Quick and Dirty Tsunami Runup Estimation a.k.a. "The Formula Based Approach" by Steven N. Ward
The formula based approach to estimate tsunami height does not require detailed
waveform simulation, but it does try to embody the same processes (generation, wave
spreading, shoaling, runup) as does the simulation based approach. Naturally, the formula
based approach uses simplified approximations of these processes. Certain parameters in
the formula based approach are obtained by fitting output of many runs of full waveform
simulator; so in that sense, the formula and simulation based approaches are linked.
According to the formula based approach, wave runup is
h = B[A0PS] (1)
where, A0 is the initial tsunami source amplitude, P is the propagation loss (less than 1.0),
S is the shoaling correction (usually more than 1.0), and B is the beaching function.
Initial Tsunami Source Amplitude A0
For an earthquake
A0 = aDu (2)
where u is the earthquake slip and  is the fraction of slip that transforms into tsunamimaking uplift. This factor depends upon the style of the fault
a = (1 - f /1800 )sin f sin j
(3)
where  and  and are the dip 0<<900 and rake angles -900< <900 of the faulting in
degrees. The most efficient mechanism for tsunami generation have  near 450 and  at
plus or minus 900. Note that equation (3) makes no attempt to model source radiation
patterns, so in this sense it represents a worst case.
Propagation Loss P
Tsunami waves decay in size as they travel due to geometrical spreading and
frequency dispersion. For an ocean of constant depth H0 (I call H0, the water depth at the
earthquake), the propagation loss is roughly
æ 2R
P = è1+ öø
D
-y
(4)
where R is the distance of measurement point from the quake, D is the initial dimension
of the tsunami pulse and
æ
Dö
y = 0.5 + 0.575exp ç -0.0175 ÷ (5)
è
H0 ø
The dimension of the initial tsunami pulse can be approximated from the earthquake fault
length L and width W from either
D = (L + W) / 2 or D = 2 LW / p (6)
The first term in exponent (5) accounts for square root of distance losses due to
geometrical spreading. The second term in (5) accounts for additional wave height losses
due to frequency dispersion. Larger dimensioned tsunami decay more slowly with
distance on this account. Typically (5) is between 0.7 and 1.0. Peak deep water tsunami
wave amplitude Ad(R) is
æ 2R
Ad (R) = A0 P = aDu è1 + öø
D
-y
(7)
By wave amplitude, I mean 1/2 peak to trough height. Propagation losses (5) assume that
the path from the quake to the observation location is unobstructed. If the waves have to
pass around headlands, squeeze through narrows, or cut into shadow zones, then (7) again
would be a worst case.
Shoaling Correction S
Equation (7) assumes oceans of constant depth H0. Toward shore, real oceans
shallow to depth HS and tsunami slow and grow to conserve energy flux. Deep water
amplitude Ad(R) needs to be corrected to account for shoaling. According to linear
theory, the shoaling correction S is
1/ 2
é Vg (w max , H0 ) ù
(8)
S=
ê V (w , H ) ú
ë g max S û
where Vg (w max ,H0 ) and Vg (w max ,HS ) are the tsunami wave group velocities at ocean
depth H0 and HS evaluated at the frequency associated with the peak tsunami height. If
you are interested in a simplified formula, equation (8) can be approximated using a long
wave assumption as: Vg (w max ,H0 ) = gH0 and Vg (w max ,HS ) = gHS so
S = [ H0 / HS ]
1/ 4
(9)
The shoaled tsunami amplitude AS(R) is defined as a function of the peak wave
amplitude Ad(R) at distance R as:
æ 2R
AS (R) = Ad (R)S = A0 PS = aDu 1+ ö
è
Dø
-y
1/ 4
é H0 ù
êë H úû
(10)
S
Here again, amplitude (10) is meant to be 1/2 peak to trough height.
Beaching Function B[AS(R)]
On shore peak wave runup height (R) is estimated using the beaching function B
h(R) = B[AS (R)] = AS4 / 5(R)H1/S 5 (11)
Many more complicated formulas for this runup correction exist that include parameters
like beach slope, wave period, etc. Again, the spirit of the formula based approach is to
keep things simple.
Combining equations (10) and (11), the peak wave runup can be calculated using:
h(R) = AS4/ 5 (R)H1/S 5 = Ad4 / 5(R)S 4/ 5 HS1/ 5 = Ad4 / 5(R)H 1/0 5 (12)
Equation (12) shows that the wave runup at a beach at distance R, does not depend on the
value of HS that you select as the shoaled depth to obtain AS(R). This makes sense, but
evaluating a AS(R) is sometimes useful.
Let's do an example:
Suppose that we have a pure dip slip earthquake (=900.) on a 45 degree dipping
fault plane  =450 that is L=50 km in length, W=20 km in down dip width with a slip of
u=4m under H0=2000 m of water.
Let's estimate the shoaled tsunami height in Hs=10 m of water at R=1000 km distance
and the estimated runup there.
From (3)
a = (1 - 450 / 1800 )sin45 0 sin 90 0 = 0.53
From (2) the initial tsunami pulse size is A0 = aDu = 0.53x4m = 2.12m
From (6) the initial tsunami pulse diameter is
D = (50km + 20km)/ 2 = 35km or D = 2 50km x20km/ p = 35.6km
From (5) the attenuation exponent is
35000 ö
æ
y = 0.5 + 0.575exp è -0.0175
= 0.923
2000 ø
From (4) the propagation loss is
æ 2 x1000km ö
P = è1+
35km ø
-0.923
= 0.024
From (7) peak deep water tsunami amplitude is
Ad (R) = 2.12m x 0.024 = 0.051m
From (9) the shoaling amplification would be
S = [2000m/10 m]
1/ 4
= 3.76
--- say 35 km..
From (10) the shoaled amplitude AS(R) is
AS (R) = 0.051m x 3.76 = 0.192m =19.2cm
Finally from (12) the on shore peak wave runup height (R)
h(R) = 0.051m4 / 52000m1/ 5 = 0.423m = 42.3cm
So for this example, I expect a wave of 19 cm amplitude (38 cm peak to trough) in 10 m
of water offshore at 1000 km distance and a run up height there of 42 cm.
Recall that, due to the assumptions above, this is likely a worst case estimate.
That is about it. You can write a little program to do this for different sources, distances,
etc.
Steve