2-D Niblett-Bostick magnetotelluric inversion - MTNet

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C = − ∑ ∑ [ −∑C = a a∑ ] ( σ σ − −∑ˆ σ ) [ −= ∑ ∑a [ σσ + − ∑ a σ ] σ. + (10) J. RODRÍGUEZ et al. 2-D Niblett-Bostick extremely cumbersome, for it is necessary to cover many where possibilities during each epoch training session. Another approach to inversion based on neural networks bypasses the learning sessions by directly providing the algorithm with the relevant physics behind the particular (13) application. Zhang and Paulson (1997) applied a simple recursive regularization algorithm associated with a Hopfield artificial neural network (HANN) in order to E is the Ising Hamiltonian (Ising, 1925) which Tank solve the 1-D MT inverse problem with excellent results. and Hopfield (1986) showed is a never increasing quantity In the same paper the authors explored the application to as the dynamics defined by the following equation evolve the 2-D full nonlinear inversion of MT data. Our interest from given starting values: in this algorithm was triggered by the fact that there are no references in the literature attempting to continue their (14) research in this direction, even though the results presented are very promising. In this methodology there are no dedicated input-output defined units (as with the feed- sj forward network architecture). Instead, every neuron si is interconnected with every other sj neuron through a Tij weight link as detailed below. Every neuron has a twofold input-output purpose. We assume M apparent conductivity measurements and N blocks whose conductivities are unknown. According to equation (6), for given conductivity values sj, j = 1,...,N, the responses ˆsak,, k = 1,...,M, of the model can be expressed as (9) The misfit between sak (the data) and ˆsak, (the model responses) can be written as (10) Squaring as indicated and rearranging terms, this expression corresponds to (11) It can be noted that this is an unnecessarily long, and certainly very cumbersome way of representing C, the sum of squares. However, this particular form helps to recognize its correspondence to (12) Geologica Acta, 8(1), 15-30 (2010) 18 DOI: 10.1344/105.000001513 (t+l) is the updated conductivity value of sj (t) which is either a given starting value or the result of a previous iteration. The relevance of dynamic is that as iterations proceed E decreases at each step. Since E corresponds to C, the dynamics ensure that the sum of the squared differences between data and model responses decrease at each step, thus leading to the model whose responses best fit the data in a least square sense. It can be noted that with the HANN the matrix elements Tij are known and provided by the physics of the problem, since they are the elements (AT ki kj i akj ak ki ak i 1 M kikj akj i 2 2 i= 1 j¹ i= 1 k= 1 2 k= 1 i= 1 2 2k = 1k = 1 k= j1= 1 ( σ ) 2 . 2∑ ∑ ( σ ak ak) . 2 k k = 1 N N M N M M 1 M 1 1 2 = − ∑ ∑ [ −∑ a N N N ( ki 2 σ akj) 1] E −1 . Niσ j N− N ∑ ∑ [ − ∑ ak (11) akiσ i + ∑ aki 2 i= 1 j¹ i= 1 2 k= 1 i= 1 ∑ ∑ Tijσ 2 iσ k j −= 1 I iσ , k= 1 k= 1E − 2∑ ∑ Tijσ iσ j −∑ ∑ I iσ i i, 2 i i = 1 j j ¹ ¹ i i = 1 i i = 1 N N N M 1 M 1 2 M M E = − M 1 M M ∑ ∑ Tijσ iσ j − Tij = −∑ a∑ I iσ i , kia ∑ ( σ ak ) 2 kj i −1 (12) . a2 2 ∑ kiσ i + a i= 1 j¹ i= 1T ij = −∑ aiki = 1a 2 kj k= 1 i − k= 1 2∑ akiσ i + ∑ aki ki k= 1 2 k k = 1 k k = 1 M MN N M N 1 2 T 1 1 ⎛ N ij = −∑ akia kj and IE i = − ∑ a∑ ⎞ ( t+ 1) ( ) = 1 + 1 sgn ⎛ N t ⎜ ⎟ i . 2 2 ⎜ ∑ Tijσ j + I ⎞ ( t+ kiσ T 1) i + ijσ∑ iσ ja− ki∑ σ . I ak iσ i(13) , ( t) k= 1 22 ki = 1 j¹ i= 1 sgn⎜ i⎟ i = k+ = 1 i= 1 ⎟. 2 2 ⎜ ∑ Tijσ j + I i ⎝ j≠i = 1 ⎟ ⎝ j≠i = 1 ⎠ M 1 1 ⎛ N M M ⎞ 1 ( t+ 1) ( t+ 1) ( t) 2 σ T σ ( t+ 1) σ sgn⎜ ⎟ i = ij = −+ ∑ a i kia . 2i 2 ⎜ ∑ kj Tand ijσ + I j Ii − i ⎟ ∑ akiσ i + (14) ∑ akiσ ak . k= 1 (t ) ⎝ 1 ⎠ 2 j≠i = k= 1 k= 1 σ (t ) σ j j ( t+ 1) σ T ij 1 1 ⎛ N ⎞ i ij ( t+ 1) ( t) sgn⎜ ⎟ T i = + . σ a( x, T ) = ∫ Fh( x, x', z ', σa, T ) σ( x', (t ) z ') dx' dz '. (8) ( AA T σ ( AA ) ij 2 2 ⎜ ∑ Tijσ j + I i ⎟ j ij ⎝ j≠i = 1 ⎠ T σ a( x, T ) = ∫ Fh( x, x', z ', ij σa, T ) σ( x', z ') dx' dz '. ( t+ 1) σ (8) a( x, T ) = ∫ Fh( x, xT', z ', σa, T ) σ( x', σ iz ') dx' dz '. (8) F ( AA ) ij h (t ) ) = σ ∫ F h F σ h( x, x', z ', σa, T ) σ( x', z ') dx' dz '. (8) j a F σ h( x, ax ', z ', σa, T ) σ ( x', z ') dx' dz '. (8) T ij T σ j ( AA ) ij TN , σ j ij (8) σ , M σ T a( x, T ) = j , j = 1,..., N, ∫ Fh( x, x', z ', σa, T ) σ( x', z ') dx' dz '. (8) 2 ij ˆ σ k M N A)ij, ak , σ j , = j1= ,..., 1,..., N, ˆ σ a , k 1,..., M. which in turn come from the integral equation. ak = ∑ kjσ j N = (9) 2 ˆ σ ak , k = 1,..., Mj= 1 ˆ σ ak = ∑ akjσ j , k = 1,..., M. (9) j= 1 N Strictly speaking, equation (14) applies only to binary σ , ˆ σ ak = ∑ akjσ j , k = 1,..., M. variables (9) that can take values of zero and one, as in the HANN ak N j= 1 architecture (Hopfield, 1982). The corresponding equations for σˆ ˆ σ 2 ak = akσ ∑ a M kjσ j , k = 1,..., M. (9) N M N 1ak , 1 an assemblage of such variables, to make up for arbitrary values j= 1 ,..., M 2 M N 2 ∑ = N a, ∑ ( σ ˆ ak 1− σ ak ) = ∑ 2[ σ 1ak −∑ akjσ j ] . 2 (10) kjσ j of conductivity, are very similar and are given in Appendix D. σˆ 2 C , k = = 1,..., ( M σ . k= 1 ∑ 2ˆ ak −σ ) [ ] . 2 kak = σ (9) = 1 ∑ 2 j= 1 ak −∑ akjσ j (10) j= 1 ak 1 ,..., M k= 1 k= 1 j= 1 M M N 1 2 1 2 The application of equation (14) is straightforward ˆ N N CM= N ( σ N ) M [ M ] . 1 (9) N N∑ M ak −σ ak = σ 1 N∑ ak − a M ∑ kjσ M j (10) when M>N, for the process reduces to the standard least 2 − ∑ C∑= −[ 1 ˆ − σ M ak = 2 k= 1 ∑ akia kj −] σ iσ j −∑ [ − 2 k= 1 − ∑ −a 1 2j = 1 kiσ i + ∑ + akiσ ak ] σ i + squares + problem. This corresponds to the over constrained 2 ∑ ∑∑ akjσN j , k = 1,..., M. (9) 2 1 [ ∑ akia kj ] σ iσ 2 σ ˆ j i= 1 j¹ i= 1 k= 1 i= 1 2∑ [ ∑ akiσ i ∑ akiσ ak ] σ ak − σ ak ) = i M2 ∑ [ j= σ1 ak −∑ akjσ j ] . (10) i= 1 j¹ i= 1 N k= 1 i= k1= 1 2 k= 1 k= 1 k= 1 2 1 2 k= 1 j= 1 2 case when we have more data than unknown parameters. ˆ σ ) [ N N M N M M ak = ∑ 1 σ ak −∑ akjσ j ] . (10) 1 2 2 In general, however, M
J. RODRÍGUEZ et al. 2-D Niblett-Bostick ( t) ( t) σ = σ . (15) j w l ( j+ l) l= −n of the data in such a way that the averaging functions resemble boxcar functions for the averages to have the usual meaning. The resulting models are not intended to produce responses that fit the data, but to address the non-uniqueness character of the inverse problem. Within the limitations of linearization, they can be called true averages. More rigorous approaches for computing averages have been developed for the 1-D magnetotelluric problem (e.g. Weidelt, 1992), but their extension to higher dimensions are far from trivial. Simulated averages The shortcut that we take consists of computing models that are averages of themselves everywhere with a given neighborhood, and that at the same time their responses fit the data at an optimal level. Consider that (15) The dynamics of the HANN in (14) transforms to (16) Following Zhang and Paulson (1997), we will refer to this as the regularized version of the HANN or RHANN. The updated conductivities sj Geologica Acta, 8(1), 15-30 (2010) 19 DOI: 10.1344/105.000001513 (t+l) are obtained from averages of the original values sj (t) . The choice of the appropriate filter depends on the application. Zhang and Paulson (1997) found binomial filters useful to stabilize their solution. In our case, we use boxcar functions, for this leads to the simplest and most intuitive type of averages. The filter, with 2n+1 uniform weights wk = (2n+1) -1 its performance in 1-D, and compare its results with existing methods of average estimations. In the following lines, we test the performance of equation (16) as it applies to the solution of equation (6). The elements of the matrix are computed using equation (7) which represents the 1-D version of the proposed approximation. The synthetic data for the tests, shown in Fig. 1A, correspond to apparent conductivity responses of the model shown in Fig. 1B. The tests are intended to demonstrate that the proposed approach for the computation of averages has the same properties as the simple formula for averages in Gómez- Treviño (1996). The formula is (18) n ( t) ( t) σ where represents the average of conductivity j = ∑ w lσ ( j+ l) . (15) 1 2 n l= −n between zi ( t) ( t) = 0.707da(Ti),i = 1,2. The averages are assigned σ j = ∑ w lσ ( j+ l) . (15) to the mean geometrical depth z = √ z z . Any two data 1 2 l= −n points can be used in the formula regardless of how far 1 1 ⎧ N n ⎫ ( t+ 1) ( t) apart they are in the sounding curve. If the points are i = + sgn⎨ ∑ Tij[ ∑ wlσ ( j+ l) ] + I i ⎬. (16) 2 2 ⎩ j≠i1 = 1 1 l= −n ⎧ N n ⎭ ⎫ ( t+ 1) ( t) contiguous, the windows are narrow and the average σ i = + sgn⎨ ∑ Tij[ ∑ wlσ ( j+ l) ] + I i ⎬. (16) 2 2 model, the average of the real earth, has the highest ⎩ j≠i = 1 l= −n ⎭ possible resolution. If, on the other hand, the chosen t+ 1) points are wide apart, the windows are themselves wide, t ) for z and z tend to separate. As the windows widen, the 1 2 − n,..., n models tend to flatten, as they should. This intuitively 1 = , k = −n,..., n appealing feature of spatial averages is built into equation 2n + 1 (18) in spite of its simplicity. Figures 1C and 1D show N n 1 1 ( t) ( t) + sgn{ this feature developing when considering averages taken ∑ [( 1− b ) Tijσ j + β ∑ Tijwlσ ( j+ l) ] + I i}. (17) 2 2 N n from data values 1 and 5 periods apart, respectively, for ( t+ 1) j≠i = 1 1 l= −n ( t) ( t) σ i = + sgn{ ∑ [( 1− b ) Tijσ j + β ∑ Tijwlσ ( j+ l) ] + I i}. (17) 2 2 the synthetic data presented in Fig. 1A. j≠i = 1 l= −n σ , k = -n,..., n (square a ( T2 ) δ a ( T2 ) −σ a ( T1 ) δ a ( T1 ) σ ( z1, z2 ) >= window), can be of different widths. , We introduce (18) a further A corresponding sequence of full filtered models (β =1) parameter β δ ato ( Taccommodate 2 ) σ− δ ( T ) a ( Ta 2 ) 1 δ a ( Ta 2traditional ) −σ a ( T1 ) trade-off δ a ( T1 ) < σ ( z between is presented in Figs. 1E and 1F when the RHANN algorithm 1, z2 ) >= , (18) two contrasting features. δWe modified the dynamics to is applied to the same set of data. It can be observed that a ( T2 ) −δ a ( T1 ) the behavior of the averages follows that of the averages i ), ( i = 1, 2 1, 2 ) > shown in Figs. 1C and 1D. There are some differences that can be traced back to basic differences between the two approaches. One is that equation (18) is based on the (17) original Niblett-Bostick approximation that assumes a boxcar function as the kernel of equation (5). The RHANN When β =1 Hwe ( xget , 0) the full filtered solution. On the other hand, algorithm uses the kernel as indicated in equation (5) which x 2 | when σ a | = ωµ β =0 0 | we return to | , the original unfiltered solution (A.1) given includes a small negative sidelobe (Gómez-Treviño, 1987b) E y ( x, 0) by equation (14). In this way, by varying this parameter we and extends to infinity. This is a somewhat less restrictive can gradually control the effect of the averaging process. approximation than the rather simpler boxcar function. The δE ( x, 0) H ( x, 0) other difference originates in the fact that the windows in y δ x | σ a | = −2 | σ a | Re[ − ]. (A.2) each approach are necessarily different, because in the first E ( x, 0) H ( x, 0) APPLICATIONS y x case they cannot be uniform, for they are determined by the data, and in the second case they are uniform for all depths. 1-D averages The overall effect is a somewhat better performance for the approximation given by equation (5) judging by the Before proceeding to the application in 2-D of the above resemblance of the average models to the original model numerical averaging technique, it is convenient to illustrate from which the data were drawn. 2 ∇ E y − iωµ 0σ ( x, z) E y = 0, (A.3) z − 2i δ E y ( x, z) = E( 0) e , (A.4) 2 2 δ = . (A.5) ωµ 0σ z z = 0 . 707δ a ( Ti ), i = 1, 2 1 2 z z n ( t) ( t) σ j = ∑ w lσ ( j+ l) . (15) l= −n 1 1 ⎧ N n ⎫ ( t+ 1) ( t) σ i = + sgn⎨ ∑ Tij[ ∑ wlσ ( j+ l) ] + I i ⎬. (16) 2 2 ⎩ j≠i = 1 l= −n ⎭ ( t+ 1) σ i (t ) σ j 1 wk = , k = −n,..., n 2n + 1 N n ( t+ 1) 1 1 ( t) ( t) σ i = + sgn{ ∑ [( 1− b ) Tijσ j + β ∑ Tijwlσ ( j+ l) ] + I i}. (17) 2 2 j≠i = 1 l= −n σ a ( T2 ) δ a ( T2 ) −σ a ( T1 ) δ a ( T1 ) < σ ( z1, z2 ) >= , (18) NEXES δ a ( T2 ) −δ a ( T1 ) < ( 1, 2 ) > H x ( x, 0) 2 | σ a | = ωµ 0 | | , (A.1) E y ( x, 0) x , z) δE y ( x, 0) δH x ( x, 0) δ | σ a | = −2 | σ a | Re[ − ]. (A.2) E y ( x, 0) H x ( x, 0) ( x , z) x ( x, z) y ( x, z) 2 ∇ E y − iωµ 0σ ( x, z) E y = 0, (A.3) z − 2i δ E y ( x, z) = E( 0) e , (A.4) 2 2 δ = . (A.5) ωµ 0σ ( x , z) z z σ zi = 0 . 707δ a ( Ti ), i = 1, 2 1 2 z z 1 1 ⎧ N n ⎫ ( t+ 1) ( t) σ i = + sgn⎨ ∑ Tij[ ∑ wlσ ( j+ l) ] + I i ⎬. (16) 2 2 ⎩ j≠i = 1 l= −n ⎭ ( t+ 1) σ i (t ) σ j 1 wk = , k = −n,..., n 2n + 1 N n ( t+ 1) 1 1 ( t) ( t) σ i = + sgn{ ∑ [( 1− b ) Tijσ j + β ∑ Tijwlσ ( j+ l) ] + I i}. (17) 2 2 j≠i = 1 l= −n σ a ( T2 ) δ a ( T2 ) −σ a ( T1 ) δ a ( T1 ) < σ ( z1, z2 ) >= , δ a ( T2 ) −δ a ( T1 ) (18) < ( 1, 2 ) > n ( t) ( t) σ j = ∑ w lσ ( j+ l) . (15) l= −n ( 1) 1 1 ⎧ N n ⎫ t+ ( t) i = + sgn⎨ ∑ Tij[ ∑ wlσ ( j+ l) ] + I i ⎬. (16) 2 2 ⎩ j≠i = 1 l= −n ⎭ − n,..., n N n 1 1 ( t) ( t) + sgn{ ∑ [( 1− b ) Tijσ j + β ∑ Tijwlσ ( j+ l) ] + I i}. (17) 2 2 j≠i = 1 l= −n σ a ( T2 ) δ a ( T2 ) −σ a ( T1 ) δ a ( T1 ) σ ( z1, z2 ) >= , (18) δ a ( T2 ) −δ a ( T1 ) z = 3 i ), i = 1, 2 ANNEXES H x ( x, 0) 2 | σ a | = ωµ 0 | | , (A.1) E y ( x, 0) σ ( x, z) H x ( x, 0) 2 | σ a | = ωµ 0 | | , (A.1) E y ( x, 0) δE y ( x, 0) δH x ( x, 0) δ | σ a | = −2 | σ a | Re[ − ]. (A.2) E y ( x, 0) H x ( x, 0) δE y ( x, 0) δH x ( x, 0) | σ a | = −2 | σ a | Re[ − ]. (A.2) E y ( x, z) E y ( x, 0) H x ( x, 0) H x ( x, z) E y ( x, z) z z σ zi = 0 . 707δ a ( Ti ), i = 1, 2 1 2 z z z = ANNEXES H x ( x, 0) 2 | σ a | = ωµ 0 | | , (A.1) E y ( x, 0) σ ( x, z) δE y ( x, 0) δH x ( x, 0) δ | σ a | = −2 | σ a | Re[ − ]. (A.2) E y ( x, 0) H x ( x, 0) E y ( x, z) H x ( x, z) E y ( x, z) 2 ∇ E y − iωµ 0σ ( x, z) E y = 0, (A.3) z − 2i δ E y ( x, z) = E( 0) e , (A.4) 2 2 δ = . (A.5) ωµ 0σ H x ( x, z) ∇ × E = −iωµ 0H n ∑
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2-D Niblett-Bostick magnetotelluric inversion - MTNet

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