alessia at geodynamics.org alessia at geodynamics.org
Fri Nov 21 05:26:13 PST 2008

Author: alessia
Date: 2008-11-21 05:26:13 -0800 (Fri, 21 Nov 2008)
New Revision: 13367

Modified:
Log:
New cross-correlation formula

===================================================================
--- seismo/3D/ADJOINT_TOMO/flexwin/latex/acknowledgements.tex	2008-11-21 02:28:43 UTC (rev 13366)
+++ seismo/3D/ADJOINT_TOMO/flexwin/latex/acknowledgements.tex	2008-11-21 13:26:13 UTC (rev 13367)
@@ -2,10 +2,12 @@
This is contribution No.~10003 of the Division of Geological
\& Planetary Sciences (GPS) of the California Institute of Technology.
We acknowledge support by the National Science Foundation under grant EAR-0711177.
+ANY OTHER FUNDING?  E.G. SURF SUPPORT FOR DANIEL?
The numerical simulations for this research were performed on the GPS Dell cluster.
The facilities of the IRIS Data Management System, and specifically the IRIS Data Management Center, were used for access to waveform and metadata required in global scale examples of this study. The IRIS DMS is funded through the National Science Foundation and specifically the GEO Directorate through the Instrumentation and Facilities Program of the National Science Foundation under Cooperative Agreement EAR-0004370.
Additional global scale data were provided by the GEOSCOPE network.
We thank the Hi-net Data Center (NIED), especially Takuto Maeda and Kazushige Obara, for their help in providing the seismograms used in the Japan examples.
For the southern California examples, we used seismograms from the Southern California Seismic Network, operated by California Institute of Technology and U.S.G.S.
The FLEXWIN code makes use of filtering and enveloping algorithms that are part of SAC (Seismic Analysis Code, Lawerence Livermore National Laboratory) provided for free to IRIS members.  We thank Brian Savage for adding interfaces to these algorithms in recent SAC distributions.
-{\bf CHT mod:} Thanks reviewers?  Acknowledge SURF funding for Daniel?
+We thank Jeroen Ritsema and an anonymous reviewer for insightful comments that
+helped improve the manuscript.

===================================================================
(Binary files differ)

===================================================================
--- seismo/3D/ADJOINT_TOMO/flexwin/latex/method.tex	2008-11-21 02:28:43 UTC (rev 13366)
+++ seismo/3D/ADJOINT_TOMO/flexwin/latex/method.tex	2008-11-21 13:26:13 UTC (rev 13367)
@@ -249,44 +249,48 @@
groups of phases on the synthetic seismogram.
The next stage is to evaluate the degree of similarity between the observed and
synthetic seismograms within these windows, and to reject
-those that fail basic fit-based criteria.
+those that fail basic fit-based criteria.   For each window, we consider
+the windowed waveforms $\widetilde{d}(t)$ and $\widetilde{s}(t)$ to be the
+product of $d(t)$ and $s(t)$ with
+a boxcar function that is unity between start and end times of the
+window and zero elsewhere.

-{\bf CHT mod:}
-In this section we consider records that have been windowed as
+%{\bf CHT mod:}
+%In this section we consider records that have been windowed as
+%%
+%\begin{eqnarray}
+%d_k(t) &=& W_k(t)\,d(t)
+%\\
+%s_k(t) &=& W_k(t)\,s(t)
+%\end{eqnarray}
%
-\begin{eqnarray}
-d_k(t) &=& W_k(t)\,d(t)
-\\
-s_k(t) &=& W_k(t)\,s(t)
-\end{eqnarray}
+%where $k$ denotes the index of the measurement window in the full dataset, and $W_k(t)$ is the corresponding boxcar window between times $t_{1k}$ and $t_{2k}$:
+%%
+%\begin{align}
+%W_k(t) &=
+%  \begin{cases}
+%    0 & \text{$t < t_{1k}$}, \\
+%    1 & \text{$t_{1k} \le t \leq t_{2k}$}, \\
+%    0 & \text{$t > t_{2k}$}.
+%  \end{cases}
+%\end{align}
%
-where $k$ denotes the index of the measurement window in the full dataset, and $W_k(t)$ is the corresponding boxcar window between times $t_{1k}$ and $t_{2k}$:
-%
-\begin{align}
-W_k(t) &=
-  \begin{cases}
-    0 & \text{$t < t_{1k}$}, \\
-    1 & \text{$t_{1k} \le t \leq t_{2k}$}, \\
-    0 & \text{$t > t_{2k}$}.
-  \end{cases}
-\end{align}
-%
-To avoid clutter, we will omit the $k$-subscript, but the reader is reminded that the data and synthetics are zeroed outside the window $[t_1,\,t_2]$.
+%To avoid clutter, we will omit the $k$-subscript, but the reader is reminded that the data and synthetics are zeroed outside the window $[t_1,\,t_2]$.

The quantities we use to define well-behavedness of data within a window are
signal-to-noise ratio ${\rm SNR}_W$, normalised cross-correlation value between
observed and synthetic seismograms $\mathrm{CC}$,
cross-correlation time lag $\Delta \tau$, and amplitude ratio $\Delta \ln A$.  The signal-to-noise ratio for single windows is defined as an amplitude
-ratio, ${\rm SNR}_W=A_{\rm window}/A_{\rm noise}$, where $A_{\rm window}$ and
-$A_{\rm noise}$ are the maximum absolute values of the observed seismogram $|d(t)|$ in the window
-and in the noise time-span respectively (the noise time-span is the same as that for equation~\ref{eq:noise}).  The cross-correlation value $\mathrm{CC}$ is defined as the maximum value of the
-cross-correlation function $\mathrm{CC}={\rm max}[\Gamma(t)]$, where
-{\bf CHT: I DO NOT THINK THAT THIS REFLECTS THE IMPLEMENTATION.}
+ratio, ${\rm SNR}_W=A_{\rm window}/A_{\rm noise}$, where $A_{\rm window}$ is the maximum of $|\widetilde{d}(t)|$, and
+$A_{\rm noise}$ is the maximum value of $|d(t)|$ in the noise time-span (the noise time-span is the same as that for equation~\ref{eq:noise}).  The cross-correlation value $\mathrm{CC}$ is defined as the maximum value of the normalised
+cross-correlation function, $\mathrm{CC}={\rm max}[\Gamma(t)]$, where
+%{\bf CHT: I DO NOT THINK THAT THIS REFLECTS THE IMPLEMENTATION.}
$$-\Gamma(t) = \frac {\int s(t'-t)d(t')\,\rmd t'}{ \left[\int s^2(t')\,\rmd t' \int d^2(t')\,\rmd t' \right]^{1/2}} +\Gamma(t) = \frac {\int \widetilde{s}(t') \widetilde{d}(t'+t)\,\rmd t'} + { \left[\int \widetilde{s}^2(t')\,\rmd t' \int \widetilde{d}^2(t'+t)\,\rmd t' \right]^{1/2}}$$
-quantifies the similarity in shape between the $s(t)$ and $d(t)$
+quantifies the similarity in shape between the $\widetilde{s}(t)$ and $\widetilde{d}(t)$
waveforms.  The time lag $\Delta \tau$ is defined as the value of $t$
at which $\Gamma$ is maximal, and quantifies the delay in time between a
synthetic and observed phase arrival. The amplitude ratio $\Delta \ln A$ is
@@ -298,7 +302,7 @@
%
$$\Delta\ln{A} = \ln(A_{\rm obs}/A_{\rm syn}) -= 0.5 \ln \left[ \frac{\int d^2(t) \,\rmd t}{\int s^2(t) \,\rmd t} \right] \label{eq:dlnA_def} += 0.5 \ln \left[ \frac{\int \widetilde{d}^2(t) \,\rmd t}{\int \widetilde{s}^2(t) \,\rmd t} \right] \label{eq:dlnA_def}$$
%
(note that \citet[][Eq.~3]{DahlenBaig2002} is the first-order approximation of equation~\ref{eq:dlnA_def}).