# Submarine Landslide Tsunami Linearity

## This page documents tests on the scaling of tsunami waves with Earthquake volume

As part of our work developing a Probabilistic Tsunami Hazard Assessment (PTHA) for submarine landslide generated tsunamis we are investigating the scaling of tsunami waves with volume of the landslide.

For the initial test we are using the TOPICS submarine-landslide-generated-tsunami initialisation code. There are some questions as to its applicability in the shallow canyon environment but it will do for the initial investigation.

In TOPICS both the volume is specified and the length/width and maximum thickness. This overspecifies the problem (although as it is maximum thickness there is some room for a range of volumes to be consistent with the given dimensions).

We are using the standard Nicholson Canyon setup used in XXXXXXXX with four different volumes: 100,000,000 m^3, 400,000,000 m^3, 333,333,333 m^3 and 1,000,000,000 m^3. The maximum thickness is held constant at 200 m. The landslide length and width are taken as equal and are 2,400 m, 1,500 m, 1,400 m and 760 m respectively for the four cases.

In order to investigate the influence of different dimensions given the same volume we are doing three simulations with volume set as 400,000,000 m^3 and maximum thickness given as 50 m, 100 m and 200 m.

There does not seem to be a strong relationship between volume and water height (i.e. changing the thickness has a large effect on wave height at the places of interest. Some areas appear to be inundated while others aren't. Is this correct or is this a bug of the inundation programme?

One issue that has just been uncovered is that the centroid of the landscape was in fact incorrect and was the middle of the canyon instead. So the first question is how much difference will this make to the results. Joshu Mountjoy has given me 4 new scenarios which are:

`# SCENARIO 1    # The InitSubmarineLandslide event of the topics module is used to generate the initial    # conditions (elevation D, initial velocities D_U and D_V) from the geometrical parameters    # of the landslide.    GModule topics    Init {} { D = 0 }    InitSubmarineLandslide D {        x = 174.75        y = -41.453        alpha = 310        depth = 160.0        theta = 2        length = 2300        width = 2300        thickness = 100        volume = 500000000        gamma = 2.0    }     # We apply the initial conditions at the start of the simulation.    # NB: The initial conditions are representative of the wave generated by the landslide    # at a characteristic time after the start of the landslide    Init { start = 0 } { P = (Zb < 0 ? MAX (0., D - Zb):0) U = (D_U) V = (D_V) }`
`# SCENARIO 2    # The InitSubmarineLandslide event of the topics module is used to generate the initial    # conditions (elevation D, initial velocities D_U and D_V) from the geometrical parameters    # of the landslide.    GModule topics    Init {} { D = 0 }    InitSubmarineLandslide D {        x = 174.782        y = -41.473        alpha = 295        depth = 230.0        theta = 4        length = 1700        width = 1400        thickness = 100        volume = 250000000        gamma = 2.0    }     # We apply the initial conditions at the start of the simulation.    # NB: The initial conditions are representative of the wave generated by the landslide    # at a characteristic time after the start of the landslide    Init { start = 0 } { P = (Zb < 0 ? MAX (0., D - Zb):0) U = (D_U) V = (D_V) }`
`# SCENARIO 3    # The InitSubmarineLandslide event of the topics module is used to generate the initial    # conditions (elevation D, initial velocities D_U and D_V) from the geometrical parameters    # of the landslide.    GModule topics    Init {} { D = 0 }    InitSubmarineLandslide D {        x = 174.787        y = -41.449        alpha = 150        depth = 120.0        theta = 15.        length = 1100        width = 1000        thickness = 50        volume = 55000000        gamma = 2.2    }     # We apply the initial conditions at the start of the simulation.    # NB: The initial conditions are representative of the wave generated by the landslide    # at a characteristic time after the start of the landslide    Init { start = 0 } { P = (Zb < 0 ? MAX (0., D - Zb):0) U = (D_U) V = (D_V) }`
`# SCENARIO 4    # The InitSubmarineLandslide event of the topics module is used to generate the initial    # conditions (elevation D, initial velocities D_U and D_V) from the geometrical parameters    # of the landslide.    GModule topics    Init {} { D = 0 }    InitSubmarineLandslide D {        x = 174.823         y = -41.461         alpha = 150        depth = 120.0         theta = 15.        length =  500         width = 4500        thickness =450         volume = 1000000000        gamma = 2.2    }     # We apply the initial conditions at the start of the simulation.    # NB: The initial conditions are representative of the wave generated by the landslide    # at a characteristic time after the start of the landslide    Init { start = 0 } { P = (Zb < 0 ? MAX (0., D - Zb):0) U = (D_U) V = (D_V) }`

A couple of issues have come up with this. Firstly TOPICS is not quite initialising correctly as D_U and D_V that it initialises are vertically averaged velocity, not flux which is defined in the St Venant equations as vertically averaged velocity times elevation above datum. This should not be difficult to adjust it is just a matter of setting U=D*D_U and V=D*D_V in the initialisation process.

Secondly the GfsOutputGRD routine only works properly in serial - there are artefacts in parallel. This is not a big deal, just output .gfs files when you want and ascii grid and post process it

### Outputting data at certain points

William Power provided the file Cook_points.csv to output time series or maximum wave heights at. This Includes all of Marlborough sounds which might be more than is needed so we may wish to cut it down.

As a start to cutting it down we will just use a box 174 - 175.5 in longitude, -41.8 - -41.15 in latitude. This will include some of the inner Marlborough sounds but not to much.

First I needed to dos2unix the file to get rid of dos endoflines Then I used the following awk file convert_Cook.awk

`BEGIN{    FS=","}{    x=\$1    y=\$2    if ((x > 174) && (x < 175.5) && (y > -41.8) && (y < -41.15)) {	print \$1, \$2, " 0"    }}`

awk -f convertCook < Cook_points.csv > Cook_points.out

The .awk file timeseries.awk given below post-processes the output data to give time and the sea level heights. X and Y are also given in case these are mixed up in parallel runs.

`BEGIN{    FS=" |:"}{    if (\$1 == "#") {	# get column indices of relevant fields	for (i=1; i <= NF; i++) {	    if (\$i == "H")		iH = \$(i-1);	    if (\$i == "x")		ix = \$(i-1);	    if (\$i == "y")		iy = \$(i-1);	}    }    if (NR > 1) {	print \$1, \$ix, \$iy, \$iH    }}`

Which is run using the command

`GfsOutputLocation {step = 1} { awk -f timeseries.awk > timeseries_interp1.out } Cook_points.out { interpolate = 1}`

Currently I am exploring whether or not it is best to interpolate (1 or 0). I am also trying to set it so that if the land is not inundated then the height is set to zero - I am not sure whether this is best or whether it is better to output wave height and also Z for that point and compare this. I tried only outputting if P>0 but that made minimal difference.

There is an issue with the time series in that in some places (especially where there is a steep transition e.g. coastal cliffs. We are getting outputs of positive wave heights even though we are well higher than the water should be. My guess is that it is a mismatch in the way that H is calculated compared to Zb and P. By ensuring that the coast is refine to the highest level looks like it may have solved this problem. We may want to be a bit more selective about how we do this and rather only refine at the points where we are reading the time series off at.

### Superposition of two landslides

Anther question I am investigating is whether we can linearly add the results of two landslides that occur simultaneously. In order to test this I will use scenarios 1 & 2 and scenarios 3 & 4. I will run two simulations. I will run the two separately as well as simultaneously (This I will achieve by outputting the ascii grid for each initialisation, summing them and then reinitialising a run with the sum of the two). I will compare maximum wave height at the end of the simulation as well as time series at certain points.

There are some issues with the time series output but where these do no occur then we get good agreement between the simultaneous timeseries and the sum of the two time series. When interested in maximum wave height. Two end members for comparing this are the sum or the maximum of the two maximum wave heights. Generally we would expect the sum to overestimate the result.

Using Pmax outputted onto an ascii raster at the highest refinement level I compared the additive and maximum approaches. I will refer to the difference output12-max12 as Diff_max and output12-sum12 as Diff_sum. So positive differences occur when we underestimate the maximum wave height. mean(Diff_max)=0.0167 std(Diff_max)=0.1303 max(Diff_max)=7.4179 min(Diff_max)=-2.7561

I also looked at only the places where output12 was greater than one (i.e. the maximum wave height was greater than one because we are interested in how well we are doing in places where there is a tsunami and don't want the signal to be swamped by other areas. (Approximately 7% of the cells had output12 greater than 1) mean(Diff_max_g1)=0.1298 std(Diff_max_g1)=0.3451 max(Diff_max_g1)=7.4179 min(Diff_max_g1)=-2.1230

Likewise for the diff to the sum mean(Diff_sum)=-0.2027 std(Diff_sum)= 0.2264 max(Diff_sum)=4.0107 min(Diff_sum)=-8.0031

mean(Diff_sum_g1)=-0.7180 std(Diff_sum_g1)=0.4634 max(Diff_sum_g1)=4.0107 min(Diff_sum_g1)=-8.0031

So the maximum approach is closer. It may on average slightly underestimate the maximum wave height but it is a lot closer than the sum approach which for maximum wave height greater than 1 has a mean difference greater than one standard deviation away from zero.

Both approximations skew away from normal at the tails. Sum tends to be left skewed - i.e. overestimating the max wave height. Max tends to be right skewed.