Simulation of DART Buoy Data
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- Rafe Benson
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1 Simulation of DART Buoy Data goal: develop procedure for simulating detided DART buoy data that can be used to test inversion algorithm, train users etc. simulated data has three stochastic components: background noise seismic noise distorted tsunami signal (as explained later, these will be added to a deterministic component) 1
2 Background Noise: I background noise is what is observed at DART buoy when nothing is going on properties can be deduced from data prior to arrival of tsunami wave (in absence of seismic noise) buoy detided data (meters) in the above, background noise observed up to time 2.0 2
3 Background Noise: II let X t be a stochastic process that models the background noise for simplicity, will assume that X t obeys a first order autoregressive (AR(1)) process with zero mean: X t = φ B X t 1 + ɛ (B) t, t =0, 1,...,N 1, where ɛ (B) t is a Gaussian white noise process with zero mean and variance σb 2 (can estimate φ B and σb 2 from data) will assume that φ B and σb 2 are the same for each buoy, but noise processes at different buoys are uncorrelated can show that variance of X t is σb 2 /(1 φ2 B ) can initialize simulations by letting X 0 = ɛ (B) 0 σ B / (1 φ 2 B ) 3
4 Background Noise: III use first 33 data points (33 minutes) from buoy to get estimates ˆφ = 0.3 and σb 2 = simulated background noise for buoy detided data (meters) 4
5 Background Noise: III use first 33 data points (33 minutes) from buoy to get estimates ˆφ = 0.3 and σb 2 = simulated background noise for buoy detided data (meters) 4
6 Background Noise: III use first 33 data points (33 minutes) from buoy to get estimates ˆφ = 0.3 and σb 2 = simulated background noise for buoy detided data (meters) 4
7 Background Noise: III use first 33 data points (33 minutes) from buoy to get estimates ˆφ = 0.3 and σb 2 = simulated background noise for buoy detided data (meters) 4
8 Background Noise: III use first 33 data points (33 minutes) from buoy to get estimates ˆφ = 0.3 and σb 2 = simulated background noise for buoy detided data (meters) 4
9 Seismic Noise example of seismic noise (after detiding) detided seismic data (sampled every 15 seconds) pressure (millimeters) hours model as AR process Y t with amplitude modulation? used buoy geometry to adjust for different buoys? 5
10 Distorted Tsunami Signal: I motivation: residuals from fitted models exhibit low frequency patterns due to errors in unit-slip models etc consider example from 26 July seminar (fitted model is blue dashed line, data is black solid line) 6
11 Least Squares Estimate Â2 of Slip for a12 & meters A = residuals (meters)
12 Distorted Tsunami Signal: II let Z t be a stochastic process that models the distortions assume Z t is an AR(1) process with time varying parameters: Z t = φ D,t Z t 1 + ɛ (D) t, t =0, 1,...,N 1, where ɛ (D) t is a Gaussian white noise process with zero mean and variance σd,t 2 will assume that φ D,t and σd,t 2 depend upon a small number of parameters parameters are same for all buoys (reasonable?) distortions at different buoys are uncorrelated (?) 8
13 Distorted Tsunami Signal: III simulated distortion based upon analysis of residuals from fit to buoy simulated tsunami distortion for buoy distortion (meters) 9
14 Distorted Tsunami Signal: III simulated distortion based upon analysis of residuals from fit to buoy simulated tsunami distortion for buoy distortion (meters) 9
15 Distorted Tsunami Signal: III simulated distortion based upon analysis of residuals from fit to buoy simulated tsunami distortion for buoy distortion (meters) 9
16 Distorted Tsunami Signal: III simulated distortion based upon analysis of residuals from fit to buoy simulated tsunami distortion for buoy distortion (meters) 9
17 Distorted Tsunami Signal: III simulated distortion based upon analysis of residuals from fit to buoy simulated tsunami distortion for buoy distortion (meters) 9
18 Putting It All Together simulated DART buoy data given by K A k g k (t a k )+X t + Y t + Z t where k=1 g k (t) isthekth unit source model for a given DART buoy; A k is the slip for the kth unit source a k is either zero or a uniformily distributed random variable over an interval of ±3 minutes (same for all buoys?) X t is the background noise Y t is the seismic noise Z t is the distorted tsunami signal 10
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