Modified Greens operator

X Z/2 (D), where I/2 D) is the Hilbert space of the vector functions determined in domain D and integrable in D with the norm  

The remarkable property of this operator is that according to inequality (9.9), 

Equation (9.137) can be treated as an integral equation with respect to the 

The solution of integral equation (9.139) is similar to equation (9.125) and can also be obtained using the method of successive iterations, which is governed by the equations 

These iterations are always convergent, because the operator O is a contraction operator  

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Quasi-linear approximation of the modified Green s operator We will obtain a more accurate approximation even on the first step if we assume that the anomalous field E inside the inhomogeneous domain is not equal to zero, as it was supposed in the previous section, but is linearly related to the background field E by some tensor A ... 

Taking into account the definition of the modified Green s operator (9.135) and formula (9.170), we obtain... 

The critical problem of both the Born iterative and the distorted-Born iterative methods is the convergence of the algorithms. To ensure convergence we can use a modified Green s operator G ", which has the contraction property... 

The modified Green s operator is introduced by formula (9.135), which we reproduce here ... 

The iterative Born inversion based on the modified Green s operator involves subsequently determining A from equation (10.42) for specified E, and then updating E from equation (10.47) for predetermined A5, etc. 

Gas wells that are refractured and recompleted will not be considered to be modified if well owners and operators use green completions rather than flaring to reduce emissions, and they meet notification and reporting requirements for new wells. 

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