Commit 849cb82d authored by Florian Wittkamp's avatar Florian Wittkamp

Added references to D. Koehn's dissertation

Added references to D. Koehn's dissertation. Large parts of the theory are extracted from his dissertation.
parent 0c927f9c
......@@ -3,7 +3,7 @@
\chapter{Theoretical Background}
%------------------------------------------------------------------------------------------------%
This chapter is mainly based on the dissertation of Daniel Köhn (\cite{koehn:11}), who originally has written this manual.
\section{Equations of motion for an elastic medium}\label{elastic_fd_model}
The propagation of waves in a general elastic medium can be described by a system of coupled linear partial differential equations. They consist of the equations of motion
\EQ{m:1}{\begin{split}
......@@ -27,7 +27,6 @@ This expression is called {\bf{Stress-Displacement}} formulation. Another common
\rm{\frac{\partial \epsilon_{ij}}{\partial t}}&\rm{=\frac{1}{2}\biggl(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}\biggr)}
\end{split}}
This expression is called {\bf{Stress-Velocity}} formulation. For simple cases \ER{2:20} and \ER{2:20:1} can be solved analytically. More complex problems require numerical solutions. One possible approach for a numerical solution is described in the next section.
\newpage
\section{Solution of the elastic wave equation by finite differences}\label{elastic_FD_Code}
\subsection{Discretization of the wave equation}
For the numerical solution of the elastic equations of motion, Eqs. \ER{2:20} have to be discretized in time and space on a grid. The
......@@ -182,7 +181,7 @@ A comparison between the exponential damping and the PML boundary is shown in Fi
\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{figures/ABS_PML_comp_shots.pdf}
\caption{\label{comp_EXP_PML} Comparison between exponential damping (left column) and PML (right column) absorbing boundary conditions for a homogeneous full space model.}
\caption{\label{comp_EXP_PML} Comparison between exponential damping (left column) and PML (right column) absorbing boundary conditions for a homogeneous full space model. (\cite{koehn:11})}
\end{center}
\end{figure}
\end{enumerate}
......@@ -251,7 +250,7 @@ Holberg) of FD operators. For the Holberg coefficients n is calculated for a min
\epsfig{file=figures/homogenous_grid_n_16_5.pdf, width=6 cm}\epsfig{file=figures/homogenous_grid_n_16_10.pdf, width=6 cm}
\epsfig{file=figures/homogenous_grid_n_4_5.pdf, width=6 cm}\epsfig{file=figures/homogenous_grid_n_4_10.pdf, width=6 cm}
\epsfig{file=figures/homogenous_grid_n_2_5.pdf, width=6 cm}\epsfig{file=figures/homogenous_grid_n_2_10.pdf, width=6 cm}
\caption{\label{grid_disp_pics} The influence of grid dispersion in FD modeling: Spatial sampling of the wavefield using n=16 (top), n=4 (center) and n=2 gridpoints (bottom) per minimum wavelength $\rm{\lambda_{min}}$.}
\caption{\label{grid_disp_pics} The influence of grid dispersion in FD modeling: Spatial sampling of the wavefield using n=16 (top), n=4 (center) and n=2 gridpoints (bottom) per minimum wavelength $\rm{\lambda_{min}}$. (\cite{koehn:11})}
\end{center}
\end{figure}
\clearpage
......@@ -284,7 +283,7 @@ FDORDER & h (Taylor) & h (Holberg) \\ \hline
\begin{center}
\epsfig{file=figures/courandt_1.pdf, width=7 cm}\epsfig{file=figures/courandt_2.pdf, width=7 cm}
\epsfig{file=figures/courandt_3.pdf, width=7 cm}\epsfig{file=figures/courandt_4.pdf, width=7 cm}
\caption{\label{courandt_pics} Temporal evolution of the Courant instability. In the colored areas the wave amplitudes are extremly large.}
\caption{\label{courandt_pics} Temporal evolution of the Courant instability. In the colored areas the wave amplitudes are extremly large. (\cite{koehn:11})}
\end{center}
\end{figure}
\clearpage
% ------------------------------------
\chapter{The adjoint problem}
% ------------------------------------
This chapter is mainly based on the dissertation of Daniel Köhn (\cite{koehn:11}), who originally has written this manual.\\
\newline
The aim of full waveform tomography is to find an ''optimum'' model which can explain the data very well. It should not only explain the
first arrivals of specific phases of the seismic wavefield like refractions or reflections, but also the amplitudes which contain information
on the distribution of the elastic material parameters in the underground. To achieve this goal three problems have to be solved:
......@@ -31,7 +32,7 @@ has a special physical meaning. It represents the residual elastic energy contai
\begin{figure}[!bh]
\begin{center}
\includegraphics[width=15 cm]{figures/data_res_sketch_1.pdf}
\caption{Definition of data residuals $\rm{\mathbf{\delta u}}$.}
\caption{Definition of data residuals $\rm{\mathbf{\delta u}}$. (\cite{koehn:11})}
\label{sketch_data_res}
\end{center}
\end{figure}
......@@ -49,7 +50,7 @@ along the search direction $\rm{\mathbf{\delta m}_{1}}$ with the step length $\r
\begin{figure}[!bh]
\begin{center}
\includegraphics[width=12.5cm]{figures/sketch_grad_1.pdf}
\caption{Schematic sketch of the residual energy at one point in space as a function of two model parameters $\rm{m_1}$ and $\rm{m_2}$. The blue dot denotes the starting point in the parameter space, while the red cross marks a minimum of the objective function.}
\caption{Schematic sketch of the residual energy at one point in space as a function of two model parameters $\rm{m_1}$ and $\rm{m_2}$. The blue dot denotes the starting point in the parameter space, while the red cross marks a minimum of the objective function. (\cite{koehn:11})}
\label{sketch_grad}
\end{center}
\end{figure}
......@@ -105,7 +106,7 @@ Eq. \ER{dL2_dm} can be related to the mapping of small changes from the data to
\begin{figure}[ht]
\begin{center}
\includegraphics[width=10cm]{figures/mapping_data_model.pdf}
\caption{Mapping between model and data space and vice versa.}
\caption{Mapping between model and data space and vice versa. (\cite{koehn:11})}
\label{mapping_data_model}
\end{center}
\end{figure}
......@@ -424,7 +425,7 @@ The aim is to find the minimum of this function loacted at the point [1,1] which
\begin{center}
\includegraphics[width=15cm]{figures/Rosenbrock_1.pdf}\\
\includegraphics[width=15cm]{figures/Rosenbrock_2.pdf}\\
\caption{Results of the convergence test for the Rosenbrock function. The minimum is marked with a red cross, the starting point with a blue point. The maximum number of iterations is 16000. The step length $\rm{\mu_n}$ varies between $\rm{2e-3}$ (top) and $\rm{6.1e-3}$ (bottom).}
\caption{Results of the convergence test for the Rosenbrock function. The minimum is marked with a red cross, the starting point with a blue point. The maximum number of iterations is 16000. The step length $\rm{\mu_n}$ varies between $\rm{2e-3}$ (top) and $\rm{6.1e-3}$ (bottom). (\cite{koehn:11})}
\label{Rosenbrock_constant}
\end{center}
\end{figure}
......@@ -432,7 +433,7 @@ gradient of the Rosenbrock function is large. After reaching the narrow valley t
\begin{figure}[ht]
\begin{center}
\includegraphics[width=15cm]{figures/sl_case1_final}
\caption{Line search algorithm to find the optimum step length $\rm{\mu_{opt}}$: The true misfit function (yellow line) is approximated by a parabola fitted by 3 points.}
\caption{Line search algorithm to find the optimum step length $\rm{\mu_{opt}}$: The true misfit function (yellow line) is approximated by a parabola fitted by 3 points. (\cite{koehn:11})}
\label{sl_case1_final}
\end{center}
\end{figure}
......@@ -502,7 +503,7 @@ This approach leads to a smoother decrease of the objective function, but also i
\begin{figure}[ht]
\begin{center}
\includegraphics[width=16cm]{figures/Rosenbrock_3}
\caption{Results of the convergence test for the Rosenbrock function. The minimum is marked by a red cross, the starting point by a blue point. The maximum number of iterations is 4000. The optimum step length is calculated at each iteration by the parabola fitting algorithm. Note the criss-cross pattern of the updates in the narrow valley near the minimum.}
\caption{Results of the convergence test for the Rosenbrock function. The minimum is marked by a red cross, the starting point by a blue point. The maximum number of iterations is 4000. The optimum step length is calculated at each iteration by the parabola fitting algorithm. Note the criss-cross pattern of the updates in the narrow valley near the minimum. (\cite{koehn:11})}
\label{Rosenbrock_variable}
\end{center}
\end{figure}
......@@ -544,7 +545,7 @@ I use the very popular choice $\rm{\beta_n = max[0,\beta_n^{PR}]}$ which provide
\begin{figure}[ht]
\includegraphics[width=17cm]{figures/Rosenbrock_4}\\
\caption{Results of the convergence test for the Rosenbrock function using the conjugate gradient method, where the optimum step length is calculated with the parabolic fitting algorithm. The minimum is marked by a red cross, the starting point by a blue point. The maximum number of iterations is 2000.}
\caption{Results of the convergence test for the Rosenbrock function using the conjugate gradient method, where the optimum step length is calculated with the parabolic fitting algorithm. The minimum is marked by a red cross, the starting point by a blue point. The maximum number of iterations is 2000. (\cite{koehn:11})}
\label{Rosenbrock_cg}
\end{figure}
......
......@@ -112,10 +112,12 @@
\section*{Authors}
The IFOS2D code (formerly DENISE) was at first developed by Daniel K\"ohn, Denise De Nil and Andr$\rm{\acute{e}}$ Kurzmann at the Christian-Albrechts-Universit\"at Kiel and TU Bergakademie Freiberg (Germany) from 2005 to 2009.\\
\newline
This documentation was originally written by Daniel K\"ohn.\\ Large parts of the shown theory is extracted from his dissertation \cite{koehn:11}.\\
\newline
The forward code is based on the viscoelastic FD code fdveps (now SOFI2D) by \cite{bohlen:02}.\\
\newline
Different external libraries for timedomain filtering are used.\\
Different external libraries for time domain filtering are used.\\
The copyright of the source codes are held by different persons:\\
\newline
cseife.c, cseife.h, lib\_stfinv, lib\_aff, lib\_fourier:\\
......@@ -160,13 +162,14 @@ The Authors of IFOS2D are listed in file \path{AUTHORS}.
%------------------------------------------------------------------------------------------------%
We thank for constructive discussions and further code improvements:\\
\newline
\newline
Daniel K\"ohn (Christian-Albrechts-Universit\"at Kiel), \\
Anna Przebindowska (Karlsruhe Institute of Technology), \\
Olaf Hellwig (TU Bergakademie Freiberg), \\
Dennis Wilken and Wolfgang Rabbel (Christian-Albrechts-Universit\"at Kiel).
\newline
\noindent The development of the code was suppported by the Christian-Albrechts-Universität Kiel, TU Bergakademie Freiberg, Deutsche Forschungsgemeinschaft (DFG), Bundesministerium für Bildung und Forschung (BMBF), the Wave Inversion Technology (WIT) Consortium and the Verbundnetz-Gas AG (VNG).
\noindent The development of the code was supported by the Christian-Albrechts-Universität Kiel, TU Bergakademie Freiberg, Deutsche Forschungsgemeinschaft (DFG), Bundesministerium für Bildung und Forschung (BMBF), the Wave Inversion Technology (WIT) Consortium and the Verbundnetz-Gas AG (VNG).
\noindent The code was tested and optimized at the computing centres of Kiel University, TU Bergakademie Freiberg, TU Chemnitz, TU Dresden, the Karlsruhe Institute of Technology (KIT) and the Hochleistungsrechenzentrum Nord (HLRN 1+2).
\newline
......
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment