Real-time Modelling of Tsunami Data. Applied Physics Laboratory Department of Statistics University of Washington Seattle, Washington, USA

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Stanley Byrd
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1 Real-time Modelling of Tsunami Data Applied Physics Laboratory Department of Statistics University of Washington Seattle, Washington, USA collaborative effort with NOAA Center for Tsunami Research

2 Background - I even before disasterous Sumatra tsunami in December 2004, destructive potential of earthquake-generated tsunamis was wellknown due to rate at which a tsunami advances across the ocean, possible to lessen its effect on some coastal communities through advance warnings in USA, warning centers in Alaska and Hawaii are responsible for issuing timely bulletins about impending tsunamis seismometers give first indication of a potential tsunami event starting time of earthquake event location of epicenter initial estimate of magnitude 1

3 Background - II seismic information alone is not enough to accurately forecast potential impact of tsnumai led to development of Deep-ocean Assessment and Reporting of Tsunamis (DART r ) buoys 2

6 Background - III prior to December 2004, there were six DART r buoys deployed in Pacific since then, many more have been added mindful of Percival s First Law Real-time computer demonstrations are doomed to fail! let s look at network of buoys and recent data by going to and movie of Nov Kuril Island event downloadable from 3

7 Place pointer on station to display corresponding plot or click on station to view station page.

8 Place pointer on station to display corresponding plot or click on station to view station page.

9 Format of Data buoy records one pressure measurement every sec (represents an average over sec) -min stream: when nothing is going on, buoy transmits one -sec measurement every min -sec stream: when triggered by a seismic event, transmits all recorded measurements 1-min stream: during most of tsunami event, transmits averages of four consecutive -sec measurements 4

10 Short-term Inundation Forecast for Tsunamis (SIFT) SIFT is a computer application that uses DART r data and precomputed geophysically-based predictions to assess magnitude of tsunami while tsunami event is evolving SIFT currently used at warning centers in Alaska and Hawaii, but is still under development at NOAA Center for Tsunami Research statistical issues involved in processing data within SIFT signal extraction: need to remove tidal component before precomputed predictions can be used assessment of uncertainty: need error bars on estimates of magnitude of tsunami variable selection: help operators select predictors 5

11 wave height Buoy Data for Nov Kuril Islands Event buoy days (from start of event) 6

12 Approaches for Signal Extraction (Detiding) three approaches, each with certain strengths and weaknesses 1. empirical orthogonal functions (Tolkova, 2009) 2. localized polynomial fits 3. two-stage approach, the second of which involves Kalman filter/smoother will concentrate on third approach 7

13 Model for Data with Tides assume measured data at time t can be expressed as LX x(t) = µ(t) + c l (t) + (t) where l=1 µ(t) + P l c l(t) represents tides; µ(t) is unpredictable tidal component (has slowly varying mean level); c l (t) = C l cos(2πf l t + φ l ) is lth part of predictable tidal component (sinusoid with amplitude A l, frequency f l and phase φ l ); and (t) is detided data 8

14 Model for -Min Stream Prior to Tsunami: I let t = 0 denote starting time (in days) of tsunami events can write C l cos(2πf l t + φ l ) = A l cos(2πf l t) + B l sin(2πf l t), so assume data x(t) at times t < 0 can be expressed as LX x(t) = µ + A l cos(2πf l t) + B l sin(2πf l t) + t where l=1 µ is an unknown overall mean level; f l is a known tidal frequency; A l and B l are unknown amplitudes; and t is a residual term use linear least squares to estimate µ, A l and B l denote these estimates as ˆµ, Â l and ˆB l 9

15 Model for -Min Stream Prior to Tsunami: II if using 14 days of prior data, let L = 4, and set f 1, f 2, f 3 and f 4 to so-called M2, S2, O1 and K1 frequencies (other rules apply if use, e.g., 1 day or 29 days of prior data) estimate tidal component at time t using LX ˆx(t) = ˆµ + Â l cos(2πf l t) + ˆB l sin(2πf l t) l=1 for t < 0, can compare estimated tide to actual data for t > 0, can detide data from 1 min and/or sec streams by subtracting off ˆx(t) 10

16 wave height Day Model (M2, S2, O1 and K1) buoy days (from start of event) 11

17 wave height Day Model and Extrapolation buoy days (from start of event) 12

18 detided wave height Data Detided using 14 Day Model buoy days (from start of event) 13

19 Second-Stage Detiding via Kalman Filter/Smoother after first-stage detiding, can write x(t) ˆx(t) = µ(t) + (t), where µ(t) is unpredictable tidal component extract µ(t) by formulating a state-space model, which allows use of Kalman filter/smoother: µ(t) = µ(t 1) + v(t) v(t) = v(t 1) + ζ(t) where ζ(t) is Gaussian white noise with mean zero and variance σζ 2 (can be set using historical data) 14

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210 Modelling of Detided DART r Buoy Data: I geophysically-based propagation predictions are based on a simplification provided by so-called unit sources unit sources are rectangles of area km 2 covering portions of globe from which tsunami-generating earthquakes can occur 16

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212 Modelling of Detided DART r Buoy Data: II propagation predictions must be precomputed (not enough time to compute them once a tsunami is in progress) database has been established with precomputed predictions for each pairing of particular unit source and particular buoy predictions assume earthquake is located in center of unit source and is of magnitude 7.5 entry in database predicts what will observed over time at a particular buoy given earthquake from a particular unit source 17

213 Model for Buoy and Unit Source a12 modeled data (meters) hours since event 18

214 Modelling of Detided DART r Buoy Data: III will use buoy data to adjust predictions to handle earthquakes greater or less than 7.5 in magnitude consider case of 2 buoys (1, 2) and three unit sources (a, b, c) let x 1 and x 2 be relevant detided data from buoys 1 and 2 let g 1,a etc. be prediction of what buoy 1 should see from earthquake at unit source a leads to following model for buoy data: α x1 g1,a g x = 1,b g 1 1,c α x 2 g 2,a g 2,b g ,c α 2 3 Gα + error terms j assumed to obey first-order autoregressive model 19

215 Modelling of Detided DART r Buoy Data: IV solve for α using constrained least squares: minimize k k 2 = kx Gαk 2 subject to α 0 constraint needed to get physically reasonable solution ˆα note: if kth element of ˆα is set to zero, kth unit source effectively eliminated from model sum of elements of ˆα can be used to estimate tsunami magnitude T M uncertainty in T M assessable using covariance matrix for ˆα following example involves 11 buoys and 3 units sources 20

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217 Concluding Comments SIFT application currently relies on experienced operators to select appropriate unit sources desirable for future versions of SIFT to include statisticallyoriented selection of unit sources (related to variable selection in linear regression) selection of data from buoy also currently done manually by operators need for statistical guidance here also 21

218 Thanks to... Ross Darnell for invitation to speak colleagues at NOAA Center for Tsunami Research in Seattle Don Denbo Marie Eblé Edison Gica Hal Mofjeld Mick Spillane Elena Tolkova Vasily Titov Crime Scene for 2nd place finish (got $13 in $2 sweep!!!) 22

Extraction of tsunami source parameters via inversion of DART r buoy data

Natural Hazards manuscript No. (will be inserted by the editor) Extraction of tsunami source parameters via inversion of DART r buoy data Donald B. Percival Donald W. Denbo Marie C. Eblé Edison Gica Harold