Recent Advances in PTHA Methodology

Randall J. LeVeque, University of Washington

F. I. González, University of Washington

Loyce M. Adams, University of Washington

Knut Waagan, University of Washington

Guang Lin, Pacific Northwest National Laboratory

NRC Workshop on

Probabilistic Flood Hazard Assessment

January 29–31, 2013

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Crescent City PTHA Pilot Study

Preliminary results will be shown from ongoing pilot study.

Supported by BakerAECOM, as part of a coastal modeling/mapping

effort funded by the FEMA Region IX office as part of the new

California Coastal Analysis and Mapping Project (CCAMP).

Simulations done with GeoClaw model (shallow water

equations) www.clawpack.org/geoclaw

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Crescent City, CA

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Four sample event realizations

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Problem formulation

Desired output: Map of target region showing, e.g.:

• Depth of flooding that occurs with given probability,

e.g. p = 0.01 for “100-year flood”.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Problem formulation

Desired output: Map of target region showing, e.g.:

• Depth of flooding that occurs with given probability,

e.g. p = 0.01 for “100-year flood”.

• Probability of flooding for given “exceedance value”.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Problem formulation

Desired output: Map of target region showing, e.g.:

• Depth of flooding that occurs with given probability,

e.g. p = 0.01 for “100-year flood”.

• Probability of flooding for given “exceedance value”.

Input: Potential tsunami sources, e.g. earthquakes

• Finite list of possible “events” and return times.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Problem formulation

Desired output: Map of target region showing, e.g.:

• Depth of flooding that occurs with given probability,

e.g. p = 0.01 for “100-year flood”.

• Probability of flooding for given “exceedance value”.

Input: Potential tsunami sources, e.g. earthquakes

• Finite list of possible “events” and return times.

Single “realization”,

or several, with conditional probabilities.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Problem formulation

Desired output: Map of target region showing, e.g.:

• Depth of flooding that occurs with given probability,

e.g. p = 0.01 for “100-year flood”.

• Probability of flooding for given “exceedance value”.

Input: Potential tsunami sources, e.g. earthquakes

• Finite list of possible “events” and return times.

Single “realization”,

or several, with conditional probabilities.

• Or: Probability distribution of possible realizations.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Problem formulation

Desired output: Map of target region showing, e.g.:

• Depth of flooding that occurs with given probability,

e.g. p = 0.01 for “100-year flood”.

• Probability of flooding for given “exceedance value”.

Input: Potential tsunami sources, e.g. earthquakes

• Finite list of possible “events” and return times.

Single “realization”,

or several, with conditional probabilities.

• Or: Probability distribution of possible realizations.

• Cumulative probability distribution of tide stage.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Some recent advances

• Improved methodology for tidal uncertainty.

• Approaches to mapping probabilities in addition to depth.

• Proposed new (mathematical) methodology if given a

probabilistic description of slip on fault planes.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Some recent advances

• Improved methodology for tidal uncertainty.

• Approaches to mapping probabilities in addition to depth.

• Proposed new (mathematical) methodology if given a

probabilistic description of slip on fault planes.

Some limitations:

• Proper probability density for slip distribution is hard to

determine — geophysics problem.

• How to add in the possibility of submarine landslides

affecting tsunami size?

• Many other uncertainties, e.g. friction coefficient and other

aspects of mathematical model / numerical method.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Hazard Curves

First define hazard curve: for each location (x, y):

P(ζ) = P(ζ; x, y) = Prob[inundation ≥ ζ in one year].

Example: If only one possible event E1 with recurrence time T1

(Poisson rate ν1 = 1/T1), that floods to level H1(x, y), then

P(ζ) =







1 if ζ = 0,

1 − e

−ν1

if 0 < ζ < H1,

0 if ζ > H1.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Hazard Curves

Example: If only one possible event E1 with recurrence time T1

(Poisson rate ν1 = 1/T1), that floods to level H1(x, y), then

P(ζ) =







1 if ζ = 0,

1 − e

−ν1

if 0 < ζ < H1,

0 if ζ > H1.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Hazard Curves

Example: Three possible events E1, E2, E3 with recurrence

times T1, T2, T3, that flood to levels H1, H2, H3.

Where, for example,

P(E3 or E1) = 1 − (1 − p3)(1 − p1) = 1 − e

−(ν3+ν1)

.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Probabilistic maps of flooding depth

Standard view: Map of flooding depth for fixed probability

Preliminary Results — Not for Use

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Probabilistic maps of flooding depth

Standard view: Map of flooding depth for fixed probability

Preliminary Results — Not for Use

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Probabilistic maps of flooding depth

Alternative map: Probability of exceeding fixed depth

Preliminary Results — Not for Use

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Probabilistic maps of flooding depth

Alternative map: Probability of exceeding fixed depth

Preliminary Results — Not for Use

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Probabilistic maps of flooding depth

Alternative map: Probability of exceeding fixed depth

Preliminary Results — Not for Use

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Probabilistic maps of flooding depth

Alternative map: Probability of exceeding fixed depth

Preliminary Results — Not for Use

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Effect of tides on inundation at Crescent City

Bathymetry uses vertical datum MHW (Mean High Water).

Tidal range:

MHHW ≈ MSL + 97 cm.

MHW ≈ MSL + 77 cm.

MLW ≈ MSL − 75 cm.

MLLW ≈ MSL − 113 cm.

How much difference does modifying tide stage have on

inundation?

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Sample inundation from CSZ M9 earthquake

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Sample inundation from CSZ M9 earthquake

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Tidal uncertainty

Suppose we determine for a given event that largest waves are

all seen within a period of ∆t hours after arrival of first wave.

Then we can use tide record to determine the cumulative

probability that the tide stage will be above s at some time

between t and t + ∆t. (Where t is assumed to be a random

time in tide cycle when first wave hits.)

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Tidal uncertainty

For example, if ∆t = 2 hours is appropriate for this event:

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
The pattern method

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Effect of tidal uncertainty on hazard curves

Example: Three events with no tidal uncertainty:

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Effect of tidal uncertainty on hazard curves

Example: Three events with tidal uncertainty:

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Hazard curve with many events + tides

Preliminary Results — Not for Use

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Cascadia Subduction Zone (CSZ)

• 1200 km long off-shore fault stretching from northern California to

southern Canada.

• Last major rupture: magnitude 9.0 earthquake on January 26, 1700.

• Tsunami recorded in Japan with run-up of up to 5 meters.

• Historically there appear to be magnitude 8 or larger quakes every 500

years on average.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Sample seafloor deformations

R. C. Witter and Y. Zhang and K. Wang and G. R. Priest and C. Goldfinger and L. L. Stimely

and J. T. English and P. A. Ferro, Simulating tsunami inundation at Bandon, Coos County,

Oregon, using hypothetical Cascadia and Alaska earthquake scenarios, Oregon Department of

Geology and Mineral Industries Special Paper 43, 2011. (15 realizations)

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
CSZ fault geometry and possible slip regions

3 Possible slip regions (Art Frankel, USGS) and fault geometry of

Pollitz, McCrory, Wilson, Svarc, Puskas, Smith,

http://doi.wiley.com/10.1111/j.1365-246X.2010.04546.x

and Blair, McCrory, Oppenheimer, http://pubs.usgs.gov/ds/633/

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Generating stochastic source realizations

For a given event (e.g. CSZ Mw 9.1) we need to use more than

one possible slip distribution.

To simplify, consider an event with a single fault plane for which

we know strike, dip, rake, and the integral of slip (from Mw).

Assuming uniform slip gives one possible realization but not

realistic.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Generating stochastic source realizations

For a given event (e.g. CSZ Mw 9.1) we need to use more than

one possible slip distribution.

To simplify, consider an event with a single fault plane for which

we know strike, dip, rake, and the integral of slip (from Mw).

Assuming uniform slip gives one possible realization but not

realistic.

Can split plane into m pieces and distribute slip.

Let di = slip on ith subfault, for i = 1, 2, . . . , m.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Generating stochastic source realizations

For a given event (e.g. CSZ Mw 9.1) we need to use more than

one possible slip distribution.

To simplify, consider an event with a single fault plane for which

we know strike, dip, rake, and the integral of slip (from Mw).

Assuming uniform slip gives one possible realization but not

realistic.

Can split plane into m pieces and distribute slip.

Let di = slip on ith subfault, for i = 1, 2, . . . , m.

How to generate many realistic slips di?

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Karhunen-Loève Modes of CSZ

Eigenfunctions of Covariance assuming correlation lengths of

40% along-strike and down-dip.

P. M. Mai and G. C. Beroza, A spatial random field model to characterize complexity in

earthquake slip, J. Geophys. Res. 107(2002).

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Karhunen-Loève expansion

Let (λk, vk) be eigenvalue/vectors of C, for k = 1, 2, . . . , m.

Order so that λ1 ≥ λ2 ≥ · · · ≥ λm.

Eigenvectors for larger k are more oscillatory.

Similar to Fourier decomposition but for arbitrary geometry.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Karhunen-Loève expansion

Let (λk, vk) be eigenvalue/vectors of C, for k = 1, 2, . . . , m.

Order so that λ1 ≥ λ2 ≥ · · · ≥ λm.

Eigenvectors for larger k are more oscillatory.

Similar to Fourier decomposition but for arbitrary geometry.

Then realizations d = [d1, . . . , dm] can be generated by

d = μ +

Xm

k=1

zk

p

λk vk

where the zk are chosen to be independent from N(0, 1)

(normally distributed with mean 0 and variance 1).

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Karhunen-Loève expansion

d = μ +

Xm

k=1

zk

p

λk vk

Advantages:

• Easy to generate independent N(0,1) random numbers.

• If correlation length is large then eigenvalues decay rapidly.

So can trucate sum to small number of terms.

Reduction in the dimension of the stochastic space to explore!

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Karhunen-Loève expansion

d = μ +

Xm

k=1

zk

p

λk vk

Advantages:

• Easy to generate independent N(0,1) random numbers.

• If correlation length is large then eigenvalues decay rapidly.

So can trucate sum to small number of terms.

Reduction in the dimension of the stochastic space to explore!

Moreover, Once we apply Okada model to d, oscillatory

components get further smoothed.

Even lower dimensional space of seafloor deformations?

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Uncertainty quantification techniques

Why is dimension reduction important?

May be able to do better than brute-force Monte-Carlo by doing

smaller number of simulations at carefully chosen points in

stochastic space, represented by vectors

[z1, z2, . . . , zr] (r = number of stochastic dimensions)

and building up polynomial approximation to response surface

(stochastic collocation).

Then use known probability distribution of the zk to compute

probability of response.

Hard to do in large number of dimensions, but can explore

techniques such as sparse grids, importance sampling, MCMC,

multi-level MC.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Dealing with Uncertainties

Aleatoric uncertainties: (Mathematical problem)

Given the correct probability density of slip distribution, there

are techniques to reduce dimension and sample efficiently.

Epistemic uncertainties: (Geophysical problem)

• What is correct distribution/recurrence of different

magnitude quakes?

• What is correct fault geometry?

• Is there tapering of slip?

• Is K-L expansion with Gaussian weights correct?

• What is correct correlation function?

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Some recent advances

• Improved methodology for tidal uncertainty.

• Approaches to mapping probabilities in addition to depth.

• Proposed new (mathematical) methodology if given a

probabilistic description of slip on fault planes.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013
Some recent advances

• Improved methodology for tidal uncertainty.

• Approaches to mapping probabilities in addition to depth.

• Proposed new (mathematical) methodology if given a

probabilistic description of slip on fault planes.

Some limitations:

• Proper probability density for slip distribution is hard to

determine — geophysics problem.

• How to add in the possibility of submarine landslides

affecting tsunami size?

• Many other uncertainties, e.g. friction coefficient and other

aspects of mathematical model / numerical method.

R. J. LeVeque, University of Washington NRC PFHA Workshop, January, 2013