The method of Tsunami waveform phase correction is explained.
Shino Watada 2014/Aug/22
2015/Jul/30 A correction is added.
2017/Jul/02 A comment is added.
2017/Nov/17 A correction is added.

2017/Dec/20 Dispersion tables for density-stratified ocean modelsby Ho et al. 2017 are added.
2021/May/12 Dispersion tables for short-period tsunamis by Sandanbata et al. 2021 are added.

This directory contains data files and a GMT shell script to create
Figure 7 of the paper below.
Watada, S., S. Kusumoto, and K. Satake (2014), 
Traveltime delay and initial phase reversal of distant tsunamis 
coupled with the self-gravitating elastic Earth,
J. Geophys. Res. Solid Earth, 119, doi:10.1002/2013JB010841.

### CORRECTION in Watada et al 2014 JGR ####
1. In figure 1 caption, a) for the 2011 Tohoku-Oki eq  and b) the 2010 Chile eq.
2  In figure 3, "on March 2013,.." should be "on March 2011,.."
3. In figure 6, the Gaussian peak frequency f_a=0.0015 Hz.
4. In Page 4298, 7 lines before section 4.3, 
"... at a period of 330 s."  should be 
"... at a period of 670 s."
### END of CORRECTION in Watada et al 2014 JGR ####

Improved tsunami dispersion data including the effect of
density stratification of seawater is added.
See Figure 4 of the following paper.
Ho, T.-C., K. Satake, and S. Watada (2017), 
Improved phase corrections for transoceanic tsunami data
in spatial and temporal source estimation:
Application to the 2011 Tohoku Earthquake
J. Geophys. Res. Solid Earth, 122, doi:10.1002/2017JB015070 

Density stratification structure of seawater is given in
Supporting Information Figure S1 of Ho et al. 2017 JGR.

Short-period tsunami dispersion data is added
See Figure 1 or Figure 4 of the following paper.
Sandanbata, O., S. Watada, T.-C. Ho, K. Satake (2021),
Phase delay of short-period tsunamis in the density-stratified
compressible ocean over the elastic Earth,

Geophys. J. Int., 226, 3, 1975-1985, doi:10.1093/gji/ggab192

## How to Download a text file from the URL site ##

In your browser open the following URL.

https://www.eri.u-tokyo.ac.jp/people/watada/tsunami/PREM_tsunami_dispersion/Watada_etal_2014_Fig7/

Then add the filename at the end of URL.

For example to download mode.dat_2km_yn

open the following URL.

https://www.eri.u-tokyo.ac.jp/people/watada/tsunami/PREM_tsunami_dispersion/Watada_etal_2014_Fig7/mode.dat_2km_yn

## END of how to Download a text file from the URL site ##

## File list ##
0README.txt: this file
## Developed in Watada et al. 2014 ##
gravity_value.txt: gravity value at the sea surface and sea bottom of
                   2, 4, and 6 km deep oceans
mode.dat_2km_yn: table of tsunami mode of PREM with 2km deep ocean
mode.dat_4km_yn: same above but for the 4km deep ocean
mode.dat_6km_yn: same above but for the 6km deep ocean
plot.sh: GMT version 4 script to create postscript file of Figure 7.
## Developed in Ho et al. 2017 ##
mode.dat_2km_strf: table of tsunami mode of density stratified 2km deep ocean
mode.dat_4km_strf: same above but for 4km deep ocean
mode.dat_6km_strf: same above but for 6km deep ocean
mode.dat_8km_strf: same above but for 8km deep ocean
## Developed in Sandanbata et al. 2021 ##
mode.dat_0.1km_strf_sp.dat: table for short-period tsunami of 0.1km deep ocean
mode.dat_0.5km_strf_sp.dat: same above but for 0.5km deep ocean
mode.dat_1.0km_strf_sp.dat: same above but for 1.0km deep ocean
mode.dat_2.0km_strf_sp.dat: same above but for 2.0km deep ocean
mode.dat_4.0km_strf_sp.dat: same above but for 4.0km deep ocean
mode.dat_6.0km_strf_sp.dat: same above but for 6.0km deep ocean
mode.dat_8.0km_strf_sp.dat: same above but for 8.0km deep ocean
mode.dat_10.0km_strf_sp.dat: same above but for 10.0km deep ocean
## END of Files ##

mode.dat_2km_yn
mode.dat_4km_yn
mode.dat_6km_yn
mode.dat_2km_strf
mode.dat_4km_strf
mode.dat_6km_strf
mode.dat_8km_strf
mode.dat_0.1km_strf_sp.dat
mode.dat_0.5km_strf_sp.dat
mode.dat_1.0km_strf_sp.dat
mode.dat_10.0km_strf_sp.dat
mode.dat_2.0km_strf_sp.dat
mode.dat_4.0km_strf_sp.dat
mode.dat_6.0km_strf_sp.dat
mode.dat_8.0km_strf_sp.dat

are in the same format.

These tsunami modes for PREM (and modified PREM) have been computed by the 
normal mode method described in
Watada, S., and H. Kanamori (2010), Acoustic resonant oscillations
between the atmosphere and the solid earth during the 1991 Mt. Pinatubo
eruption, J. Geophys. Res., 115, B12319, doi:10.1029/2010JB007747.

### auxiliary files to explain the phase correction method ###
cut.txt : a part of the Gaussian-shaped long-wave waveform 
          in Figure 6b.
cut_corrected.txt : PREM phase-correction applied to the cut.txt 
corrected.txt : Figure 6d (PREM phase-correction applied to Figure 6a.)
sample_cut.pdf: Numerical demonstration of the arbitral timewindow length.

                In Figure a), three lines from top correspond to,
                a part of Gaussian-shaped long-wave (cut.txt),
                phase-correction applied to cut.txt based on equation (7)
                (cut_corrected.txt), and phase correction applied based on
                equations (1) and (2) (corrected.txt), respectively.

                In Figure b), three lines in Figure a) are overlaid each
                other and magnified. Two dispersed waveforms are the same
                in the beginning, but a phase shift between them appears
                at round time of 14 hours. This is because the equation (7)
                holds only for long-waves. As the wave period becomes shorter 
                the wave deviates more from the long-waves. At time 14 hour,
                the wave period of the dispersed wave is about 6 min 
                (360 sec) and the wave deviates from long-waves.

### END of auxiliary files to explain the phase correction method ###


## Example of tsunami mode table (mode.dat_4km_yn)##
#l omg period V_phase V_group PREM 4km ocean
2 6.84207618e-05 91831.56 1.74363469e+02 1.93236984e+02
3 9.91228375e-05 63387.87 1.80431885e+02 1.97213270e+02
....
4800 8.57786413e-02 73.25 1.13841417e+02 5.85912786e+01
5000 8.75947270e-02 71.73 1.11602041e+02 5.71287922e+01
## END of Example of tsunami mode table ##

Each column from left means that:
l: angular order of tsunami eigenmode
omg: angular frequency of tsunami eigenmode
period: eigen period of tsunami mode == 2*PI/omg in sec
V_phase: phase velocity of tsunami mode in meter/sec
V_group: group velocity of tsunami mode in meter/sec

cut.txt
cut_corrected.txt
corrected.txt
are in the same format.

### Example of waveform data ###
....
4.7550e+04 1.0076e-01
4.7580e+04 1.2170e-01
4.7610e+04 1.4327e-01
4.7640e+04 1.6489e-01
4.7670e+04 1.8586e-01
...
### END of Example of waveform data ###

The first column from left is the time (sec), 
second column is the amplitude (m) at location 9000km 
rom source.  g=9.8231 m/s^2, D=4km.


###
### Steps to apply phase correction to distant tsunami
###

 Please note that the phase correction method can be applied to
 the direct tsunamis that can be modeled as unidirectionally
 propagating waves.
 Later part of tsunami waveforms containing reflected waves
 from the coasts are modeled less accurately. 

1) Prepare long-wave tsunami simulation waveform data.

 The data timewindow start time is arbitral. See sample_cut.pdf
 in which two timewindow lengths are compared.

sample_cut.pdf demonstrates that different timewindows containing
 the same long-wave signal result in the same phase-corrected waveforms.
 Top trace is a part of the Gaussian-shaped long-wave shown in Figure 6b.
 Middle trace is the phase-corrected waveform of the Top trace.
 Bottom trace, same to Figure 6d, is the waveforms for which
 phase-correction is applied to the entire waveforms starting
 at the origin time.

 Detrending and tapering should be applied to the waveform data
 in the timewindow. 
 Zero-padding to the waveform data is applied, to make the 
 number of data is the power of 2.

2) Pick a reference depth D. D=4km is OK for global tsunami propagation.

3) Find phase velocities of PREM (or density-stratified seawater)
 tsunami mode for given frequencies by interpolating the tsunami mode table.
 The frequencies are the determined from the total length and sampling interval
 of a given tsunami waveform data.
 At those frequencies, we measure the phase of the waveform time series
 by the Fast Fourier Transofrm (FFT) method.

 If a frequency is outside of a frequency range in the tsunami mode table, 
 set the amplitude and phase at the frequency to be zero and no
 spectra is computed. Resultant tsunami waveforms do not have signals
 at these frequencies outside the frequency range. 

4) At each frequency, use equation (10) of Watada et al. 2014 to compute the
 phase velocity difference of a reference 4km deep ocean.
 Distance L is assumed simply, for example, to be the great circle distance
 between the tsunami source to the observation site.

5) At each frequency, use equation (14) to convert the phase velocity
 difference to the phase difference.

6) Apply the phase correction to the simulated long-wave waveforms.
 This can be done in the frequency domain to add a phase correction to
 the phase spectrum computed by the FFT method.

 Please note that, to obtain the phase corrected time series by the
 inverse FFT (IFFT), phase-corrected phase spectrum must satisfy
 a condition that spectrum at a negative frequency is the complex
 conjugate of the spectrum at the positive frequency.

phase-corrected Fourier spectrum Y(i), i=0,1,...., N, N is a power of 2.
 Complex_conjugate of Y(N-i) = Y(i)
 This means that Y(0) and Y(N/2) are real numbers.

 IFFT of Y(i) will result in a X(i), real part of X(i) is the
 phase-corrected waveforms.


### Final note ###
please look at sample_cut.pdf. It demonstrates that the phase-correction
method works well for a part of long-wave waveform data.
It also demonstrates the phase-correction becomes inaccurate as the wave-type
deviates from shallow water waves to deep water waves.
When phase-correction is applied to short-period tsunamis, i.e. deep water
waves, the method proposed by Sandanbata et al. is recommended.
## END of Final note ###

End

n/Watada_etal_2014_Fig7/