SEG Houston 2009 International Exposition and Annual Meeting

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1 Fonations of the metho of M fiel separation into pgoing an ongoing parts an its application to MCSM ata Michael S. Zhanov an Shming Wang*, Universit of Utah Smmar The renee interest in the methos of electromagnetic fiel ecomposition into pgoing an ongoing parts is generate b the practical problem of removing the effect of M airaves on marine controlle sorce electromagnetic ata collecte in shallo ater. In this paper e consier the principles of solving this problem sing the classical methos of M fiel separation into eternal an internal parts. We emonstrate that the most general approach to pgoing/ongoing fiel ecomposition is base on the theor of Stratton-Ch tpe integrals. This approach allos s to separate the fiel observe on an arbitrar srface. In the case of flat observational srface, the Stratton-Ch metho is eqivalent to the ecomposition techniqe base on the Forier transform in the spatial freqenc omain. We present also a novel metho of M fiel separation sing the metho of horiontal graients of the M fiel. The ne techniqe is teste on snthetic MCSM ata. We emonstrate also that the metho represents a sefl tool for a rapi qalitative interpretation of MCSM ata. Introction The problem of electromagnetic fiel separation into pgoing an ongoing parts is one of the olest problems of geophsics (e.g., Chapman an Bartels, 94). The most general approach to the soltion of this problem is base on the formalism of the Stratton-Ch tpe integrals, introce b Zhanov (98, 988). Later on this techniqe as also etene to the separation of M fiels measre at the sea bottom (Zhanova an Zhanov, 999). Recentl e have observe a renee interest in the problem of pgoing/ongoing M fiel separation, generate b increase research an evelopment of marine controlle sorce electromagnetic (MCSM) methos. A problem that arises in the MCSM metho is that M energ ma travel from the sorce to the receiver along man paths. For eample, seafloor receivers of M ata measre not onl a response from the sea-bottom geoelectrical formations (the pgoing fiel ) bt the also measre a irect part of the primar fiel from the sorce, the fiel travelling from the sorce to the sea srface an reflecte back to the sea bottom (a so-calle airave), an the natral magnetotellric fiel. The latter parts of the total M fiel form the ongoing fiel becase the sorces of these fiels are locate above the receivers (above the seafloor). The methos of M fiel ecomposition into pgoing an ongoing components in application to MCSM ata are iscsse in the recent paper b Amnsen et al., 6. The presente a constrctive metho of the ecomposition of the M fiel given on a horiontal plane, an provie practical eamples of its application for D an D moels. In the crrent paper e emonstrate ho the theor of the Stratton-Ch tpe integrals can be se for ecomposition of the M fiel measre on an arbitrar srface, e.g., on a seafloor ith a variable bathmetr. In a case here the ata are measre on a horiontal plane, the most effective techniqe for fiel separation can be evelope sing a spatial Forier transform in the (k, ω ) omain (Berichevsk an Zhanov, 984; Zhanova an Zhanov, 999). In this paper, e introce a ne metho of the fiel separation base on sing horiontal graients of the observe M fiels. This metho allos s to evelop a fast an accrate metho for separation of 3D M fiels into pgoing an ongoing fiels, hich can be effectivel se for interpretation of the MCSM fiel ata. Application of the Stratton-Ch Tpe Integrals for M Fiel Separation A general integral metho of M fiel ecomposition into pgoing an ongoing components has been evelope b Zhanov (98, 988). We ill emonstrate belo that this theor provies the fonations for the soltion of the problem of M fiel separation into pgoing an ongoing parts. Let s consier a tpical MCSM srve concte in an area ith rogh bathmetr. The conctivit of seaater is knon an it is eqal to σ. The freqenc omain M fiel is generate b an electric bipole transmitter T locate at some epth ithin the sea-ater laer. This fiel is measre b a sstem of receivers locate at the seabottom. We can represent the electromagnetic fiel, as a sm of the pgoing, an ongoing, components: =, =. () SG oston 9 International position an Annal Meeting 85

2 Fonations of M fiel separation The problem of fiel ecomposition can be formlate as follos: fin the pgoing an ongoing components at the srface of observation S from the total M fiel,, observe also at S. We appl the theor of the Stratton-Ch tpe integrals to solve this problem. Let s take a point at the srface of observation, r S, an ra a sphere O R of rais R ith the center at r (see Figre ). We enote b R a pieceise smooth close srface forme b the semisphere O R an S R, R = O R S R. This srface R is a close bonar of the pper part, D R, of the ball bone b a sphere O R, locate above srface S. We can introce no the Stratton-Ch integrals over a piece-ise smooth close srface R : C C ( r ) = [(n ) G (n ) G iωµ (n ) G ] s, R R ( r ) = [(n ) G (n ) G σ (n ) G ] s, R R (3) here r D R, n is the nit vector of an inar pointing normal to D R, an G is the fnamental Green s fnction for the elmholt eqation in a homogeneos fll space ith the seaater conctivit σ. Figre : A scheme illstrating the applications of the Stratton-Ch tpe integrals to the separation of the pgoing an ongoing fiels. Using the properties of Stratton-Ch tpe integrals ( Zhanov 988), one can emonstrate that (r ) = (r ) CS (r ), (4) an (r ) = (r ) CS (r ). (5) Similar epressions can be erive for magnetic fiel components as ell: () an (r ) = (r ) CS (r ), (6) (r ) = (r ) CS (r ). (7) In formlas (4) throgh (7), epressions C S(r ) an C S(r ) stan for the Stratton-Ch tpe integrals over the entire observation srface S, etermine at a singlar point r S in terms of the Cach principal vale. qations (4) throgh (7) escribe a general integral transformation of the total M fiel, observe on the arbitrar srface S ithin the ater laer, into pgoing an ongoing parts. These eqations serve as a theoretical fonation of the metho of p/on ecomposition of the M fiel. lectromagnetic Fiel Separation into Upgoing an Dongoing Parts Using oriontal Graients We shol note that the metho of M fiel ecomposition base on Stratton-Ch tpe transforms reqires measrements of the total M fiel on the entire srface of observation S. In practice, hoever, e measre these fiels at a iscrete nmber of observation points locate ithin a limite part of S. That is h it is important to evelop simplifie methos of fiel ecomposition hich ol reqire measrements at jst a fe points. The simplest a to solve this problem is base on an assmption that the observe ata can be approimate b a verticall propagate plane M ave. This approach, for eample, as iscsse in Amnsen et al. (6) an it is iel se for ecomposition of the fiel MCSM ata. oever, one can proce a more accrate bt still rapi ecomposition metho assming jst slo horiontal variations of the M fiel observe ithin the area of observation. In this case, the M fiel ithin the seaater can be epresse approimatel as (Zhanov et al., 996): ( = ( ep[ )] ( ep[ )], ( = ( ep[ )] ( ep[ )], (8) here b, b is the epth of the sea bottom, is the epth of the location of the receivers, an k = k = iωµσ. We assme that the vector fnctions,,, var relativel slol ith the epth. The goal is to fin the pgoing an ongoing components of the M fiel. This problem can be solve b ifferentiation of both sies of eqations (8) ith respect to the vertical coorinate : SG oston 9 International position an Annal Meeting 86

3 Fonations of M fiel separation ( = ik ep[ )] ik )], ( = ik ep[ )] ik )]. (9) Mltipling both sies of eqation (8) b ik an aing (9), e have: ( = ik )], () ( = ik )]. seafloor. The M fiel in the moel is generate b a horiontal electric bipole transmitter toing behin the ship at an elevation of 5 m above the seafloor. The receivers measring all si components of the M fiel (or for horiontal components onl) are locate at the seafloor along several srve lines ith a separation of 5 meters in both an irections. From the last epression e fin immeiatel that: ( = [( ( ]ep[ )], ik ik ( = [( ( ]ep[ )]. () Ths the problem of fiel separation is rece to comptations of the vertical erivatives of the electric an magnetic fiels. The last problem can be solve sing Maell s eqations. It can be shon that the vertical erivatives of electric an magnetic fiels, / an /, can be epresse sing horiontal graients of these fiels. For eample, e have for electric fiel: iωµ =, = = Note that the epressions ik ik ik ik iωµ ik ik,., (),, etc. represent the horiontal graients of the electric an magnetic fiel components. After e etermine the vector coefficients,, an,, e can fin the pgoing an ongoing fiels themselves, e.g.: ( = )], (3) ( = )]. Figre : The geoelectrical section of the moel is forme b a seaater laer an conctive sea-bottom seiments. A 3D is embee in the sea bottom at a epth of 5 m belo the seafloor. We shol note that the problem of pgoing/ongoing fiel separation can be solve sing the ata along one srve line onl. oever, the most profon reslt can be seen in the maps of the pgoing an ongoing parts of the fiel (see Figres 3 an 4). One can clearl see in these figres that the separation of the pgoing part of the total electric fiel reslts in effective imaging of the horiontal location of the reservoir. A similar effect is observe in the maps of the horiontal component of the magnetic fiel after a ecomposition transformation. Nmerical amples of Marine M Data Decomposition In orer to check the effectiveness of the separation techniqe evelope above base on the horiontal graients, e have concte a nmber of nmerical eperiments ith snthetic M ata. We present belo jst one of these reslts. For a moel shon in Figre, the geoelectrical section of this moel is forme b a seaater laer ith a resistivit of.4 ohm-m an a thickness of 34 m, an conctive sea-bottom seiments ith a resistivit of ohm-m, respectivel. A 3D reservoir ith ohm-m resistivit is embee in the sea bottom at a epth of 5 m belo the Figre 3: The top panels sho maps of the amplite (left top panel) an phase (right top panel) of the total in-line electric fiel at a freqenc of.5. The bottom panels present maps of the amplite (left bottom panel) SG oston 9 International position an Annal Meeting 87

4 Fonations of M fiel separation an phase (right bottom panel) of pgoing in-line electric fiel separate sing the metho of horiontal graients. Note that in Figres 3 an 4 a ashe hite rectangle otlines the location of a resistive reservoir. From these reslts, it is clear, the pgoing fiel (real an/or imaginar parts, amplite an/or phase) separate sing the evelope ecomposition techniqe, reflects the location of the target mch better than the total M fiel ata. Nmerical ata, shon above, ere generate for a freqenc of.5. We also se other freqencies from. to an other locations of the transmitting bipole sorce to test or techniqe. We have obtaine similar reslts for all these freqencies an ifferent transmitter locations. transformation of MCSM ata for a qalitative evalation of the location of the tpical eploration targets, e.g., an C reservoir. Acknolegments The athors acknolege the spport of the Universit of Utah Consortim for lectromagnetic Moeling an Inversion (CMI). Figre 4: Moel : The top panels sho maps of the amplite (left top panel) an phase (right top panel) of the total magnetic fiel at a freqenc of.5. The bottom panels present maps of the amplite (left bottom panel) an phase (right bottom panel) of pgoing magnetic fiel separate sing the metho of horiontal graients. Conclsions The moern techniqe of M fiel ecomposition into pgoing an ongoing parts has its roots in the classical methos of geomagnetic fiel separation into eternal an internal parts. The most general approach to the soltion of this problem is base on the theor of the Stratton-Ch tpe integrals. This approach makes it possible to separate the MCSM ata observe on the seafloor in areas ith rogh bathmetr. oever, Stratton-Ch tpe transformation, in a general case, reqires measrement of the ata in the relativel large area of observation. We have evelope a metho of solving this problem hich is free of these limitations. It is base on the calclations of the horiontal graients of the fiel, an therefore it can be se for a ecomposition of MCSM ata measre at a fe points onl. The nmerical st shos that this novel metho can be se as a rapi SG oston 9 International position an Annal Meeting 88

5 DITD RFRNCS Note: This reference list is a cop-eite version of the reference list sbmitte b the athor. Reference lists for the 9 SG Technical Program pane Abstracts have been cop eite so that references provie ith the online metaata for each paper ill achieve a high egree of linking to cite sorces that appear on the Web. RFRNCS Amnsen, L., L. Løseth, R. Mittet, S. llingsr, an B. Ursin, 6, Decomposition of electromagnetic fiels into pgoing an ongoing components: Geophsics, 7, no. 5, G G3. Berichevsk M. N., an M. S. Zhanov, 984, Avance theor of eep geomagnetic soning: lsevier. Chapman, S, an J. Bartels, 94, Geomagnetism, vol. : Clarenon Press. Zhanov, M. S., 98, Cach integral analogs for the separation an contination of electromagnetic fiels ithin concting matter: Geophsical Srves, 4, 5 36., 988, Integral transforms in geophsics: Springer-Verlag eielberg. Zhanov, M. S., P. Tranin, an J. Booker, 996, Unergron imaging b freqenc omain electromagnetic migration: Geophsics, 6, Zhanova, O. N., an M. S. Zhanov, 999, Methos for the analsis an interpretation of the sea floor electromagnetic fiels, in K. K. Ro S. K. Verma, an K. Mallick, es., Deep electromagnetic eploration: Springer an Narosa Pblishing ose. SG oston 9 International position an Annal Meeting 89

1 Differential Equations for Solid Mechanics

1 Differential Equations for Solid Mechanics 1 Differential Eqations for Solid Mechanics Simple problems involving homogeneos stress states have been considered so far, wherein the stress is the same throghot the component nder std. An eception to