alessia at geodynamics.org alessia at geodynamics.org
Thu Nov 20 09:56:34 PST 2008

Author: alessia
Date: 2008-11-20 09:56:34 -0800 (Thu, 20 Nov 2008)
New Revision: 13359

Modified:
Log:
Finished revising manuscript

===================================================================
--- seismo/3D/ADJOINT_TOMO/flexwin/latex/appendix.tex	2008-11-20 16:36:23 UTC (rev 13358)
+++ seismo/3D/ADJOINT_TOMO/flexwin/latex/appendix.tex	2008-11-20 17:56:34 UTC (rev 13359)
@@ -1,6 +1,6 @@
\appendix
\section{Tuning considerations\label{ap:tuning}}
-FLEXWIN is not a black-box application, and as such cannot be blindly applied
+FLEXWIN is not a black-box application, and as such cannot be applied blindly
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
@@ -15,7 +15,7 @@
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 need to be repeated and refined several times before
+(the process may 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
@@ -50,28 +50,33 @@
We suggest finer adjustments to $w_E(t)$ be made after $r0(t)$,
$CC_0(t)$, $\Delta T_0(t)$ and $\Delta \ln A_0(t)$ have been configured.

-$r0(t), CC_0(t)$, $\Delta T_0(t)$, $\Delta \ln A_0(t)$ : These parameters --
+$r_0(t)$, $\mathrm{CC}_0(t)$, $\Delta \tau_{\rm ref}$, $\Delta +\tau_0(t)$, $\Delta \ln A_{\rm ref}$ and $\Delta \ln A_0(t)$ : These parameters ---
window signal-to-noise ratio, normalized cross-correlation value between
observed and synthetic seismograms, cross-correlation time lag, and amplitude
-ratio -- control the degree of well-behavedness of the data within accepted
-windows.  The user first sets constant values for these four parameters, then
+ratio --- control the degree of well-behavedness of the data within accepted
+windows (\stgd).  The user first sets constant values for these four parameters, then
adds a time dependence if required.  Considerations that should be taken into
account include the quality of the Earth model used to calculate the synthetic
seismogram, the frequency range, the dispersed nature of certain arrivals (e.g.
{\em for t corresponding to the group velocities of surface waves, reduce
$CC_0(t)$}), and a-priori preferences for picking certain small-amplitude seismic phases
(e.g. {\em for t close to the expected arrival for $P_{\rm diff}$, reduce $r_0(t)$}).
+$\Delta \tau_{\rm ref}$ and $\Delta \ln A_{\rm ref}$ should be set to zero at first, and only
+reset if the synthetics contain a systematic bias in travel-times or amplitudes.

-$c_{0-4}$ : These parameters are active in \stgc\ of the algorithm, the stage in which the suite of all possible data windows is pared down using criteria on the shape of the STA:LTA $E(t)$ waveform alone.  Detailed descriptions of the behavior of each parameter are available in section~\ref{sec:stageC} and will not be repeated here.  We suggest the user start by setting these values to those used in our global example (see Table~\ref{tb:example_params}).  Subsequent minimal tuning should be performed by running the algorithm on a subset of the data and closely examining the lists of windows rejected at each stage to make sure the user agrees with the choices made by the algorithm.
+$c_{0-4}$ : These parameters are active in \stgc\ of the algorithm, the stage in which the suite of all possible data windows is pared down using criteria on the shape of the STA:LTA $E(t)$ waveform alone.  Detailed descriptions of the behavior of each parameter are available in Section~\ref{sec:stageC} and will not be repeated here.  We suggest the user start by setting these values to those used in our global example (see Table~\ref{tb:example_params}).  Subsequent minimal tuning should be performed by running the algorithm on a subset of the data and closely examining the lists of windows rejected at each stage to make sure the user agrees with the choices made by the algorithm.

+$w_{\mathrm{CC}}$, $w_{\rm len}$ and $w_{\rm nwin}$ : These parameters control the overlap resolution stage of the algorithm (\stge), and are discussed in detail in Section~\ref{sec:stageE}.  Values of $w_{\mathrm{CC}}= w_{\rm len} = w_{\rm nwin} = 1$ should be reasonable for most applications.
+
The objective of the tuning process summarily described here should be to maximize the selection of windows around desirable features in the seismogram, while minimizing the selection of undesirable features, bearing in mind that the desirability or undesirability of a given feature is subjective, and depends on how the user subsequently intends to use the information contained within he data windows.

\subsection{Examples of user functions\label{ap:user_fn}}

-As concrete examples of how the time dependence of these tuning parameters can be used, we present here the functional forms of these time dependencies used for the three example tomographic scenarios described in the text (Windowing Examples, section~\ref{sec:results}).
+As concrete examples of how the time dependence of the tuning parameters can be exploited, we present here the functional forms of the time dependencies used for the three example tomographic scenarios described in the text (Section~\ref{sec:results}).
%CHT modified:
-In each example we use information (predicted arrival times) derived from 1D Earth models to help guide certain user functions in the windowing algorithm. Note, however, that the actual selection of individual windows is based primarily on the details of the waveforms, and not on information from 1D Earth models.
+In each example we use predicted arrival times derived from 1D Earth models to help modulate certain parameters. Note, however, that the actual selection of individual windows is based on the details of the waveforms, and not on information from 1D Earth models.

\subsubsection{Global scenario\label{ap:user_global}}

@@ -131,7 +136,7 @@
\end{cases}
\end{align}

-For the \trange{6}{30} data, the fit between the synthetic and observed surface waves is expected to be poor, as the 3D model used to calculate the synthetics cannot produce the required complexity. We therefore want concentrate on body wave arrivals only, and avoid surface wave windows altogether by modulate $w_E(t)$ as follows:
+For the \trange{6}{30} data, the fit between the synthetic and observed surface waves is expected to be poor, as the 3D model used to calculate the synthetics cannot produce the required complexity. We therefore want concentrate on body wave arrivals only, and avoid surface wave windows altogether by modulating $w_E(t)$ as follows:
%
\begin{align}
w_E(t) & =

===================================================================
--- seismo/3D/ADJOINT_TOMO/flexwin/latex/discussion.tex	2008-11-20 16:36:23 UTC (rev 13358)
+++ seismo/3D/ADJOINT_TOMO/flexwin/latex/discussion.tex	2008-11-20 17:56:34 UTC (rev 13359)
@@ -1,18 +1,18 @@
\section{Using FLEXWIN for tomography}
\label{sec:discuss}

-The window selection algorithm we describe in this paper was designed to solve the problem of automatically picking windows for tomographic problems in which phase separation and identification are not necessary: 3D-3D numerical tomography, of which the adjoint tomography proposed by \cite{TrompEtal2005} and \cite{TapeEtal2007} is an example. For these problems, our algorithm provides a window-selection solution that is midway between full-waveform selection -- which carries the risk of including high-noise portions of the waveform that would contaminate the tomography -- and the selection of known phases or phase-groups based on a-priori arrival times -- which carries the risk of missing the information contained in the non-traditional phases produced by fully 3D structures.
+The window selection algorithm we describe in this paper was designed to solve the problem of automatically picking windows for tomographic problems in which phase separation and identification are not necessary: 3D-3D numerical tomography, of which the adjoint tomography proposed by \cite{TrompEtal2005} and \cite{TapeEtal2007} is an example. For these problems, our algorithm provides a window-selection solution that is midway between full-waveform selection --- which carries the risk of including high-noise portions of the waveform that would contaminate the tomography --- and the selection of known phases or phase-groups based on a-priori arrival times --- which carries the risk of missing the information contained in the non-traditional phases produced by fully 3D structures.

-FLEXWIN may also be used to select windows for tomographic problems in which separation of seismic arrivals is necessary and occurs naturally (with certain frequency and epicentral distance conditions) by virtue of differences in travel-times.  It can straightforwardly be adapted to studies of distinct body wave phases \citep[e.g.][]{RitsemaVanHeijst2002} or to emulate the wavepacket selection of \cite{PanningRomanowicz2006} by modulating the $w_E(t)$ water-level using predicted phase arrival times, and selecting appropriate values for the signal-to-noise, cross-correlation and amplitude limits.  The method can also be used to pre-select windows for studies of fundamental mode surface waves \citep[e.g. those based on the methods of][]{TrampertWoodhouse1995, EkstromEtal1997, LevshinRitzwoller2001} by modulating $w_E(t)$ to exclude portions of the waveform that do not correspond to the correct group velocity window or epicentral distance range.  Given the dispersed nature of surface waves, synthetics produced by 1D starting models often are considerably different in shape from the data, so the $CC$ and $\Delta T$ conditions (but not the signal-to-noise or $\Delta \ln A$ conditions) should relaxed in the window selection.  These windows should then be passed on to specific algorithms used to extract the dispersion information.
+FLEXWIN may also be used to select windows for tomographic problems in which separation of seismic arrivals is necessary and occurs naturally (under certain frequency and epicentral distance conditions) by virtue of differences in travel-times.  It can straightforwardly be adapted to studies of distinct body wave phases \citep[e.g.][]{RitsemaVanHeijst2002} or to emulate the wavepacket selection of \cite{PanningRomanowicz2006} by modulating the $w_E(t)$ water-level using predicted phase arrival times, and selecting appropriate values for the signal-to-noise, cross-correlation and amplitude limits.  The method can also be used to pre-select windows for studies of fundamental mode surface waves \citep[e.g. those based on the methods of][]{TrampertWoodhouse1995, EkstromEtal1997, LevshinRitzwoller2001} by modulating $w_E(t)$ to exclude portions of the waveform that do not correspond to the correct group velocity window or epicentral distance range.  Given the dispersed nature of surface waves, synthetics produced by 1D starting models often are considerably different in shape from the data, so the $CC$ and $\Delta T$ conditions (but not the signal-to-noise or $\Delta \ln A$ conditions) should relaxed in the window selection.  These windows should then be passed on to specific algorithms used to extract the dispersion information.
For this class of natural separation tomographic problems, the advantages of using FLEXWIN over manual or specifically designed automated windowing would be the encapsulation of the selection criteria entirely within the parameters of Table~\ref{tb:params} (and their time-dependent modulation), leading to greater clarity and portability between studies using different inversion methods.

%FLEXWIN is not indicated for tomographic problems in which the extraction and separation of information from overlapping portions of a single timeseries is required, for example studies of higher mode surface wave dispersion for which specific methods -- mode branch stripping \citep{vanHeijstWoodhouse1997}, separation of secondary observables \citep{CaraLeveque1987, Debayle1999}, partitioned waveform and automated multimode inversion \citep{Nolet1990, LebedevEtal2005}, non-linear direct search \citep{YoshizawaKennett2002b, VisserEtal2007}  -- have been developed.
%{\bf CHT modify}
-FLEXWIN is not intended for tomographic problems in which the extraction and separation of information from overlapping portions of a single timeseries is required, for example studies of higher mode surface wave dispersion for which specific methods have been developed, for example, mode branch stripping \citep{vanHeijstWoodhouse1997}, separation of secondary observables \citep{CaraLeveque1987, Debayle1999}, partitioned waveform and automated multimode inversion \citep{Nolet1990, LebedevEtal2005}, and non-linear direct search \citep{YoshizawaKennett2002b, VisserEtal2007}.
+FLEXWIN is not intended for tomographic problems in which the extraction and separation of information from overlapping portions of a single timeseries is required, for example studies of higher mode surface wave dispersion for which specific methods have been developed: mode branch stripping \citep{vanHeijstWoodhouse1997}, separation of secondary observables \citep{CaraLeveque1987, Debayle1999}, partitioned waveform and automated multimode inversion \citep{Nolet1990, LebedevEtal2005}, and non-linear direct search \citep{YoshizawaKennett2002b, VisserEtal2007}.

-The full power of FLEXWIN can only be unleashed for problems -- such as adjoint tomography -- which do not require the separation (natural or otherwise) of seismic phases.  The specificity of adjoint tomography, among the 3D-3D tomographic methods, is to calculate the sensitivity kernels by interaction between the wavefield used to generate the synthetic seismograms and an adjoint wavefield whose source term is derived from measurements of misfit between the synthetic and observed seismograms \cite{TrompEtal2005, LiuTromp2006}.  The manner in which the adjoint sources are constructed is specific to each type of measurement (e.g. waveform difference, cross-correlation time-lag, multi-taper phase and amplitude anomaly), but once formulated can be applied indifferently to any part of the seismogram.  Adjoint methods have been used to calculate kernels of various body- and surface-wave phases with respect to isotropic elastic parameters and interface depths \citep{LiuTromp2006}, and with respect to anisotropic elastic parameters \citep{SieminskiEtal2007a,SieminskiEtal2007b}.  Adjoint methods allow us to calculate kernels for each and every wiggle on a given seismic record, thereby giving access to virtually all the information contained within.
+The full power of FLEXWIN can only be unleashed for problems --- such as 3D-3D tomography --- which do not require the separation (natural or otherwise) of seismic phases.  The specificity of adjoint tomography, among the 3D-3D tomographic methods, is to calculate the sensitivity kernels by interaction between the wavefield used to generate the synthetic seismograms and an adjoint wavefield whose source term is derived from measurements of misfit between the synthetic and observed seismograms \cite{TrompEtal2005, LiuTromp2006}.  The manner in which the adjoint sources are constructed is specific to each type of measurement (e.g. waveform difference, cross-correlation time-lag, multi-taper phase and amplitude anomaly), but once formulated can be applied indifferently to any part of the seismogram.  Adjoint methods have been used to calculate kernels of various body- and surface-wave phases with respect to isotropic elastic parameters and interface depths \citep{LiuTromp2006}, and with respect to anisotropic elastic parameters \citep{SieminskiEtal2007a,SieminskiEtal2007b}.  Adjoint methods allow us to calculate kernels for each and every wiggle on a given seismic record, thereby giving access to virtually all the information contained within.

It is becoming clear, as more finite-frequency tomography models are published, that better kernels on their own are not the answer to the problem of improving the resolution of tomographic studies.  \cite{TrampertSpetzler2006} and \cite{BoschiEtal2007} investigate the factors limiting the quality of finite-frequency tomography images, and conclude that incomplete and inhomogeneous data coverage limit in practice the improvement in resolution that accurate finite-frequency kernels can provide.  The current frustration with the data-induced limitations to the improvements in wave-propagation theory is well summarized by \cite{Romanowicz2008}.  The ability of adjoint methods to deal with all parts of the seismogram indifferently means we can incorporate more information from each seismogram into a tomographic problem, thereby improving data coverage.

===================================================================
--- seismo/3D/ADJOINT_TOMO/flexwin/latex/figures_paper.tex	2008-11-20 16:36:23 UTC (rev 13358)
+++ seismo/3D/ADJOINT_TOMO/flexwin/latex/figures_paper.tex	2008-11-20 17:56:34 UTC (rev 13359)
@@ -16,7 +16,7 @@
$\mathrm{CC}_0(t)$          & acceptance level for normalized cross-correlation\\
$\Delta\tau_0(t)$  & acceptance level for time lag \\
$\Delta\ln{A}_0(t)$   & acceptance level for amplitude ratio \\
-$\Delta\tau_{\rm ref}$ & reference time lag measurement \\
+$\Delta\tau_{\rm ref}$ & reference time lag \\
$\Delta\ln{A}_{\rm ref}$ & reference amplitude ratio \\
\hline
\multicolumn{2}{l}{Fine tuning parameters:} \\ [5pt]
@@ -46,6 +46,7 @@
% CHECK THAT THE MOMENT IS LISTED IN N-M, NOT DYNE-CM
% CARL HAS FORMULAS TO CONVERT FROM A MOMENT TENSOR TO M0 TO MW
101895B		& 28.06		& 130.18	& 18.5	& 5.68e19 & 7.1	& Ryukyu Islands \\
+200808270646A & -10.49 & 41.4400 & 24.0 & 4.68e+17 & 5.7 & Comoros Region \\
050295B		& -3.77		& -77.07	& 112.8	& 1.27e19 & 6.7	& Northern Peru \\
060994A		& -13.82	& -67.25	& 647.1	& 2.63e21 & 8.2	& Northern Bolivia \\
\hline
@@ -219,9 +220,13 @@
(a)~Top: observed and synthetic seismograms (black and red
traces); bottom: STA:LTA timeseries $E(t)$.  Windows chosen by the algorithm
are shown using light blue shading.  The phases contained these windows are:
-(1) $PP$, (2) $PS+SP$, (3) $SS$, (4) $SSS$, (5) $S5$, (6) $S6$, (7) fundamental
+(1)~$PP$, (2)~$PS+SP$, (3)~$SS$, (4)~$SSS$, (5)~$S5$, (6)~$S6$, (7)~fundamental
mode Rayleigh wave.
-(b)~Ray paths corresponding to the body wave phases present in the data windows.
+(b)~Ray paths corresponding to the body wave phases present in the data windows in~(a).
+(c)~Window selection results for event 200808270646A from Table~\ref{tb:events} recorded
+at OTAV ($0.24$\degN, $78.45$\degW, $\Delta=119$\deg, vertical component). Phases contained within selected windows:
+(1)~$S_{\rm diff}$ and~$PS+SP$, (2)~$SS$, (3)~fundamental mode Rayleigh wave.
+(d)~Ray paths corresponding to the body wave phases present in the data windows in~(c).
}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

===================================================================
--- seismo/3D/ADJOINT_TOMO/flexwin/latex/introduction.tex	2008-11-20 16:36:23 UTC (rev 13358)
+++ seismo/3D/ADJOINT_TOMO/flexwin/latex/introduction.tex	2008-11-20 17:56:34 UTC (rev 13359)
@@ -61,7 +61,7 @@
associated with distinct energy arrivals, we need to analyse the character of the synthetic waveform itself.  This analysis is similar to that used on observed waveforms
in automated phase detection algorithms for the routine location of
earthquakes.
-In designing our time-window selection algorithm, we have taken a tool used in this detection process -- the long-term / short-term average ratio -- and applied it to the definition of time windows around distinct seismic phases.
+In designing our time-window selection algorithm, we have taken a tool used in this detection process --- the long-term / short-term average ratio --- and applied it to the definition of time windows around distinct seismic phases.

The choices made in time-window selection for tomography are
interconnected with all aspects of the tomographic inversion process,

===================================================================
--- seismo/3D/ADJOINT_TOMO/flexwin/latex/method.tex	2008-11-20 16:36:23 UTC (rev 13358)
+++ seismo/3D/ADJOINT_TOMO/flexwin/latex/method.tex	2008-11-20 17:56:34 UTC (rev 13359)
@@ -58,8 +58,8 @@

Detection and identification of seismic phase arrivals is routinely performed by automated
earthquake location algorithms \citep[e.g.][]{Allen1982, EarleShearer1994, AsterRowe2000, BaiKennett2000, SleemanVanEck2003}. We have taken a tool used in most implementations of the
-automated detection process -- the short-term long-term average ratio \citep[STA:LTA, e.g.][]{WithersEtal1998,BaiKennett2001}
--- and adapted it to the task of defining time windows around seismic phases.
+automated detection process --- the short-term long-term average ratio \citep[STA:LTA, e.g.][]{WithersEtal1998,BaiKennett2001}
+--- and adapted it to the task of defining time windows around seismic phases.
Given a synthetic seismogram $s(t)$, we derive an
STA:LTA timeseries using an iterative algorithm applied to the envelope of the synthetic.
If we denote the Hilbert transform of the synthetic seismogram by
@@ -396,6 +396,6 @@
The best subset of candidate windows within each group is the one with the
highest combined score $S$.  The final set of windows is
given by concatenating the best subsets of candidate windows for each group.
-Figure~\ref{fg:res_abkt} shows an example of final windows selected on real
+Figure~\ref{fg:res_abkt}a shows an example of final windows selected on real
data.

===================================================================
--- seismo/3D/ADJOINT_TOMO/flexwin/latex/results.tex	2008-11-20 16:36:23 UTC (rev 13358)
+++ seismo/3D/ADJOINT_TOMO/flexwin/latex/results.tex	2008-11-20 17:56:34 UTC (rev 13359)
@@ -4,7 +4,7 @@
applied to real data.  These examples illustrate the robustness and flexibility of the
algorithm.  We have applied the algorithm to three
tomographic scenarios, with very different geographical extents and distinct period ranges:
-a global tomography (\trange{50}{150}),
+a long-period global tomography (\trange{50}{150}),
a regional tomography of the Japan subduction zone, down to 700~km (\trange{6}{120}), and
a regional tomography of southern California, down to 60~km (\trange{2}{40}).
For each of these scenarios, we compare
@@ -37,23 +37,26 @@
which the mantle is given by the S20RTS model of \citet{RitsemaEtal2004},
and the crust by the CRUST2.0 model of \citet{BassinEtal2000}.
The degree-20 $S$-wave velocity model S20RTS defines isotropic perturbations to
-radially anisotropic PREM \citep{DziewonskiAnderson1981}; $P$-wave velocity and
-density anomalies are scaled to the $S$-wave anomalies.  CRUST2.0 specifies a
+radially anisotropic PREM \citep{DziewonskiAnderson1981}; the SPECFEM3D implementation of S20RTS takes
+$P$-wave velocity anomalies from the degree-12 $P$-wave velocity model of \citet{RitsemaVanHeijst2002}.
+CRUST2.0 specifies a
seven-layer crustal seismic velocity and density profile for each cell on a
2\deg\ grid.  The S20RTS+CRUST2.0 combination produces synthetics that are a
good match to observed seismograms for periods longer than 25~s.  For our
examples, we shall be working in the period range \trange{50}{150}.

-Here we discuss windowing results for shadow-zone seismograms of three earthquakes listed
-in Table~\ref{tb:events}: a shallow event in the Ryukyu Islands, Japan
-(101895B), an intermediate depth event in northern Peru (050295B), and a strong
+Here we discuss windowing results for shadow-zone seismograms of four earthquakes listed
+in Table~\ref{tb:events}: a shallow event in the Ryukyu Islands, Japan (101895B),
+a smaller magnitude shallow event in the
+Comoros region, between Mozambique and Madagascar (200808270646A),
+an intermediate depth event in northern Peru (050295B), and a strong
deep event in northern Bolivia (060994A).  We focus on shadow zone
seismograms as these contain a large number of often poorly time-separated
phases, and pose a greater windowing challenge than more commonly used
teleseismic seismograms.

Windowing results for these seismograms (one example per earthquake) are shown
-in Figures~\ref{fg:res_abkt}a and~\ref{fg:examples}a,c. The first observation
+in Figures~\ref{fg:res_abkt}a,c and~\ref{fg:examples}a,c. The first observation
we make when looking at these examples is that the synthetics match the data
well, indicating that the Earth model S20RTS+CRUST2.0 provides a good 3D image
of how the Earth is seen by \trange{50}{150} seismic waves.  The fit is far from
@@ -72,13 +75,13 @@
windows around actual seismic phases, we have identified the
seismic arrivals contained within the chosen data windows, using standard
PREM-based travel-time curves.  We have found that most of the features within
-the windows in Figures~\ref{fg:res_abkt}a and~\ref{fg:examples}a,c correspond
+the windows in Figures~\ref{fg:res_abkt}a,c and~\ref{fg:examples}a,c correspond
to known seismic phases, which are listed in the
corresponding figure captions.  We have also traced the body wave ray paths
-corresponding to these phases and show them in Figures~\ref{fg:res_abkt}b and~\ref{fg:examples}b,d;
+corresponding to these phases and show them in Figures~\ref{fg:res_abkt}b,d and~\ref{fg:examples}b,d;
these ray path plots serve to illustrate the considerable
amount of information contained in a single seismogram, even a long period
-seismogram, when all the usable seismic phases are considered.
+seismogram, when all the usable seismic phases are considered.  Fewer useable seismic phases are windowed for the smaller magnitude event in Figure~\ref{fg:res_abkt}c.

Not all the features within a given seismogram are identifiable as seismic
phases. For example, the second window in Figure~\ref{fg:examples}b seems to
@@ -86,8 +89,7 @@
feature retains its character  and is clearly identifiable as $sS_{\rm diff}$,
while the second feature looses its character entirely  and is more readily
assimilated to a generic $S$ wave coda than to a distinct seismic phase.
-Although this feature is not a seismic phase in the high-frequency sense, it is
-nevertheless present in both observed and synthetic seismograms, and
+This feature is present in both observed and synthetic seismograms, and
undoubtedly contains information.  The particularity of our windowing algorithm
is to treat such features as information, without trying to identify their
sources.  A scheme that permits the computation of