[cig-commits] r12971 - in seismo/3D/ADJOINT_TOMO/flexwin/latex: . figures figures/stalta
alessia at geodynamics.org
alessia at geodynamics.org
Fri Sep 26 00:49:17 PDT 2008
Author: alessia
Date: 2008-09-26 00:49:17 -0700 (Fri, 26 Sep 2008)
New Revision: 12971
Added:
seismo/3D/ADJOINT_TOMO/flexwin/latex/figures/stalta/
seismo/3D/ADJOINT_TOMO/flexwin/latex/figures/stalta/stalta_analytic.gnuplot
Modified:
seismo/3D/ADJOINT_TOMO/flexwin/latex/appendix.tex
seismo/3D/ADJOINT_TOMO/flexwin/latex/flexwin_paper.pdf
seismo/3D/ADJOINT_TOMO/flexwin/latex/method.tex
Log:
Started post-review revisions. Added appendix on tuning considerations.
Modified: seismo/3D/ADJOINT_TOMO/flexwin/latex/appendix.tex
===================================================================
--- seismo/3D/ADJOINT_TOMO/flexwin/latex/appendix.tex 2008-09-25 13:58:03 UTC (rev 12970)
+++ seismo/3D/ADJOINT_TOMO/flexwin/latex/appendix.tex 2008-09-26 07:49:17 UTC (rev 12971)
@@ -1,4 +1,64 @@
\appendix
+\section{Tuning considerations\label{ap:tuning}}
+FLEXWIN is not a black-box application, and as such cannot be blindly applied
+to any given dataset or tomographic scenario. The data windowing required by
+any given problem will differ depending on the inversion method, the scale of
+the problem (local, regional, global), the quality of the data-set and that of
+the model and method used to calculate the synthetic seismograms. The user
+must configure and tune FLEXWIN for the given problem. In this appendix we
+shall discuss some general considerations the uesr should bear in mind during
+the tuning process. for more detailed information on tuning, and for further
+examples of tuning parameter sets, we refer the reader to the user manual that
+accomanies the FLEXWIN code distribution.
+
+The order in which the parameters in Table~\ref{tb:params} are discussed in the
+main text of this paper follows the order in which they are used by the
+algorithm, but is not necessarily the best order in which to consider them for
+tuning purposes. We suggest the following as a practical starting sequence
+(the process will likely need to be repeated and refined several times before
+converging on the optimal set of parameters for a given problem and data-set).
+
+$T_{0,1}$ : In setting the corner periods of the band-pass filter, the
+user is deciding on the frequency content of the information to be used int he
+tomographic problem. Values of these corner periods should reflect the
+information content of the data, the quality of the Earth model and the
+accuracy of the simulation used to generate the synthetic seismogram. The
+frequency content in the data depends on the spectral characteristics of the
+source (higher frequencies for small magnitude local events, lower frequencies
+for large magnitude events), on the instrument responses (flat to 120s for
+STS-2s, but xxxxxxxxx for Hi-Net instruments), and on the attenuation
+characteristics of the medium. As $T_{0,1}$ depend on the source and station
+characteristics, which may be heterogeneous in any given data-set, these filter
+periods can be modified dynamically by constructing an appropriate user
+function (e.g. {\em if station is in list of stations with instrument X then
+reset T0 and T1 to new values}).
+
+$r_{P,A}$ : In setting the signal-to-noise ratios for the entire seismogram the
+user is applying a simple quality control on the data. Note that these criteria
+are applied after filtering. No windows will be defined on data that fail this
+quality control.
+
+$w_E(t)$ : The short-term long-term average ratio $E(t)$ of a constant signal
+converges to a constant value when
+the length of the time-series is greater than the effective averaging length of
+the long-term average. This value would be 1 if $C_S=C_L$ in equations
+(\ref{eq:sta}) and (\ref{eq:lta}). For $C_S$ and $C_L$ given by equation
+(\ref{eq:Cs_Cl}), $E(t)$ of a constant signal converges to a value close to
+0.08, with only a weak dependence on $T_0$. We suggest the user start with a constant
+level for $w_E(t)$ equal to this convergence value. The time dependence of
+$w_E(t)$ should then be adjusted to exclude those portions of the waveform the
+user is not interested in, by raising $w_E(t)$ to 1. Finer adjustments may be
+made to include or exclude low amplitude local maxima in $E(t)$ from the window
+creation process, but we suggest such adjustments be made after $r0(t)$,
+$CC_0(t)$, $\Delta T_0(t)$ and $\Delta \ln A_0(t)$ have been configured.
+
+
+
+$r0(t), CC_0(t)$, $\Delta T_0(t)$, $\Delta \ln A_0(t)$ : Constraints on the quality of
+fit. This is straightforward (ish). \ldots
+
+$c{0-4}$ : err... \ldots
+
\section{User functions\label{ap:user_fn}}
Functional forms of the time-dependent parameters in Table~\ref{tb:params}. They are defined by the user, and can depend on quantities related to the focal parameters and to the relative positions of the earthquake source and the receiver.
Added: seismo/3D/ADJOINT_TOMO/flexwin/latex/figures/stalta/stalta_analytic.gnuplot
===================================================================
--- seismo/3D/ADJOINT_TOMO/flexwin/latex/figures/stalta/stalta_analytic.gnuplot (rev 0)
+++ seismo/3D/ADJOINT_TOMO/flexwin/latex/figures/stalta/stalta_analytic.gnuplot 2008-09-26 07:49:17 UTC (rev 12971)
@@ -0,0 +1,58 @@
+#!/usr/bin/gnuplot
+
+# functions
+dt=1.0
+Cs(T)=10**(-dt/T)
+Cl(T)=10**(-dt/(12*T))
+
+# functions for constant signal
+# x = time (backwards)
+# T = central period
+stalta_const(x,T)=log(Cl(T))/log(Cs(T))*(Cs(T)**x-1)/(Cl(T)**x-1)
+
+# functions for constant + boxcar (after end of boxcar has passed point)
+# x = time (backwards)
+# T = central period
+# b = ratio of boxcar to constant signal amplitudes
+# w = width of boxcar as a multiple of T
+x2(x1,T,w)=x1+w*T
+S_boxcar(x,T,b,w,xconv)=1/log(Cs(T)) * (Cs(T)**xconv-1) + \
+ b/log(Cs(T)) * (Cs(T)**x2(x,T,w)-Cs(T)**x)
+L_boxcar(x,T,b,w,xconv)=1/log(Cl(T)) * (Cl(T)**xconv-1) + \
+ b/log(Cl(T)) * (Cl(T)**x2(x,T,w)-Cl(T)**x)
+stalta_boxcar(x,T,b,w,xconv)=S_boxcar(x,T,b,w,xconv)/L_boxcar(x,T,b,w,xconv)
+
+set term postscript eps mono
+# plots
+set output "stalta_constant.eps"
+set xlabel "Time / s"
+set ylabel "STA:LTA"
+set title "STA:LTA for a constant signal"
+plot [x=0:1000] stalta_const(x,5) t "T = 5sec", \
+ stalta_const(x,10) t "T = 10sec", \
+ stalta_const(x,20) t "T = 20sec", \
+ stalta_const(x,50) t "T = 50sec"
+
+
+set output "stalta_constant_convergence.eps"
+set xlabel "Period / s"
+set title "Convergence value of STA:LTA for a constant signal"
+plot [T=5:50] stalta_const(1000,T) t "Value after 1000 sec"
+
+set output "stalta_boxcar_width.eps"
+set xlabel "Time after end of boxcar / s"
+set title "STA:LTA for a boxcar, T=20s, amplitude 10xbackground"
+plot [x=0:800] stalta_boxcar(x,20.0,10.0,1.0,1000.0) t "width = 1 T", \
+ stalta_boxcar(x,20.0,10.0,2.0,1000.0) t "width = 2 T", \
+ stalta_boxcar(x,20.0,10.0,5.0,1000.0) t "width = 5 T", \
+ stalta_boxcar(x,20.0,10.0,10.0,1000.0) t "width = 10 T"
+
+set output "stalta_boxcar_amp.eps"
+set title "STA:LTA for a boxcar, T=20s, width 1xT"
+plot [x=0:800] stalta_boxcar(x,20.0,1.0,1.0,1000.0) t "aplitude ratio = 1", \
+ stalta_boxcar(x,20.0,2.0,1.0,1000.0) t "amplitude ratio = 2", \
+ stalta_boxcar(x,20.0,5.0,1.0,1000.0) t "amplitude ratio = 5", \
+ stalta_boxcar(x,20.0,10.0,1.0,1000.0) t "amplitude ratio = 10", \
+ stalta_boxcar(x,20.0,100.0,1.0,1000.0) t "amplitude ratio = 100"
+
+quit
Property changes on: seismo/3D/ADJOINT_TOMO/flexwin/latex/figures/stalta/stalta_analytic.gnuplot
___________________________________________________________________
Name: svn:executable
+ *
Modified: seismo/3D/ADJOINT_TOMO/flexwin/latex/flexwin_paper.pdf
===================================================================
(Binary files differ)
Modified: seismo/3D/ADJOINT_TOMO/flexwin/latex/method.tex
===================================================================
--- seismo/3D/ADJOINT_TOMO/flexwin/latex/method.tex 2008-09-25 13:58:03 UTC (rev 12970)
+++ seismo/3D/ADJOINT_TOMO/flexwin/latex/method.tex 2008-09-26 07:49:17 UTC (rev 12971)
@@ -77,8 +77,8 @@
series with timestep $\delta t$, calculate its short term average
$S(t_i)$ and its long term average $L(t_i)$ as follows,
\begin{align}
-S(t_i) & = C_S \; S(t_{i-1}) + e(t_i) , \\
-L(t_i) & = C_L \; L(t_{i-1}) + e(t_i) ,
+S(t_i) & = C_S \; S(t_{i-1}) + e(t_i) , \label{eq:sta}\\
+L(t_i) & = C_L \; L(t_{i-1}) + e(t_i) , \label{eq:lta}
\end{align}
and obtain their ratio:
$E(t_i) = S(t_i)/L(t_i)$.
@@ -92,15 +92,28 @@
$C_S$ and $C_L$ to depend on the dominant period of the waveform under
analysis. We have followed their lead in setting
\begin{equation}
-C_S = 10^{- \delta t / T_0} \qquad {\rm and} \qquad C_L = 10^{-\delta t / 12 T_0},
+C_S = 10^{- \delta t / T_0} \qquad {\rm and} \qquad C_L = 10^{-\delta t / 12 T_0}, \label{eq:Cs_Cl}
\end{equation}
where the use of $T_0$, the low-pass corner period of our band-pass filter,
substitutes that of the dominant period.
+The STA:LTA of a constant signal converges to a constant value when the length
+of the time-series is greater than the effective averaging length of the
+long-term average. An energy
+arrival in $e(t)$ causes the $E(t)$ to rise sharply, then drop to below the
+convergence value after the arrival, before stabilizing again. The maximum
+height reached by the $E(t)$ for a given $T_0$ depends on the amplitude of the
+arrival in $e(t)$ and on its duration: higher amplitudes and shorter durations
+cause higher $E(t)$ values. The depth to which the $E(t)$ drops after the
+end of the arrival depends on the same parameters: higher amplitudes and longer
+durations cause deeper drops in $E(t)$ that take longer to return to the
+convergence value for a constant signal.
+
An example of a synthetic seismogram and its corresponding envelope and STA:LTA timeseries $E(t)$ is
-shown in Figure~\ref{fg:stalta}. Before the first arrivals on the synthetic
-seismogram, the $E(t)$ timeseries warms up and rises to a plateau. At each
-successive seismic arrival on the synthetic, $E(t)$ rises to a
+shown in Figure~\ref{fg:stalta}. {\bf Before the first arrivals on the synthetic
+seismogram, the $E(t)$ timeseries warms up and rises to a plateau whose
+amplitude is determined by the low level numerical noise on the synthetic.} At
+each successive seismic arrival on the synthetic, $E(t)$ rises to a
local maximum. We can see from Figure~\ref{fg:stalta} that these local maxima
correspond both in position and in width to the seismic phases in the
synthetic, and that the local minima in $E(t)$ correspond to the
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