Frechet derivative calculation

In Chapter 9 we developed the expressions for the Frbchet differentials of the forward modeling electromagnetic operator (9.51) and (9.52), which we reproduce here for convenience  

In the last formulae a — db + Ad is a conductivity distribution, for which we calculate the forward modeling operator variation 6d is the corresponding variation of the conductivity a, which is obviously equal to the variation of the anomalous conductivity, 6d = 6Ad. Tensors Gg jf are electric and magnetic Green s tensors calculated for the given conductivity a. Vector E in expressions (10.54) and (10.55) represents the total electric field, E = E -t-E for the given conductivity d. 

Note that in the RCG algorithm (10.53) the expression FE,Hn E,Hn denotes the result of an application of the adjoint Fr6chet derivative operator to the corresponding electric or magnetic residual field R ,//n = — dE,n on the n-th 

The expressions for the adjoint Pr6chet derivative operators are given by formulae (10.25) and (10.26). Based on these formulae, we can write  

Therefore, from (10.56) and (10.57) we find thdt the result of applying the adjoint Frechet operator to the residual field is just equal to the scalar product of the complex conjugate of the electric field E , computed at the n-th iteration, with 

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Formulae (9.59) and (9.60) play an important role in electromagnetic inversion. 9.1.7 Frechet derivative calculation using the differential method We now present another way of calculating the Frechet derivative using the differential method, proposed by McGillivray et al., 1994. 

Frechet derivative calculation using finite difference methods The problem of the Frechet derivative, or sensitivity matrix calculation for electromagnetic field has been examined in many publications and was discussed in Chapter... 

We can reduce significantly the number of the required iterations using the reciprocity principle (Madden, 1972 Rodi, 1976 Madden and Mackie, 1989 McGillivray and Oldenburg, 1990 de Lugao and Wannamaker, 1996 and de Lugao et ah, 1997) for Frechet derivative calculations. Due to reciprocity, the field is equal to the... 

Calculation of the first variation (Frechet derivative) of the electromagnetic field for 2-D models... 

McGillivray, P. R., and D. W. Oldenburg, 1990, Methods for calculating Frechet derivatives and sensitivities for the nonlinear inverse problem a comparative study Geophys. Prosp., 38, 499-524. 

Note that comparing formulae (10.12), (10.13) and (9.51), (9.52), we see that the expressions in the right hand sides of the Born approximations can be represented as the Frechet derivative operators calculated for the background conductivity db and the anomalous conductivity A5 ... 

The last formula provides an analytical expression for computing the Frechet derivative for the forward modeling operator. Note that, in this case, the amount of calculation for the forward modeling solution and for the Frechet derivative is equivalent to computing the Born approximation. 

Calculation of the first variation (Frechet derivative) of the vector wavefield We begin this section with an analysis of the equation for the vector wavefield variation. This equation can be derived by applying the perturbation operator to both sides of the vector Helmholtz equation (14.67), expressed in terms of the slowness function s(r). 

Comparing formulae (15.5), (15.6) and (14.29), (14.84), we see that Born approximations can be expressed as the Frechet derivative operators (or Frechet differentials) calculated for the background square slowness s and the anomalous square slowness As ... 

In the section of this chapter describing the Kirchhoff inversion method and general non-linear inversion techniques, we have demonstrated that the calculation of the Kirchhoff adjoint operator (15.142) and the adjoint Frechet derivative operator... 

The theory also explains the value of 5. Unfortunately, that part of the story requires more sophisticated apparatus than we are prepared to discuss (operators in function space, Frechet derivatives, etc.). Instead we turn now to a concrete example of renormalization. The calculations are only approximate, but they can be done explicitly, using algebra instead of functional equations. 

For any orbital-functional model, an optimal effective (local) potential (OEP) can be constructed following a well-defined variational formalism [24,25]. If a Frechet derivative existed for the exchange-correlation energy E,lc for ground states, it would be obtained in an OEP calculation, while the minimum energy and corresponding reference state would coincide with OFT results. Thus numerically accurate OEP calculations test the locality hypothesis. 

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