Bibby, H.M. & Bannister, S.C., 2008. Three-dimensional modeling of magnetotelluric data from the Rotokawageothermal field, Taupo Volcanic Zone, New Zealand, Geophys. J. Int.,173, 740–750.

Heise, W., Caldwell, T.G., Bibby, H.M. & Bennie, S.L., 2010. Three-dimensional electrical resistivity image of magma beneath an active con-tinental rift, Taupo Volcanic Zone, New Zealand, Geophys. Res. Lett.,37(10), L10301, doi:10.1029/2010GL043110.

Meju, M.A., 2002. Geoelectromagnetic exploration for natural resources:models, case studies and challenges, Surv. Geophys., 23, 133–205.

Patro, P.K. & Egbert, G.D., 2008. Regional conductivity structure ofCascadia: preliminary results from 3D inversion of USArray trans-portable array magnetotelluric data, Geophys. Res. Lett., 35, L20311,doi:10.1029/2008GL035326.

Rodi, W.L., 1976. A technique for improving the accuracy of finite elementsolutions for magnetotelluric data, Geophys. J. R. astr. Soc., 44, 483–506.

Sasaki, Y. & Meju, M.A., 2006. Three-dimensional joint inversion for mag-netotelluric resistivity and static shift distribution in complex media, J.geophys. Res., 111, B05101, doi:10.1029/2005JB004009.

Siripunvaraporn, W. & Egbert, G., 2000. An efficient data-subspace inver-sion method for 2D magnetotelluric data, Geophysics, 65, 791–803.

Siripunvaraporn, W. & Egbert, G., 2009. WSINV3DMT: vertical magneticfield transfer function inversion and parallel implementation, Phys. Earthplanet. Inter., 173, 317–329.

Siripunvaraporn, W., Egbert, G., Lenbury, Y. & Uyeshima, M., 2005. Three-dimensional magnetotelluric inversion: data-space method, Phys. Earthplanet. Inter., 150, 3–14.

Tuncer, V., Unsworth, M.J., Siripunvaraporn, W. & Craven, J.A., 2006. Ex-ploration for unconformity-type uranium deposits with audiomagnetotel-luric data: a case study from the McArthur River mine, Saskatchewan,Canada, Geophysics, 71, B201–B209.

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A P P E N D I X : P T E R RO R C A L C U L AT I O NU S I N G D E LTA M E T H O D

PT elements are expressed as:⎡⎣ pxx pxy

pyx pyy

⎤⎦ =

⎛⎝ r1 r2

r3 r4

⎞⎠

−1

×⎛⎝ i1 i2

i3 i4

⎞⎠

=⎛⎝ r4 −r2

−r3 r1

⎞⎠ ×

⎛⎝ i1 i2

i3 i4

⎞⎠/

(r1 × r4 − r2 × r3) ,(A1)

where

r1 = real(zxx), r2 = real(zxy), r3 = real(zyx), r4 = real(zyy),

i1 = imag(zxx), i2 = imag(zxy), i3 = imag(zyx), i4 = imag(zyy)

From eq. (A1), respective elements of the PT are expressed as

pxx = (r4 × i1 − r2 × i3)/det,

pxy = (r4 × i2 − r2 × i4)/det,

pyx = ( − r3 × i1 + r1 × i3)/det,

pyy = ( − r3 × i2 + r1 × i4)/ det,

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3-D inversion of MT phase tensor data 65

where

det = r1 × r4 − r2 × r3.

pxxError

First, we estimate error of pxx from error estimations of theimpedance tensor.

∂p1

∂r1= ∂

∂r1

[(r4 × i1 − r2 × i3)

(r1 × r4 − r2 × r3)

]. (A2)

By using the identity ∂

∂x

(uv

) = (v ∂u∂x −u ∂v

∂x )v2 , where u and v are func-

tional of x, eq. (A2) becomes

∂p1

∂r1= −(r4 × i1 − r2 × i3) × r4

(r1 × r4 − r2 × r3)2,

∂p1

∂r1= −pxx × r4

det.

Similarly, the following relations are derived.

∂p1

∂r2= (−i3 + pxx × r3)

det,

∂p1

∂r3= (pxx × r2)

det,

∂p1

∂r4= (i1 − pxx × r1)

det,

∂p1

∂i1= r4

det,

∂p1

∂i3= −r2

det.

If X is a functional of variables A, B, C, D, . . . such that X = f(A,B,C,D. . . . . . .), then variation of X is expressed as:

δx =(

∂ f

∂ A

)δA+

(∂ f

∂ B

)δB+

(∂ f

∂C

)δC +

(∂ f

∂ D

)δD + .........,

where higher order terms are neglected. Assuming that covariancesbetween the variables A, B, C, D,. . . . are zero, variance of X isestimated as

σ 2x =

(∂ f

∂ A

)2

σ 2A +

(∂ f

∂ B

)2

σ 2B +

(∂ f

∂C

)2

σ 2C

+(

∂ f

∂ D

)2

σ 2D + .........,

where, for example, σ 2A is variance of A.

By using the above relation, one can obtain an error propagationformula of pxx as follows

epxx2 =(

∂p1

∂r1

)2

e21 +

(∂p1

∂r2

)2

e22 +

(∂p1

∂r3

)2

e23 +

(∂p1

∂r4

)2

e24

+(

∂p1

∂i1

)2

e21 +

(∂p1

∂i3

)2

e23,

epxx = sqrt(epxx2) ,

where epxx is an error estimate of a PT element pxx and e1, e2, e3and e4 denote error estimates of impedance tensor zxx, zyx andzyy, respectively.

Similarly, we can derive the expressions to estimate errors for theother PT elements epxy, epyx and epyy as follows.

pxyError

∂p2

∂r1= −pxy × r4

det,

∂p2

∂r2= (−i4 + pxy × r3)

det,

∂p2

∂r3= pxy × r2

det,

∂p2

∂r4= (i2 − pxy × r1)

det,

∂p2

∂i2= r4

det,

∂p2

∂i4= −r2

det,

epxy2 =(

∂p2

∂r1× e1

)2

+(

∂p2

∂r2× e2

)2

+(

∂p2

∂r3× e3

)2

+(

∂p2

∂r4× e4

)2

+(

∂p2

∂i2× e2

)2

+(

∂p2

∂i4× e4

)2

,

epxy = sqrt(epxy2).

pyxError

∂p3

∂r1= (i3 − pyx × r4)

det,

∂p3

∂r2= pyx × r3

det,

∂p3

∂r3= (−i1 + pyx × r2)

det,

∂p3

∂r4= −pyx × r1

det

∂p3

∂i1= −r3

det,

∂p3

∂i3= r1

det,

epyx2 =(

∂p3

∂r1× e1

)2

+(

∂p3

∂r2× e2

)2

+(

∂p3

∂r3× e3

)2

+(

∂p3

∂r4× e4

)2

+(

∂p3

∂i1× e1

)2

+(

∂p3

∂i3× e3

)2

,

epyx = sqrt(epyx2).

pyyError

∂p4

∂r1= (i4 − pyy × r4)

det,

GJI, 192, 58–66

at National G

eophysical Research Institute on N

ovember 13, 2012

http://gji.oxfordjournals.org/D

ownloaded from

66 P.K. Patro, M. Uyeshima and W. Siripunvaraporn

∂p4

∂r2= pyy × r3

det,

∂p4

∂r3= (−i2 + pyy × r2)

det,

∂p4

∂r4= −pyy × r1

det,

∂p4

∂i2= −r3

det,

∂p4

∂i4= r1 × pyy

det,

epyy2 =(

∂p4

∂r1× e1

)2

+(

∂p4

∂r2× e2

)2

+(

∂p4

∂r3× e3

)2

+(

∂p4

∂r4× e4

)2

+(

∂p4

∂i2× e2

)2

+(

∂p4

∂i4× e4

)2

,

epyy = sqrt(epyy2).

GJI, 192, 58–66

at National G

eophysical Research Institute on N

ovember 13, 2012

http://gji.oxfordjournals.org/D

ownloaded from