Use of ssq rotational invariant of magnetotelluric impedances for estimating informative properties for galvanic distortion

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1 DOI.86/s FULL PAPER Open Access Use of ssq rottionl invrint of mgnetotelluric impednces for estimting informtive properties for glvnic distortion T. Rung Arunwn,,,4*, W. Siripunvrporn, nd H. Utd Astrct Severl useful properties nd prmeters model of the regionl men one-dimensionl (D) conductivity profile, locl nd regionl distortion indictors, nd pprent gins were defined in our recent pper using two rottionl invrints (det: determinnt nd ssq: sum of squred elements) from set of mgnetotelluric (MT) dt otined y n rry of oservtion sites. In this pper, we demonstrte their chrcteristics nd enefits through synthetic exmples using D nd three-dimensionl (D) models. First, model of the regionl men D conductivity profile is otined using the verge ssq impednce with different levels of glvnic distortion. In contrst to the Berdichevsky verge using the verge det impednce, the verge ssq impednce is shown to yield relile estimte of the model of the regionl men D conductivity profile, even when severe glvnic distortion is contined in the dt. Second, the locl nd regionl distortion indictors were found to indicte the glvnic distortion s expressed y the splitting nd sher prmeters nd to quntify their strengths in individul MT dt nd in the dtset s whole. Third, the pprent gin ws lso shown to e good pproximtion of the site gin, which is generlly climed to e undeterminle without externl informtion. The model of the regionl men D profile could e used s n initil or priori model in higher-dimensionl inversions. The locl nd regionl distortion indictors nd pprent gins could e used to exmine the existence nd to guess the strength of the glvnic distortion. Although these conclusions were derived from synthetic tests using the Groom Biley distortion model, dditionl tests with different distortion models indicted tht these conclusions re not strongly dependent on the choice of distortion model. These glvnic-distortion-relted prmeters would lso ssist in judging if proper tretment is needed for the glvnic distortion when n MT dtset is given. Hence, this informtion derived from the dtset would e useful in MT dt nlysis nd inversion. Keywords: Mgnetotellurics, Rottionl invrint, Glvnic distortion Introduction To otin relile three-dimensionl (D) inversion from mgnetotelluric (MT) dt, either distorted or undistorted, the choice of n initil or priori model is crucil. The enefit of hving good model of the regionl men one-dimensionl (D) profile s n initil or priori model hs een reported in previous studies. For exmple, the optiml model of the men D conductivity *Correspondence: t.rungrunwn@gmil.com Deprtment of Physics, Fculty of Science, Mhidol University, 7 Rm 6 Rod, Rchtwee, Bngkok 4, Thilnd Full list of uthor informtion is ville t the end of the rticle profile would minimize the lterl conductivity contrst, which could yield etter-conditioned system of equtions (Avdeev 5). Furthermore, the use of the men D profile s n priori model would result in the rpid nd stle convergence of higher-dimensionl inversion prolems (Td et l. ). If no other independent informtion is ville, the initil model cn e constructed from the men D conductivity profile, which will e descried in the following. Idelly, if we hd sufficiently lrge numer of electromgnetic (EM) oservtion sites densely distriuted over the gloe, the glol men conductivity profile could The Author(s) 7. This rticle is distriuted under the terms of the Cretive Commons Attriution 4. Interntionl License ( which permits unrestricted use, distriution, nd reproduction in ny medium, provided you give pproprite credit to the originl uthor(s) nd the source, provide link to the Cretive Commons license, nd indicte if chnges were mde.

2 Pge of 4 e defined s the zimuthl verge of the conductivity long vrile depth: σ (z) = σ(z, θ, φ)ds, () S (z) where σ(z, θ, φ) is the distriution of the Erth s electricl conductivity, S (z) is the totl surfce re of the Erth t depth z, nd ds is sphericl surfce element. Once the glol men D conductivity profile is otined in this wy (Eq. ), the D conductivity distriution t ny position within the Erth cn e expressed s comintion of the glol men D model nd the zimuthl conductivity contrst s σ(z, θ, φ) = σ (z) + σ (z, θ, φ). () The definition of the glol men D conductivity profile nd the zimuthl contrst is cler in theory, ut the estimtion of them is not esy in prctice. Although it is possile to perform glol induction studies using geomgnetic oservtory dt, there re significnt differences mong existing inverted models (e.g., Kelert et l. 9; Kuvshinov nd Semenov ; Semenov nd Kuvshinov ). Most likely, such ttempts my include ises due to the nonuniformity of their site distriutions or flse imges resulting from sptil lising (Utd nd Munekne ) ecuse the distriutions of existing geomgnetic oservtories nd MT oservtion sites re sptilly nonuniform. More importntly, the EM induction method is generlly sensitive to the conductivity eneth ech oservtion site. Prcticlly, EM explortion, e.g., MT survey, focuses on limited region where numerous oservtions re mde. In this study, we consider cse in which the corresponding induction scle length is much smller thn the rdius of the Erth; hence, the sphericity of the Erth cn e ignored. Such cse is usully clled regionl or locl induction study. From given rry of oservtions, the following regionl men D conductivity profile σ R (z) cn e otined from the rel verge of the conductivity: σ R (z) = σ(x, y, z)da () A where A is the ritrry re in which the oservtion sites re distriuted, da is surfce element, nd σ(x, y, z) is the regionl D conductivity distriution. Alterntively, we my use the logrithmic verge to define the regionl men D profile: log σ R (z) = log σ(x, y, z)da. (4) A Mthemticlly, the logrithmic scle verge of the conductivity gives less conductive structure. As with Eq. (), the regionl conductivity distriution cn e expressed s comintion of the regionl men D profile nd the conductivity contrst σ (x, y, z): σ(x, y, z) = σ R (z) + σ (x, y, z). (5) When the regionl men D conductivity profile is written s the rel verge (Eqs. or 4) of the conductivity from ny D conductivity distriution, devitions in the conductivity higher or lower thn the verge vlue re regrded s conductivity nomlies of positive or negtive contrst, respectively. σ R (z) defined either y Eqs. () or (4) is lso regrded s n optiml men D conductivity profile in the sense tht the vrince of σ (x, y, z) is minimized (Rung-Arunwn et l. 6). The regionl men D profile σ R (z) is prcticlly unknown eforehnd, lthough it cn e estimted from σ(x, y, z) inverted from the oserved dt. However, in this study, we use n lternte method in which σ R (z) is first estimted from the oserved dt in the re of interest. The conductivity model, either σ(x, y, z) or σ (x, y, z), cn then e estimted y D inversion using σ R (z) s priori informtion or strting model. This poses the prolem of how to relily estimte model of the regionl men D profile from n rry of MT oservtions in generl D sitution. B et l. () lredy presented solution to such prolem in the cse of sefloor MT study. Here, we consider the cse in which MT dt re otined from n rry of oservtions on lnd. The solution of this prolem is not strightforwrd ecuse MT dt on lnd re usully ffected y glvnic distortion, i.e., n ltertion in the MT impednce due to ner-surfce smll-scle heterogeneity tht is smller thn typicl site spcing nd confined to e shllower thn the inductive scle length of interest (Ledo et l. 998; Utd nd Munekne ; Biy et l. 5). In other words, the physicl dimensions of the distorting odies re smller thn their inductive scle length nd lso tht of the host. Berdichevsky et l. (98) proposed scheme to estimte model of the men D profile from distorted dt y verging the effective resistivity, which is equivlent to the pprent resistivity derived from the determinnt of the impednce tensor (herefter denoted s the det impednce, Z det = Z xx Z yy Z xy Z yx ). This is sttisticl pproch to smooth the effect of glvnic distortion tht is supposed to e rndom phenomenon, nd the verge in this method is referred to s the Berdichevsky verge (Rung-Arunwn et l. 6). When it ws first introduced, the det impednce ws generlly used in regionl studies (e.g., Berdichevsky et l. 98; Jones 988; Berdichevsky et l. 989). However, it ws lter dopted in two-dimensionl (D) MT pplictions (e.g., Oldenurg nd Ellis 99; Pedersen nd Engels

3 Pge of 4 5). It is lso pplied s the current chnneling indictor (Lezet nd Hk ) nd used in environmentl pplictions (Flgàs et l. 9). The det impednce hs lso een used in recent works. For exmple, Sem et l. (7) inverted the det impednces from mrine MT dt t ech oservtion point to otin D conductivity profiles eneth the Philippine Se, Arngo et l. (9) used the det impednce to interpret D MT dt, nd B et l. () nd Avdeev et l. (5) used the verge det impednce in the sme wy s Berdichevsky et l. (98). However, when the det impednce ws re-exmined on the sis of the present knowledge of glvnic distortion y pplying the Groom Biley model (Groom nd Biley 989), it ws found tht the mgnitude of the det impednce is lwys ised downwrd y the geometric distortion expressed y the sher nd splitting prmeters (see Gómez-Treviño et l. ; Rung-Arunwn et l. 6). Note tht Gómez-Treviño et l. () studied the effect of glvnic distortion on the rottionl invrints in the cse of D regionl structures. Even in the sence of site gin, the Berdichevsky verge cuses downwrd is in the pprent resistivity s compred with those from the regionl men D conductivity profile (σ R (z)) defined y Eqs. () or (4) (Rung- Arunwn et l. 6). Rung-Arunwn et l. (6) proposed nother method for estimting the model of the men D profile tht ws similr to the method of using the Berdichevsky verge ut redefined it with nother rottionl invrint: the sum of the squred elements of the impednce tensor ( (ssq impednce) Z ssq = Zxx + Z xy + Z yx + Z yy / (Szrk nd Menvielle 997). Note tht the ssq nd det impednces re identicl in the cse of D erth, ut for D nd D erth, the induction sensed y the ssq nd det impednces is different (see lso Szrk nd Menvielle 997) y Zssq Z det = (Z xx Z yy ) + (Z xy + Z yx ). In comprison with the det impednce, the ssq impednce hs een proven to e less ised y the distortion prmeters (see Gómez-Treviño et l. ; Rung-Arunwn et l. 6). An exmple of the field dt from the western prt of Thilnd is shown in Fig.. The field exmple is consistent with the theoreticl prediction presented in Rung-Arunwn et l. (6) tht the det impednce will hve smller mgnitude thn the ssq impednce. According to the prediction, the downwrd is for the det impednces is supposed to e cused y geometric (sher nd splitting) effect ecuse the phse chrcteristics re lmost identicl. Consequently, the use of verge ssq impednce is expected to more relily estimte the model of the regionl men D profile thn the use of the det impednce. Thus, this field exmple ) Fig. Mp showing the cluster of 9 MT sttions from the western prt of Thilnd (see Boonchisuk et l. ). They re plotted on the geologicl mp of this re (fter Deprtment of Minerl Resources 6). Exmple of rottionl invrint, det (gry dimonds) nd ssq (rown squres), impednces from n individul sttion (KAN-C). The det impednces hve mgnitudes smller thn the ssq impednces, which is s predicted (Rung-Arunwn et l. 6) ecuse the det impednce is ised downwrd y geometric distortion motivted us to present systemtic investigtion of the pproches proposed y Rung-Arunwn et l. (6) in this pper. Identifiction nd removl methods for glvnic distortion remin undetermined (Chp. 6 in Chve nd Jones ), lthough severl ttempts to solve the prolem of glvnic distortion hve een presented. Some studies ssumed D Erth (e.g., Bhr 988; Groom nd Biley 989), wheres others confronted the nonuniqueness of

4 Pge 4 of 4 the otined solution (e.g., Biy et l. 5). Moreover, Gómez-Treviño et l. (4) presented n pproch to estimte the D regionl impednce nd distortion prmeters, i.e., twist nd sher in the Groom Biley model, using the det nd ssq impednces. Inversion sed on the phse tensor (Cldwell et l. 4), which yields well-defined distortion-free solution, is lso promising strtegy. However, the phse tensor is only prtil solution; thus, the inverted model strongly depends on the initil model (Ptro et l. ; Tietze et l. 5). In ddition to decomposition pproches, inversion schemes tht simultneously solve the sttic shift (e.g., Sski nd Meju 6) hve ecome fesile, ut the geometric distortion is not controlled. Avdeev et l. (5) proposed D inversion with the solution of the full distortion mtrix, ut this pproch does not llow this sttic shift to e free prmeter. Although numer of pproches for hndling glvnic distortion hve een developed, n pproch for determining the presence of glvnic distortion in the oserved dt hs not een presented, except the concept of glvnic distortion indictors y Rung-Arunwn et l. (6). The ility to identify the presence of glvnic distortion either geometric or scling contined in the oserved dt nd to quntify their intensity is undoutedly importnt ecuse the ppliction of the glvnic distortion tretment to the oserved dt without knowing the presence of glvnic distortion nd doing so my either improve or deteriorte the reliility of MT dt interprettion. Rung-Arunwn et l. (6) proposed two types of glvnic distortion indictors. First, the locl nd regionl distortion indictors re used to determine the strength of the geometric distortion s expressed y the sher nd splitting effects on the sis of the fct tht the geometric distortion (sher nd splitting) hs different effects on the det nd ssq impednces. Second, the pprent gin is defined to e n pproximtion of the site gin (scling in the impednce mgnitude), which hs een presumed to e indeterminle without other independent informtion (Groom et l. 99; Biy et l. 5). These prmeters my help quntittively indicte the strength of the glvnic distortion posed in MT dt. In ddition, the employment of these two types of properties llows the effect of the site gin to e seprted from the effects of the twist, sher, nd splitting prmeters. Most importntly, we cn use these prmeters to determine the necessry tretment of glvnic distortion for given dtset, such s whether or not removl scheme should e pplied in the inversion (e.g., Sski nd Meju 6; Avdeev et l. 5). The im of this pper is to present synthetic exmples for estimting model of the regionl men D profile, the locl nd regionl distortion indictors, nd the pprent gins using the methods proposed y Rung- Arunwn et l. (6). First, the proposed methods re riefly summrized. The results from D exmples re then discussed to illustrte the sic concepts nd the ehvior of the proposed properties nd prmeters. In ddition to the estimtion of the model of the regionl men D profile from distorted sets of synthetic D impednces, the estimted model of the men D profile is compred with σ R (z) defined y Eqs. () or (4) for D exmples. The numericl results of the locl nd regionl distortion indictors nd the pprent gins re presented nd verified with synthetic vlues. In Rung- Arunwn et l. (6) nd this pper, the Groom Biley model of glvnic distortion is chosen, simply ecuse the site gin is distinguished from the geometric distortion whose opertors re normlized with their Froenius norms. Still, the choice of glvnic distortion model my e ritrry. Thus, numericl exmples nd discussion regrding the glvnic distortion model dependence of the proposed methods re provided. Theoreticl ckground This section riefly summrizes the method for estimting model of the regionl men D profile nd set of prmeters relted to the glvnic distortion, which were presented in Rung-Arunwn et l. (6). First, model of the regionl men D profile is estimted y inverting the verge ssq impednces. As hs een lgericlly proven, the ssq impednce is reltively less sensitive to the effects of the sher nd splitting prmeters e nd s, which re lso clled the geometric distortion, thn the det impednce, the mplitude of which is lwys ised downwrd y these two prmeters (Rung-Arunwn et l. 6). After re-exmintion with the Groom Biley model of glvnic distortion, the Berdichevsky verge is written s [ N Z det (ω) = Z det (r i; ω) i= i= ] N [ N ] ( ei )( s i ) N ( + ei )( + s i ) Zdet R (ω), where Z det (r i; ω) is the det impednce of the ith oserved (perhps distorted) MT impednce t the position r i ; e i nd s i re the sher nd splitting prmeters t the ith sttion, respectively; N is the totl numer of oservtions; ω is the ngulr frequency; nd Zdet R (ω) is the regionl det impednce. Note tht the twist prmeter hs no effect on the det nd ssq impednces (lso discussed in Gómez- Treviño et l. ; Rung-Arunwn et l. 6). If geometric distortion is contined in the dt, the coefficient (6)

5 Pge 5 of 4 in Eq. (6) ecomes effective nd is lwys smller thn unity. Hence, the use of the Berdichevsky verge lwys gives downwrd-ised regionl D impednce, which yields n inverted model of the structure tht is more conductive thn the true structure. On the contrry, the ssq impednce ws proven to e less ffected y geometric distortion (Gómez-Treviño et l. ; Rung-Arunwn et l. 6). Therefore, verging the ssq impednces from n rry of MT oservtions gives good pproximtion of the true regionl response: [ N Z ssq (ω) = Z ssq (r i; ω) i= where Z ssq (r i; ω) is the ssq impednce of the oserved MT impednce tensor t the position r i nd Zssq R (ω) is the regionl ssq impednce (Rung-Arunwn et l. 6). A detiled discussion regrding the geometric nd rithmetic verges of the MT impednces cn e found in Section of the Additionl file. Moreover, the vlidity of the pproximte equlity in Eq. (7) will e exmined in Section of the Additionl file. Additionlly, set of prmeters relted to the glvnic distortion is defined s follows. The locl distortion indictor (LDI) indictes the strength of the sher nd splitting prmeters t single sttion individully nd is defined s the squred rtio of the ssq impednce to the det impednce: Defined in this wy, the LDI is intrinsiclly independent of the site gin. As the twist prmeter hs no effect on the det nd ssq impednces, the presence of the twist effect cnnot e scertined from the LDI. Employing the fct tht the sher nd splitting distortion ffects the det nd ssq impednces differently (Gómez-Treviño et l. ; Rung-Arunwn et l. 6), the LDI represents the effects of the sher nd splitting prmeters s comintion, which is unlike the decomposition pproches (e.g., Groom nd Biley 989; McNeice nd Jones ; Gómez-Treviño et l. 4), where the distortion prmeters, twist nd sher in prticulr, re estimted. The regionl distortion indictor (RDI) lso indictes the strength of the sher nd splitting prmeters ut on regionl scle, i.e., it quntittively indictes how strongly distorted the dtset is on verge. It is defined s the geometric men of the LDIs: ] N Z R ssq (ω), γ i (ω) = Z ssq (r i; ω) Z det (r i; ω) ( + e i )( + s i ) ( e i )( s i ) Z R ssq (r i; ω) Z R det (r i; ω). [ N γ R (ω) = γ i (ω) i= ] N (7) (8) [ N ] N ( ei )( s i ) Z ssq R (ω) ( + e i= i )( + s i ) Z R, det (ω) (9) where Z ssq R (ω) nd Z det R (ω) re the verges of the regionl ssq nd det impednces, respectively. The pprent gin is defined s the rtio of rottionl invrint t given position to its regionl verge. As we re interested in two rottionl invrints, the corresponding pprent det nd ssq gins re derived s gi det (ω) = Z det (r i; ω) ( ei () Z det (ω) g )( s i ) i ( + ei )( + s i ) nd g ssq i (ω) = Z ssq (r i; ω) Z ssq (ω) g i, () where g i is the site gin for the ith oservtion site. Oviously, if the dt re strongly distorted, the pprent det gin underestimtes the site gin ecuse of the sher nd splitting prmeters. Thus, the pprent ssq gin is expected to e the more ccurte pproximtion of the site gin when the dt re strongly distorted. In the following sections, the chrcteristic nd ehviors of these prmeters re syntheticlly exmined. Estimtion of model of the regionl men D profile Rung-Arunwn et l. (6) proposed modifiction to the Berdichevsky verge the use of the verge ssq impednce insted of the verge det impednce to void ising from glvnic distortion. This section exmines whether the proposed method cn relily estimte model of the regionl men D profile from syntheticlly distorted dt. Here, we synthesize the Erth conductivity model y comintion of D structure nd lterl heterogeneity, s given y Eq. 5. The D prt is sed on reference model of the continentl crust nd upper mntle y Jones (999) nd hs the min fetures of resistive upper crust nd conductive lower crust (Fig. ). In this model, the upper crust extends from.5 to 4.8 km in depth, nd the lower crust extends to depth of. km. The corresponding D impednce (Fig. ) ws otined from the nlyticl solution, i.e., the recursive formuls in terms of coth functions, for the D MT prolem (e.g., Chve nd Jones ). Note tht the complex impednce is generlly represented s n pprent resistivity nd phse. Here, the period rnge ws selected to sense structure existing etween nd km in depth. Therefore, ny smll structures confined in the ner-surfce lyer of few kilometers or less, which is shllower thn the inductive scle length of present interest, re considered to e glvnic distorters (Utd nd Munekne ). We further ssume for simplicity tht glvnic distorters hve

6 Pge 6 of 4 Depth [km] Phse [deg] Resistivity [Ohm m] Fig. Lyered-erth model used in this work. Corresponding MT response (pprent resistivity nd phse) typicl size smller thn the typicl site spcing of the oservtion rry. The effect of these ner-surfce distorters is therefore rndom phenomenon, nd it cn e expressed s mthemticl model such s the Groom Biley model of glvnic distortion, which is dopted in this work. The site gin nd other distortion prmeters (twist, sher, nd splitting) re treted s rndom vriles (e.g., Avdeev et l. 5). The synthetic MT rry contins 5 MT sttions. Therefore, 5 cohorts of the site gin g nd the twist t, sher e, nd splitting s of the prmeters of the Groom Biley model were generted following norml distriution (Fig. ). The distorted impednces were then clculted y pplying these rndom prmeter vlues to the synthetic impednces. More explicitly, we ssumed tht ech set of distortion prmeters hs men of zero nd is ounded y (, +). If ny vlues re outside the ound, rndom numers were generted gin so tht the set of rndom distortion prmeters conforms with the ound. The rndom site gin ws generted on logrithmic scle without ound. To quntittively control the strength of the glvnic distortion, the stndrd devition (SD) of the norml distriution of ech prmeter ws vried. Five SD vlues of.,.,.,.4, nd.5 were used. Finlly, five MT dtsets with 5 sttions ech nd different glvnic distortion strengths were considered. D exmple First, we consider the simplest cse with D (horizontlly strtified) Erth structure where the impednce t ech site contins glvnic distortion. In this cse, the glvnic distortion cuses no phse mixing ut only sttic shift, g, which is the frequency-independent shift in the pprent resistivity (e.g., Bemish nd Trvssos 99). As n exmple of distorted dt, the ssq nd det impednces from the synthetic D impednce distorted with (g, t, e, s) = (.,.,.7,.49) re shown in Fig. 4. Here, the synthetic site gin is greter thn unity; therefore, the distorted ssq impednce is shifted upwrd. In generl, the site gin eqully ffects the det nd ssq impednces, i.e., the det impednce should lso e shifted

7 Pge 7 of 4 Normlized occurrence..5. SD=. log(gin) sher twist splitting SD=. Normlized occurrence SD=. Normlized occurrence Normlized occurrence..5. SD= Normlized occurrence..5. SD= Distortion prmeter vlue Fig. Distriutions of rndom distortion prmeters with different SDs. The normlized occurrence is the numer of occurrences divided y the mximum numer of occurrences t single prmeter vlue. Ech distriution is compred with the proility density function of the theoreticl norml distriution for the given SD (dshed lines) upwrd. However, the effect of the sher nd splitting shifts the distorted det impednce downwrd insted. As result, the mgnitude of the distorted det impednce t this site is smller thn the undistorted one. All distorted sounding curves with n SD of. re shown in Fig. 5. These curves re shifted irregulrly ecuse of the rndom distortion prmeters. After the distortion prmeters with different SDs were pplied, five MT dtsets with different glvnic distortion strengths were otined. For ech dtset, the verge det nd ssq impednces were then clculted using Eqs. (6) nd (7) (Fig. 6), respectively. Here, the error rs indicte the SD, which were clculted in the logrithmic spce, of the dt nd thus represent the level of

8 Pge 8 of 4 Z det undistorted Z ssq undistorted Z det distorted Z ssq distorted Fig. 4 Exmple of det (dimonds) nd ssq (squres) impednces from the synthetic D impednce distorted with (g, t, e, s) = (.,.,.7,.49) dispersion in the glvnic distortion strengths. At equl distortion strengths, the distorted det impednces re generlly more disperse thn the ssq impednces, s demonstrted y the lrger error rs in the det impednce results. This is result of the fct tht the det impednce is systemticlly ised downwrd y the sher nd splitting prmeters in ddition to the effect of the site gin, wheres the ssq impednce is ffected mostly y the site gin. The downwrd is of the verge det impednce ecomes noticele when the SD of the distortion prmeters is greter thn.. To otin the regionl men D profile, the verge invrint impednces were inverted with D Occm inversion, in which the second derivtive of the conductivity with respect to the depth nd conductivity is penlized (Constle et l. 987). Here, the errors in the pprent resistivity nd phse were fixed to.% nd.66, respectively. All inverted models fit the dt within root-men-squre (RMS) misfit of unity. Becuse of the downwrd is mentioned ove, the models inverted from the verge distorted det impednces tend to e more conductive thn the synthetic profile when the distortion is stronger (Fig. 7). As consequence, the D models from the downwrd-ised det impednces my misinterpret the depth of the structure. Conversely, the verge distorted ssq impednce is much less sensitive to the geometric distortion prmeters; therefore, the models inverted from the distorted ssq impednces were ll similr to tht from the undistorted impednce (Fig. 7). These numericl results confirmed the vlidity of the theoreticl prediction tht the verge ssq impednce will yield n unised estimte of the regionl D structure. Moreover, the ehviors of the det nd ssq impednces under glvnic distortion, in which the det impednce is ised downwrd y nd the ssq impednce is less sensitive to the geometric distortion, re lso consistent with the numericl results presented in Gómez-Treviño et l. (). Next, we consider cse when the regionl structure includes D nomly. D exmple To generte synthetic D dt, model of checkerord structure with resistivities of nd m nd size of 8 km 8 km ech (Fig. 8) emedded in the lower crust of the lyered-erth model used in the D exmple is constructed (Fig. ). The nomly is lrge nd systemtic nd corresponds to the inductive scle lengths (see Utd Fig. 5 Distorted det nd ssq impednces from the D exmple, where set of distortion prmeters with n SD of. ws pplied. Ech sttion is represented y different symol color

9 Pge 9 of 4 Undistorted SD=. SD=. SD=. SD=.4 SD=.5 Undistorted SD=. SD=. SD=. SD=.4 SD=.5 Fig. 6 Averge det nd ssq impednces from the D dtsets distorted with different glvnic distortion strengths Depth [km] 4 Depth [km] True model Undistorted SD=. SD=. SD=. SD=.4 SD=.5 log(resistivity [Ohm m]) 8 True model Undistorted SD=. SD=. SD=. SD=.4 SD=.5 log(resistivity [Ohm m]) Fig. 7 D models otined y inverting the verge det nd ssq impednces from the distorted D dtsets (Fig. 6). The synthetic profile is lso shown s dshed line for comprison. Note tht ll profiles re lmost identicl in ()

10 Pge of 4 nd Munekne ) pproximtely rnging from.68 to 5. km t the shortest ( s) nd longest periods ( s), respectively. The inductive effect from the nomly is expected to hve significnt effect on the response ecuse its corresponding inductive scle length is comprle to its physicl dimension, nd the D inductive effect from ech nomly is recognizle ecuse it is emedded t depth tht could e recognized within the given period rnge. An rry of 5 irregulrly distriuted MT sttions ws ssumed to cover the 6 km 6-km re of interest. The typicl site spcing ws then set to km, which is smller thn the nomly size. On verge, ech site represents n re of km km (/5 of the study re). The rndom loction (x i, y i ) of the ith sttion is given y x i = x c + s r x y () i = y c + s r y, where (x c, y c ) is the coordinte of the mesh center represented y ech MT site; s is the typicl site spcing, which is km in this cse; nd r x nd r y re uniform rndom numers ounded y (.5, +.5). In this work, the synthetic D MT responses were clculted using the softwre WSINVDMT (Siripunvrporn et l. 5; Siripunvrporn nd Egert 9). The size of the entire checkord nomly is km km, nd the horizontl mesh resolution is 5. km 5. km. The ccurcy of the clcultion with this resolution y [km] syn syn syn syn4 syn5 syn6 syn7 syn8 syn9 syn syn syn syn syn4 syn5 syn6 syn7 syn8 syn9 syn syn syn syn syn4 syn5 Ω m Ω m x [km] Fig. 8 Model of checkerord structure with resistivities of nd m nd size of 8 km 8 km emedded in the lower crust lyer (from the depth of 4.8 to. km of the lyered-erth model shown in Fig. ). An rry of 5 irregulrly distriuted MT sttions (crosses) covers the re of interest, which is 6 km 6 km (dshed frme). Here, one MT sttion represents n re of km (dshdotted frmes) ws confirmed y nother clcultion with smller mesh. The digonl elements of synthetic undistorted impednces re smller in mgnitude thn tht of the off-digonl elements y few orders of mgnitude (Figure S9 in Section of the Additionl file ). The det nd ssq impednces from this rry re shown in Fig. 9,, respectively. The frequency-dependent vrition due to the emedded nomlies cn e recognized oth for the det nd ssq impednces. To otin distorted D synthetic dt, the rndomly generted distortion tensors used in the D exmple were pplied to the synthetic dt from this rry. When the MT impednce is distorted, the digonl elements ecome significnt nd cn e comprle to the off-digonl elements (Figure S in Section of the Additionl file ). In the D sitution, the effect of the distortion prmeters on the rottionl invrints is different from tht in the D cse ecuse the ner-surfce distorter cuses sttic shift nd phse mixing, i.e., mixing mong the different elements of the MT impednce tensor. An exmple of the rottionl invrints from the distorted dt is shown in Fig.. The ssq impednce is shifted upwrd (ecuse of the site gin t this sttion) nd contins frequency dependence, s demonstrted y differences in the mgnitude nd phse derived from the distorted nd undistorted ssq impednces (Fig. ). Unlike the ssq impednce, only the mgnitude of the det impednce is ffected y the distortion. From Fig., the mgnitude of the det impednce is ised downwrd ecuse the impednce t this site is distorted y the sher nd splitting prmeters, wheres its phse remins unchnged. All det nd ssq impednces distorted y distortion prmeters with n SD of. from this rry re shown in Fig.. The difference etween those using the distorted det impednces nd those using the distorted ssq impednces is cler when they re verged (Fig. ). Here, the error rs indicte the SD. At the sme distortion strengths (SD), the verge ssq impednces hve smller SDs thn the det impednces, which is the sme s in the D cse. This lso confirms tht the ssq impednce is less sensitive to glvnic distortion. Consequently, the pprent ssq gin (Section 6) should e good pproximtion of the site gin. In ddition, the pproximtions tht the effects of the dimensionlity nd geometric distortion would e minor fter verging over numer of MT sttions, which is pplied in Eq. (7) of Rung-Arunwn et l. (6), hve een verified with the clcultion detiled in Section of the Additionl file. Next, we inverted the verge det nd ssq impednces to otin models of the regionl men D profiles for different cses with the sme criteri used in the D exmples. Becuse the verge det nd ssq impednces from the undistorted dt re similr, the models derived from them

11 Pge of Phse [deg] 6 45 Phse [deg] Fig. 9 Det nd ssq impednces from the rry of MT sttions over the D nomlies (Fig. 8) without glvnic distortion. Ech sttion is represented y different symol color re pproximtely the sme (Fig. ). These models re lso consistent with the theoreticl models of the men D profile, which were clculted y pplying Eqs. () nd (4) to the conductivity distriution within the re of interest in this setting (dshed frme in Fig. 8). However, with the presence of glvnic distortion, the models of the men D profile derived from the verge det impednces tend to e more conductive. Conversely, t ny distortion strength, the verge ssq impednces yield models of the regionl men D profile tht re close to the undistorted one. Nonetheless, we should note tht this result is otined simply ecuse the rry size is sufficiently lrger thn the typicl nomly size. In the D sitution, estimtion of the regionl men D profile could e ffected y the size of the oservtion rry nd its loction reltive to the loction of the nomly, even if the sme numer of oservtion sites is involved. We will exmine these issues in the next section. Exmintion of the consistency etween the theoreticl nd estimted models of the regionl men D profile According to the fct tht the host lyer erth or ckground is solutely unknown in relity, the estimted models of the regionl men D conductivity profiles from D models should not e compred with the synthetic lyered-erth model (the model in Fig., for exmple). Insted, it should e compred with the theoreticl regionl men D conductivity profiles, the liner nd logrithmic verges of the lterl conductivity distriution (Eqs. nd 4). Oviously, the regionl men D profiles, either theoreticl or estimted, depend on the rry size nd loction when the susurfce structure is lterlly heterogeneous. This section ims to exmine the consistency etween the defined nd estimted models of the regionl men D profile nd the effect of the

12 Pge of 4 Z det undistorted Z ssq undistorted Z det distorted Z ssq distorted Diff. of log( [Ohm m]) Z det distorted Z ssq distorted 9 4 Phse [deg.] 6 Diff. of phse [deg] 4 Fig. Exmple of det nd ssq impednces from sttion syn8 distorted with (g, t, e, s) = (.,.,.7,.49). Difference etween the distorted nd undistorted rottionlly invrint impednces consistency on the loction of the rry nd its size reltive to the nomly size through synthetic modeling. We gin use the model of the checkerord structure, s descried in Section.. In the first exmple, the oservtion rry is lso the sme, i.e., 5 MT sttions within n re of 6 km 6 km, ut the rrys re set in three different loctions t centrl, northwest, nd northest loctions (Fig. 4). At the centrl loction, the rry is concentric with the nomly intersection, wheres t the northwest nd northest loctions, the rrys re centered over the nd m nomlies, respectively. As shown in the previous sections, the det impednce is ised y the geometric distortion (sher nd splitting) so tht the ssq impednce is only considered in the following. The verge ssq impednces from these rrys were clculted nd inverted in the sme mnner s descried in Section.. The theoreticl models of the men D profile (Fig. 5) were clculted using Eq. () or (4) from the conductivity distriution within the re of the oservtion rry (e.g., the dshed frmes in Fig. 4). The MT responses from the theoreticl models (Fig. 5) were then clculted using the nlyticl solution (see Chve nd Jones ). In this sitution, where the rry is much lrger thn the typicl nomly, the regionl men D conductivity profiles from different rry loctions, oth theoreticl nd estimted, re shown to e lmost identicl to the theoreticl model. This is lso consequence of the ppliction of the verging pproch, in which the effects of the positive nd negtive nomlous conductivities re verged out. In other words, the theoreticl nd estimted models of the regionl men D profile re nerly independent of the rry loction when the rry size is much lrger thn the typicl nomly size. Next, to demonstrte the effect of the rry size, we decrese the size of ech rry to 8 km 8 km (Fig. 6),

13 Pge of Phse [deg] 6 45 Phse [deg] Fig. Det nd ssq impednces from the D exmple (s shown in Fig. 9,, respectively) distorted with set of distortion prmeters with n SD of. which is equl to the nomly size. From this setting, the estimted results re shown to e sptilly dependent (Fig. 7). Moreover, the inconsistency etween the theoreticl nd estimted models of the regionl men D profiles is evident, prticulrly in the lyer where the nomly is emedded (Fig. 7). The inconsistency ecomes more ovious if the rry size is further reduced. This is consequence of the inpproprite design of the oservtion rry, i.e., it is not lrge enough to cover the structure of interest. However, hving n rry with n pproprite size nd site spcing my e difficult in relity without ny priori knowledge ecuse the size of structure is usully unknown eforehnd. Thus, the estimtion of the men D profile with lrger rry would e more relile. In generl, the oservtion rry should e designed to cover the structure of interest if its size is known priori. However, if the nomly size is found posteriori to e comprle to or even lrger thn the size of oservtion rry, D inversion of ny pproch will fil to ccurtely imge the heterogeneity. To otin more relile results, one suggestion in such cse is to dd more MT oservtions to mke the rry size sufficiently greter thn the nomly size. Locl nd regionl distortion indictors On the sis of the fct tht the glvnic distortion hs different effects on the det nd ssq impednces, the LDI nd RDI given y Eqs. (8) nd (9), respectively, were constructed to quntify the strength of the geometric

14 Pge 4 of Undistorted SD=. SD=. SD=. SD=.4 SD= Undistorted SD=. SD=. SD=. SD=.4 SD=.5 Phse [deg] 6 45 Phse [deg] Fig. Averge det nd ssq impednces from the D dtsets distorted with different glvnic distortion strengths distortion tht cn e descried y the sher nd splitting prmeters. This section exmines the numericl results of LDIs nd RDIs derived from the synthetic D nd D exmples presented in Sections. nd., respectively. For the D cse without distortion, where the det nd ssq impednces re identicl, the LDI is unity. However, when the impednces re distorted, the LDIs from the D exmple re shifted upwrd (lrger thn unity) ut remin rel-vlued (Fig. 8). Lrger LDIs correspond to stronger geometric distortions t the MT sites. The D nomlies t the depth of interest cuse frequency-dependent difference etween the det nd ssq impednces ecuse of the inductive effect. The LDIs then ecome frequency-dependent nd complex-vlued (Fig. 8), ut the effect of the geometric distortion domintes. The RDIs lso show fetures similr to the locl ones. In the D cse, the RDIs re shifted upwrd depending on the distortion strength throughout the dtset (Fig. 9). Thus, the RDI will e le to tell whether or not simple glvnic distortion model is pplicle to the given dtset. In contrst to the LDI, the frequency-dependent fetures from the D effect re smoothed, s shown in Fig. 9, such tht the RDIs re lmost rel-vlued nd wekly frequency-dependent if the distortion is purely glvnic. For prcticl usge of the LDI, we clculte the men LDI γ i s the geometric verge of the rel prt of the LDIs over given period rnge. The rel prt chosen s the LDI is rel-vlued numer in cses of D erth. At the ith sttion, M γ i = R γ i (ω j ) j= M, ()

15 Pge 5 of 4 Depth [km] 4 Depth [km] Theo. liner Theo. log Undistorted SD=. SD=. SD=. SD=.4 SD=.5 log(resistivity [Ohm m]) 8 Theo. liner Theo. log Undistorted SD=. SD=. SD=. SD=.4 SD=.5 log(resistivity [Ohm m]) Fig. D models inverted from the verge det nd ssq impednces from the distorted D dtsets (Fig. ). The theoreticl models of the men D profiles, σ R (z), from this setting with oth liner (Eq. ) nd logrithmic (Eq. 4) scling (lck dshed lines) nd the D model from the undistorted dt (lck solid line) re shown for comprison where M is the numer of periods. The percentge error in the men LDI is clculted with P( γ i ) = γ i γ i γ i %, where the synthetic LDI is clculted using γ i = ( + e i )( + s i ) ( e i )( s i ). (4) As the LDI correctly estimted the integrted effect of sher nd splitting in D cses, their mens (Eq. 4) re not shown here. However, s shown erlier, the underlying structures ffect the LDIs (Fig. 8) if they re not D. The men LDIs from the D exmple my include some error in the estimte of the effect of the geometric distortion t ech sttion (Fig. ). The error rs of the men LDIs (Fig. ) here re set to the SD of the rel prt of the men LDI in order to represent the dispersion in the frequency-dependent prt contriuted y the underlying structure. One possile prcticl usge of the LDI is the omission of some sttions with hevily distorted impednces from the interprettion or inversion if the numer of such sites is smll. If limited numer of sites showing hevy distortion re removed, the RDI fter removl is supposed to e smll. Conversely, if the RDI still exhiits high vlue, proper tretment for the glvnic distortion, such s inversion including the glvnic distortion (e.g., DeGroot-Hedlin 995; Ogw nd Uchid 996; Sski nd Meju 6; Avdeev et l. 5) or n MT dt nlysis (e.g., Wever et l. ; Cldwell et l. 4), will e essentil. The comintion of LDIs nd RDI helps to provide insight, t lest to some extent, s to which pproch should e pplied to set of MT impednces otined from oservtion. Apprent gins From the theoreticl derivtion, the pprent ssq gin is expected to correctly estimte the site gin in D cses nd to yield good pproximtion of it in D cses, wheres the pprent det gin underestimtes the

16 Pge 6 of Northwest Northest the geometric men of the rel prt of the pprent gins over given period rnge (s with Eq. ). Only the rel prt ws used to ensure consistency with the mthemticl ssumption tht the distortion opertor is tensor of rel-vlued numers. Given tht the numer of periods where the impednces were otined t ech sttion is M, the men pprent gins cn e written s y [km] ḡ det i = M R gi det (ω j ) j= M (5) 8 Centrl Ω m Ω m x [km] Fig. 4 Checkerord structure with n nomly size of 8 km 8 km nd resistivities of (drk gry) nd (light gry) m. The checkerord structure ws emedded in the lower crust lyer (from depth of 4.8 to. km in the lyered-erth model shown in Fig. ). Three rrys of 5 MT sttions (crosses), ech with size of 6 km 6 km (dshed frmes), were plced t centrl (lck), northest (lue), nd northwest (red) positions synthetic site gin if the dt re strongly ffected y geometric distortion. In this section, we demonstrte the use of the pprent gins otined from the synthetic D nd D exmples descried in Sections. nd., respectively. From our D exmple, the pprent ssq gin (Eq. ) perfectly grees with the synthetic site gin, ut the pprent det gin (Eq. ) is ised downwrd, s expected (Fig. ). Here, the error rs of the pprent det nd ssq gins re derived from the SD when estimting the regionl verges of the det nd ssq impednces, Z det nd Z ssq in Eqs. (6) nd (7), respectively. However, in the D cse, the induction effect of the D heterogeneity, which cn e oserved in the frequency-dependent fetures of oth the mgnitude nd phse (Fig. ), is included. For exmple, the pprent ssq gin from the sttion syn8 (Fig. ) is ised downwrd nd then underestimtes the synthetic site gin in the period rnge where the induction from the underlying regionl conductive nomly is effective. Moreover, vrition in the pprent gins mong different sites due to the underlying D structure is oserved (Fig. ). However, the pprent ssq gin still grees with the synthetic site gin within the stndrd error. To meningfully interpret these results, we clculted the men pprent det nd ssq gins, ḡi det nd ḡ ssq i, using nd The men pprent gins estimted t synthetic MT sites from D nd D exmples re ll presented in Figs. nd, respectively. The percentge differences etween the men pprent gins nd the synthetic site gins re shown in Figs. nd, respectively. They re given y nd ḡ ssq i = P(ḡ det i P(ḡ ssq i M R g ssq i (ω j ) j= ) = ḡdet i ) = ḡssq i (6) where g i is the synthetic site gin t the ith sttion. In spite of the lrge site-to-site vrition in the synthetic site gin of nerly one order of mgnitude (two orders of mgnitude in terms of the sttic shift in the pprent resistivity), its estimtion error y the men pprent ssq site gins is s smll s only few percent. The men pprent gins from the D exmple re shown in Fig.. The error rs in this figure re derived from the error propgtion in clculting the men of the pprent gin t ech sttion. In this cse, the men pprent ssq gins nd synthetic site gins re the sme for every MT oservtion (Fig. ). Conversely, the pprent det gin my either underestimte or overestimte the synthetic site gin depending on the strength of the locl glvnic distortion. Unlike the D cse, the existing D nomlies my cuse further uncertinty, s the pprent ssq gin hs een demonstrted to e ffected y the induction effect from the underlying D structure. For exmple, the men pprent ssq gins from sttions over the conductive structure (e.g., sttions syn7 nd syn9) tend to e M g i g i g i, g i. (7) (8)

17 Pge 7 of 4 C: Estimted C: Theo. liner C: Theo. log NE: Estimted NE: Theo. liner NE: Theo. log NW: Estimted NW: Theo. liner NW: Theo. log Depth [km] Resistivity [Ohm m] C: Estimted C: Theo. liner C: Theo. log NE: Estimted NE: Theo. liner NE: Theo. log NW: Estimted NW: Theo. liner NW: Theo. log Phse [deg] Fig. 5 Theoreticl (dshed lines) nd estimted (solid lines) models of the men D profiles otined using the settings shown in Fig. 4. Corresponding MT responses from the theoreticl (dshed line) nd estimted (squres) models of the men D profiles. The results otined from different rry loctions Centrl (C), Northwest (NW), nd Northest (NE) re represented y lines nd symols of different colors slightly smller thn the synthetic site gins (Fig. ). In spite of this, the percentge differences (Eq. 8) still remin out % which is within the sttisticl uncertinty (Fig. ). The regionl distortion indictor in this cse (Fig. 9) shows feture consistent with the distorted D cse (Fig. 9) t periods shorter thn 5 s. If we estimte the men ssq gin from this period nd insted of Eq. (6), the percentge gin difference ecomes s smll s 5%. In previous works, the site gin is considered or regrded to e n indeterminle distortion prmeter if other independent geophysicl dt, e.g., trnsient electromgnetic (TEM) dt (Bemish nd Trvssos 99; Groom et l. 99; Biy et l. 5; Árnson 5), re not ville. However, the TEM dt my not e ville t ll MT sttions. In ddition, the sttic shift could e corrected with the TEM dt with some limittions (see Wtts et l. ; Tournerie et l. 7; Wilt nd Willims 989), e.g., when the heterogeneity is smller thn the trnsmitter loop. Utd nd Munekne () ttempted to solve this prolem y introducing Frdy s lw s constrint, ut the solution ws not prcticl. The numericl exmples presented here show tht the concept of the pprent gin cn e used to pproximte the site gin in the ssumed sitution. This pper considers the glvnic distortion cused only y smll-scle heterogeneities (smller thn the typicl site spcing nd confined within ner-surfce lyer shllower thn the inductive scle length of interest). Thus, the effect of glvnic distortion is considered s sptil lising in the MT dt. The pprent gin cn then e regrded s shift in the mgnitude of the impednce reltive to the verge vlue. For the cse where the dt re systemticlly shifted y some nersurfce structure lrger thn or comprle to the rry size (see Section 4), e.g., vlley environment such s of the Rhine Gren model (see Chp. 6 in Chve nd Jones ), the pprent ssq gin my e distriuted round

18 Pge 8 of 4 y [km] Northwest Centrl Northest Ω m Ω m x [km] Fig. 6 Sme s Fig. 4 for n rry size of 8 km 8 km some ised centrl vlues or my not e normlly distriuted on logrithmic scle. In such cses, the concept of the pprent gin should e used with cution. Dependence on the distortion model The proposed method (Rung-Arunwn et l. 6) is theoreticlly formulted on the sis of the Groom Biley model of glvnic distortion. In this pper, it is shown numericlly tht the use of the verge ssq impednce is relile method for estimting the regionl men D conductivity profile, nd the comintion of the two rottionl invrint impednces helps to detect the geometric distortion nd to pproximte the site gin. Although the Groom Biley model is well known nd dopted y numer of studies, it is not the only model. The distortion opertor C cn e prmeterized using other models (e.g., Bhr 988; Chve nd Smith 994; Smith 995; Tietze et l. 5). Therefore, the glvnic distortion model dependence of the proposed methods my e questionle. Depth [km] 4 9 C: Estimted Centrl: C: Theo. Estimted liner Northest: C: Theo. Estimted log NE: Estimted NE: Theo. liner NE: Theo. log NW: Estimted NW: Theo. liner NW: Theo. log Resistivity [Ohm m] Fig. 7 nd Sme s Fig. 5 for the settings shown in Fig. 6 C: Estimted C: Theo. liner C: Theo. log NE: Estimted NE: Theo. liner NE: Theo. log NW: Estimted NW: Theo. liner NW: Theo. log Phse [deg]

19 Pge 9 of 4 Mgnitude of LDI Mgnitude of LDI 5 5 Phse of LDI [deg.] 5 5 Phse of LDI [deg.] Fig. 8 LDIs from the distorted D nd D dt (in D exmple, D exmple sections, respectively), where set of distortion prmeters with n SD of. ws pplied. Exmples of D nd D dt t sttion syn8 distorted with (g, t, e, s) = (.,.,.7,.49) (gry circles) re compred with the synthetic vlues of the LDI t this sttion (dshed lines) In this section, we exmine the glvnic distortion model dependence of the proposed methods y using the pertured identity mtrix (PIM) model for the distortion opertor, which hs een recently introduced to model the glvnic distortion (e.g., Tietze et l. 5). Like the Groom Biley model, the PIM model hs four degrees of freedom. In the PIM model, the distortion opertor C is expressed s the perturtion of the identity mtrix, i.e., C = I + D, where I is identity mtrix, nd D is perturtion mtrix descriing the distortion. In the test, we simulte n rry of 5 MT sttions over D erth, s in Section.. The 5 cohorts of D elements re to e normlly distriuted rndom numers with vrious SD levels, nd the distortion opertors re then formed nd pplied to the D impednce tensors (Section.). The verge det nd ssq impednces from the distorted dt re clculted. Figure 4 shows the verge impednces for the cse with n SD of.5. The verge det nd ssq impednces re then inverted in the sme mnner s in Section., nd the resulting D models re shown in Fig. 5. As expected y our theory, the verge det impednce gives n underestimte of the regionl men D conductivity profile, while the verge ssq impednce depends less on the distortion. Approches for detecting the glvnic distortion re lso effective for the PIM model. The LDIs from the dtset distorted using D re shown in Fig. 4. The LDIs in the PIM model re consistent with the theoreticl expecttion tht they re greter thn unity if geometric distortion exists, nd their mgnitudes represent the distortion strength. The concept of the pprent gin is lso pplicle under the PIM model. The pprent ssq gins from different sttions nd their men vlues (Fig. 6) re clculted using Eqs. () nd (6), respectively. They seem consistent with the Froenius gin, the gin derived from the Froenius norm of the distortion tensor C (see Biy et l. 5), lthough the Froenius gins re neither normlly distriuted nor hve zero men (Additionl file :