@@ -550,9 +550,34 @@ A step length search which is based on the wolfe condition can be used which the
However, it is possible to chose a second step length search which the L-BFGS method. This one is available by the switch LBFGS\_STEP\_LENGTH=0. If you use LBFGS\_STEP\_LENGTH the wolfe condition based step length search will be deactivated automatically. Whit this search the step length 0.1, 0.5 and 1.0 will be tried and with a parabolic fit the best step length will be estimated. This search algorithm is implemented to save computation time, however the wolfe condition will not be checked, so the L-BFGS stability will not be satisfied. It is highly recommended to use the step length search based on the wolfe condition. It is likely that this option will be removed in a future releas.
For the step length estimation a parabolic line search method proposed by \cite{sourbier:09,sourbier:09b}, \cite{brossier:2009} and \cite{nocedal:1999} is implemented. For this step length estimation only two further test forward modelings are needed. The vector L2t contains the misfit values and the vector epst contains the corresponding step length. During the forward modeling of an iteration step the misfit norm of the data residuals is calculated for the shots defined by TESTSHOT\_START, TESTSHOT\_END and TESTSHOT\_INC. The value L2t(1) then contains the misfit from the forward modeling and the corresponding epst(1) value is 0.0.\\
The step lengths for the different parameters are defined as:\\
EPSILON = EPS\_SCALE * m\_max/grad\_max
EPSILON = epst[i] * m\_max/grad\_max\\
where m\_max is the maximum value of the corresponding model parameter in the whole model and grad\_max is the maximum absolute value of the gradient.\\
For a better definition of the parabola the improved line search is now trying to estimate a steplength epst(2) with L2t(2)<L2t(1). If the code is not able to find an appropiate steplength using the user-defined value EPS\_SCALE (f.e. EPS\_SCALE = 0.01 = 1\% change in terms of m\_max/grad\_max), the code divides this steplength by the variable SCALEFAC and calculates the misfit norm again. If this search fails after STEPMAX attempts DENISE exits with an error message. If the algorithm has found an appropriate value for epst(2), it is trying to estimate a steplength epst(3) with L2t(3)> L2t(2), by increasing the steplength\\
EPS\_SCALE += EPS\_SCALE/SCALEFAC.\\
If a corresponding value epst(3) can be found after STEPMAX forward modellings, DENISE can fit a parabola through the 3 points (L2t(i),epst(i)) and estimates an optimum step length at the minimum of the parabola. If the L2-value L2t(3) after STEPMAX forward models is still smaller than L2t(2) the optimum steplength estimated by parabolic fitting will be not larger than epst(3).\\
Please note: This step length search is only available wenn PCG method (GRAD\_METHOD==1) is used and will be automatically selected.
\section{Misfit definition}
{\color{blue}{\begin{verbatim}
"Gradient calculation" : "comment",
"Misfit Definition" : "comment",
"LNORM" : "2",
"NORMALIZE" : "0",
"DTINV" : "2",
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@@ -582,31 +607,6 @@ If NORMALIZE is set to 1, the synthetic data and the measured data will be norma
To reduce the memory requirements during an inversion one can define that only every DTINV time sample is used for the calculation of the gradients. To set this parameter appropriately one has to keep in mind the Nyquist criterion to avoid aliasing effects.
For the step length estimation a parabolic line search method proposed by \cite{sourbier:09,sourbier:09b}, \cite{brossier:2009} and \cite{nocedal:1999} is implemented. For this step length estimation only two further test forward modelings are needed. The vector L2t contains the misfit values and the vector epst contains the corresponding step length. During the forward modeling of an iteration step the misfit norm of the data residuals is calculated for the shots defined by TESTSHOT\_START, TESTSHOT\_END and TESTSHOT\_INC. The value L2t(1) then contains the misfit from the forward modeling and the corresponding epst(1) value is 0.0.\\
The step lengths for the different parameters are defined as:\\
EPSILON = EPS\_SCALE * m\_max/grad\_max
EPSILON = epst[i] * m\_max/grad\_max\\
where m\_max is the maximum value of the corresponding model parameter in the whole model and grad\_max is the maximum absolute value of the gradient.\\
For a better definition of the parabola the improved line search is now trying to estimate a steplength epst(2) with L2t(2)<L2t(1). If the code is not able to find an appropiate steplength using the user-defined value EPS\_SCALE (f.e. EPS\_SCALE = 0.01 = 1\% change in terms of m\_max/grad\_max), the code divides this steplength by the variable SCALEFAC and calculates the misfit norm again. If this search fails after STEPMAX attempts DENISE exits with an error message. If the algorithm has found an appropriate value for epst(2), it is trying to estimate a steplength epst(3) with L2t(3)> L2t(2), by increasing the steplength\\
EPS\_SCALE += EPS\_SCALE/SCALEFAC.\\
If a corresponding value epst(3) can be found after STEPMAX forward modellings, DENISE can fit a parabola through the 3 points (L2t(i),epst(i)) and estimates an optimum step length at the minimum of the parabola. If the L2-value L2t(3) after STEPMAX forward models is still smaller than L2t(2) the optimum steplength estimated by parabolic fitting will be not larger than epst(3).\\
Please note: This step length search is only available wenn PCG method (GRAD\_METHOD==1) is used and will be automatically selected.
\section{Abort criterion}
\label{json:abort_criterion}
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@@ -715,7 +715,7 @@ If you are using frequeny filtering (TIME\_FILT==1) during the inversion, you ca
