alessia at geodynamics.org alessia at geodynamics.org
Tue Nov 25 06:40:01 PST 2008

Author: alessia
Date: 2008-11-25 06:40:01 -0800 (Tue, 25 Nov 2008)
New Revision: 13390

Modified:
Log:
Added line in text to explain integration limits.

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

===================================================================
--- seismo/3D/ADJOINT_TOMO/flexwin/latex/cc_notes.tex	2008-11-25 01:34:58 UTC (rev 13389)
+++ seismo/3D/ADJOINT_TOMO/flexwin/latex/cc_notes.tex	2008-11-25 14:40:01 UTC (rev 13390)
@@ -31,6 +31,7 @@
$$\Gamma(t) = \frac {\int \widetilde{s}(t') \widetilde{d}(t'-t)\,\rmd t'} { \left[\int \widetilde{s}^2(t')\,\rmd t' \int \widetilde{d}^2(t'-t)\,\rmd t' \right]^{1/2}} + \label{eq:paper}$$
%
My point about this has nothing to do with what we implement.  It's that, in the absence of integration limits for convolution-like expressions, one would assume $[-\infty,\infty]$ limits.  In that case, it doesn't matter how you shift $d^2(t)$, because the result is the same:
@@ -58,6 +59,10 @@
%
This is only a start.  It is still sloppy, considering the mix of continuous and discrete notation, and the vague summation (integration) limits.  If this is the formula we are trying to implement, would you please adapt it to what we want to use, and also include explicit integration limits?  Then we can see how it is implemented.

+{\em AM:
+The effective integration limits in (\ref{eq:paper}) are the start and end times of the window $t_S$ and $t_E$ (which are times that have been chosen on the synthetic seismogram).  In the numerator, as $\widetilde{s}(t)$ is zero outside these times, then integrating over $-\infty +\infty$ is the same as integrating over $t_S$ to $t_E$.  The same is true for the $\widetilde{s}(t)$ integral in the denominator.  For the $\widetilde{d}(t)$ integral in the denominator, using the synthetic $t_S$ and $t_E$ as integration limits essentially extracts that portion of the windowed data that overlaps in time with the synthetic.  I have added a line in the text of the paper to say what the integration limits are.
+}
+
\pagebreak
\subsection*{Limiting time-shift searches in CC calculation on the basis of maximum allowed time-shift}

@@ -69,10 +74,24 @@

Okay, consider this.  If the real time shift is 5~s, but you think it should generally be no more than 4~s, then you pick 4~s as the cross-correlation.  But the max CC value is probably not going to have a passable value to have this window picked.  And if it does, then I still think the 4~s shift is much better than rejecting the window.  It would be better to use this in the inversion than nothing at all.  All it means is using a 4~s weight on the adjoint source rather than a 5~s weight.

+{\em AM:
+Here I strongly disagree.  If the time shift between data and synthetic is 5s, then taking 4s even with a reduced CC value is not a true measurement of the time-shift.  You will be asking your inversion to fit a lot of false measurements, even if you do down-weight them slightly using the CC-value.  The 4s value does not correspond to a local maximum in $\Gamma(t)$, and has no physical relation to the actual time shift between data and synthetic.  It only reflects your prior wishes on the value of the time shift.  I am OK with rejecting time-shifts that are larger than you would like them to be for the assumptions you make about your model and measurement method, but not with making up the time-shift measurements themselves.
+    }
+
All we care about is picking the right windows, and the heart of the matter is that $\Gamma(t)$ has local maxima, and typically it is the local maximum with the smaller time shift that should be preferred over the (possible) global maximum whose time shift is too large and thus the window will be rejected.  For example, you limit your search to 4~s, and the actual time shift is 1~s and $CC_{1s} = 0.80$.  However, $CC_{5s} = 0.85$, because it has found another peak that happens to line up better.  The 5~s shift is used, and then the window is rejected due to the 4~s limit.

+{\em AM:
+Yes, what we care about is picking the right windows, but in a self-consistent manner.  $\Gamma(t)$ does have local maxima, which reflect the self-similarity of different swings of a band-limited waveform representing a seismic arrival.  You cannot escape this self-similarity, you just have to deal with it.  Limiting your time-shift search will give you the wrong answer when you have a real time-shift that is larger than your search limits, as I've discussed above.  I do not think this is the way to go.  See below for an alternative.
+
+The normalization scheme we have used up to now (where we essentially consider only the windowed data that overlaps with the windowed synthetic) gives approximately equal weight to the cross-correlation values at every time-shift (as the time-shifts get away from zero, the numerator and denominator decrease in equivalent ways, given that we consider less and less of the information within the data window).  So the cross-correlation with a large time-shift (that only measures the correlation of the end portion of the data with the start of the synthetic, or the start of the data with the end of the synthetic) contributes to $\Gamma(t)$ with the same weight as the cross-correlation with small time-shift (that measures the correlation of most of the data window with most of the synthetic window).  One simple way of emphasizing the small time-shift cross-correlations is to normalize by the entire synthetic and data windows (constant normalization).   This solution has the consequence  of giving systematically lower CC values for larger time-shifts, which may lead to the rejection of some otherwise OK windows.
+}
+

+{\em AM:
+As far as I can make out, there is no easy answer to the question of the best way to evaluate the correlation between two waveforms.  We have to understand the consequences of the choices we make, and make those that make the most sense given the problem at hand.
+    }
+
%----------------------------------

\bibliographystyle{plainnat}

===================================================================
--- seismo/3D/ADJOINT_TOMO/flexwin/latex/method.tex	2008-11-25 01:34:58 UTC (rev 13389)
+++ seismo/3D/ADJOINT_TOMO/flexwin/latex/method.tex	2008-11-25 14:40:01 UTC (rev 13390)
@@ -291,7 +291,7 @@
{ \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 $\widetilde{s}(t)$ and $\widetilde{d}(t)$
-waveforms.  The time lag $\Delta \tau$ is defined as the value of $t$
+waveforms, and the integration limits are the start and end times of the window.  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
defined as the amplitude ratio between observed and synthetic