Transcript Slide 1

NC State University
Statistical Data Assimilation for Coastal Ocean Prediction
Kristen
1Department
1
Foley ,
2
Xie ,
Montserrat
Lian
Shaowu
North Carolina State University
2
Bao
of Statistics, 2 Department of Marine, Earth and Atmospheric Sciences
Motivation
The Carolinas has had a tremendous residential and commercial investment in
coastal areas during the past 10 years. However rapid development of the coastal
region produces a stressed ecosystem and an exponential increase in human and
property exposure to storm hazards. Our goal is improve the prediction of the
coastal ocean response to tropical storms and hurricanes by using data assimilation
techniques to combine available observations (from buoys, ships, satellite, etc.)
with the forecasts from the Princeton Ocean Model (POM). Our first goal is to try
and improve the prediction of storm surge which is the rise of water caused by
tropical cyclone wind and pressure.
1
Fuentes ,
Summary
 Storm surge caused by the high speed winds of a hurricane are predicted using a
numerical ocean model and observational data from water level stations.
 Using Ensemble Kalman Filtering methods we study the impact of dividing the
model domain into independent blocks to assist in computation.
 Future research includes improving the wind fields used to initialize the model. We
also plan to compare these results to the Extended KF as well as more advanced
Bayesian Filtering methods.
Notation
Xt(ti)=Xit  true (unknown) change of water elevation at time ti.
Yio  new vector of observations from water level stations.
p=5= number of observation sites within the model domain.
Xia  analysis state vector, best estimate of the unknown state at time ti
based on data assimilation methods.
Xif  output from the Princeton Ocean Model.
n=68929=number of grid point locations used by the numerical
model. Grid size = 1'longitude by 1'latitude.
Input Fields
to initialize model
Forcing
Parameters
(water level stations,
buoys, satellite, etc)
Ensemble of 10 different input fields for POM for the initial surface water
elevation for Sept. 15th hour 03. Model is initialized using both blocking schemes.
Below we see the initial fields when the domain is split into 5 independent blocks.
Figure 4: Ensemble of initial surface water elevation (in meters).
Data Assimilation
Observations
Model Inputs
Numerical Ocean
Model
(Princeton Ocean Model)
FILTERING METHODS
Final Forecast
[Hindcast /Nowcast]
(storm surge)
Figure 1: General Data Assimilation Framework
Blocking of Model Domain
The model domain is split into blocks based on the location of the 5 water level
stations. Blocks along the coast (shown in gray) are treated as independent. The
remaining grid points away for the coast (shown in white) are grouped together
as one block and are not used in the data assimilation process. We wish to
study the impact of the blocks by comparing the final analysis and forecasting
results based on these two blocking schemes.
Figure 2: Two blocking schemes used to generate the
spatial covariance of the initial water elevation fields.
Observations
Observed hourly water levels at each of the 5 water level stations are adjusted
for location and tides. These adjusted water levels can be treated as the change
in water elevation from an initial value of 0 meters and in this way can be
compared to the model output values.
Princeton Ocean Model
The Princeton Ocean Model (POM) is used to simulate the change in water level along the coast of North and
South Carolina and Georgia under strong hurricane forcing conditions. In particular a series of wind fields are
used to “spin up” the model.
Forcing Parameters
A symmetric wind model is used to initiate POM based on track information available from NOAA for the
center location and central pressure of Floyd on September 15th and 16th. The parameters of the wind model are
estimated using nonlinear least squares. We know
that hurricane wind fields are typically asymmetric
with stronger wind speeds on the right hand side of
the storm. As a next step we wish to combine a new
asymmetric wind model with a nonstationary multivariate process model for the E-W and N-S
components of the wind vectors. Improving the
quality of these initial wind fields is expected to have
a significant impact on the prediction of storm surge
along the coast.
Hurricane Floyd
The current case study is Hurricane Floyd which
was classified as a hurricane on Sept. 10, 1999
and moved up the Eastern US coast making landfall near Cape Fear NC early September 16th.
Floyd caused the worst short term flooding on
record for SE North Carolina and is the deadliest
US tropical cyclone since Agnes of 1972.
Figure 3: Estimated wind field based on track of
Hurricane Floyd, Sept. 15 hr 03 to Sept.16 hr 03.
Wind speed contour in m/s.
Ensemble Kalman Filter
The Ensemble Kalman Filter uses a Monte Carlo approach to generate samples of the state vector and carry
out an ensemble of data assimilation cycles. This ensemble is then used to estimate the forecast error
covariance.
Forecast Step:
 Sample Xj(ti-1) ~ N(Xa(ti-1), Pa(ti-1)), j=1,..., K ( K=10)
where the initial Xo(t0)=0 and Po(t0) is an exponential spatial covariance with
nugget = .001, partial sill = .036 and range = 107.1 km.
These values were estimated using Maximum Likelihood estimation based on model output using the
symmetric wind field to initialize the model compared to output based on a model run initialized by wind
field inputs derived from satellite and buoy observations provided by NOAA Hurricane Research Division.
 Compute Xjf(ti)=Mj[Xj(ti-1)], j=1,...,K where M[ ] represents the forward integration of the current water
level by the Princeton Ocean Model.
 Calculate the sample mean and covariance: Xˆ f(ti), Pˆf(ti)
Updated Water Elevation Fields
The ensemble of water level fields are updated using the Ensemble Kalman Filter
to combine model output and observed values. The EnsKF calculations were
based on a block diagonal structure for the forecast error covariance matrix as
defined by the blocking shown in Figure 2. (Plots below based on 5 blocks.)
Figure 5: Updated ensemble of surface elevation fields for Sept. 15th hr 18.
Forecasting
The analysis water elevation fields are updated in the numerical model which is
then used to integrate the analysis field forward in time, 1 hour, 3 hours and 6
hours ahead of the updating time. Below we see the effect of the EnsKF step on
the model output compared to the observed values for three of the water level
stations. The top row shows the results when 3 blocks were used. The next row
of graphs shows the results when 5 blocks are used. In general the output based
on both blocking schemes is similar but the analysis using fewer blocks appears
to better adjust the water level value based on what is observed at each location.
* observed value used
in update step
____observed value
- - - model value
____ensemble mean
….. ensemble member
Analysis Step
 Perturb the observations at time ti by adding random noise (in this case Gaussian):
Yj*(ti)=Yo(ti)+ηj, ηj~N(0,R), j=1,…,K
where R is a diagonal matrix. The observation error variance is based on the variance of one
second samples reported by each water level station for each hour.
 Update each member of the forecast sample with the Kalman Filter update to obtain Xaj(ti), j=1,...,K.
 This now serves as a sample for time ti and the process repeats. The ensemble sample mean and
covariance can be used as estimates of the updated state vector and error covariance: Xa(ti), Pa(ti)
Figure 6: Comparison of model output to observed value
(top row = 3 blocks; bottom row=5 blocks).
Acknowledgments: This research is supported in part by the NSF VIGRE program and by the
Carolina Coastal Ocean Observing and Prediction System (Caro-COOPS) program funded by
NOAA. Many thanks to Leonard Pietrafesa and Jerry Davis for their continued support.