Transcript Slide 1

Assimilation of HF Radar Data
into Coastal Wave Models
Lee Siddons and Lucy Wyatt
Department of Applied Mathematics
University of Sheffield, UK
NERC-funded PhD work also supervised by
Clive W Anderson (University of Sheffield)
Judith Wolf (Proudman Oceanographic Laboratory)
Overview
• OSCR HF radar at Holderness
• SWAN Wave Model
• Data Assimilation and Algorithms
• Results
• Future Work
OSCR measurements
Surface current
Wind direction
Wave height and
peak direction
Peak wave period
and direction
Energy spectrum
m2/Hz
OSCR
Mean direction
spectrum
Directional frequency
spectrum m2/Hz/rad
buoy
OSCR wind
direction
Wave Modelling
SWAN (Simulating WAves Nearshore)
• The Action Balance equation:
A
t
 Cx
A
x
 Cy
A
y
 Sin
• A is the action that is a function of frequency and
direction i.e.
f A(f  )  E (f  )
• C is the wave group velocity in relevant direction.
• S is the forcing to the system. Winds, non-linear
interactions etc.
See Holthuijsen L. H. et al (1999)
SWAN Application to Holderness
Boundary Data
1.2km
Action
1.2km
B
o
u
n
d
a
r
y
D
a
t
a
Data Assimilation
• The Analysis - Combination of model
output (Background state) and
observations in an optimal way.
• Taking into account model and
observational errors.
• Making assumptions about model,
observation and analysis errors.
Data Assimilation Formulation
Model
Observations
Initial State
x
t
k 1
 Mx   k 1
t
k 1
 Hx
y
t
k
t
k 1
  k 1
x  x  0
t
0
b
0
Since the state is a random variable, the estimate of the
state is found from its probability density function (pdf).
The State of the Ocean
The state of the ocean is often described in
terms of a few wave parameters, for example:
• Significant Wave Height Hs
Hs  4 *
 E f , df d
• Mean wave period T1
1
T1 

Mf
 E(f  )dfd
 f E(f  )dfd
These are the state variables used in our assimilations
Assimilation Algorithms
There are two main approaches of assimilation.
Sequential Assimilation
Only considers observations from the past and to the
time of the analysis. Some examples of sequential algorithms are:
• Optimal interpolation
• Kalman Filters
Variational Assimilation
Observations from the future can be used at the time of the analysis.
Some examples of variational algorithms are:
• Three-dimensional variational assimilation – 3DVAR
• Four-dimensional variational assimilation – 4DVAR
The Kalman Filter
The Kalman filter is derived by minimising the estimation error of the
analyzed state with respect to the Kalman gain matrix
xka  xkb  Kk ( y k  Hxkb )
ek  xk  xka
Pka  E [ek ekT ]
At time k, the best linear unbiased estimate of the true state, x k ,
from the observations, y k , and the model forecast, x kb , is given by
the analysed state x ka . It also provides information about the
uncertainty of the estimate.
Pkb  MPka1M T  Q
xka  xkb  PkbH T [HPkbH T  R ]1( y k  Hxkb )
Pka  Pkb  K k HPkb
Variational Assimilation
The aim of variational assimilation is to find the optimal estimate of the
state by minimisation of a cost function.
Three-dimensional variational assimilation – 3DVAR
The 3dvar cost function is as follows:
1
1
T 1
J ( x)  ( x  xb ) Pb ( x  xb )  ( y  H ( x)) T R 1 ( y  H ( x))
2
2
The solution that minimises the cost function is sought by iteratively
evaluating the cost function and its gradient using a suitable descent
algorithm.
Ensemble Kalman Filter (EnKF)
• EnKF introduced by Evensen - to avoid the
computational load associated with
P  MP M  Q
b
k
a
k 1
T
• Sequential method where the error statistics
are predicted using Monte Carlo or ensemble
integration.
• An ensemble of model states in integrated
forward in time and statistical information is
calculated from the ensemble.
Assimilation with Ideal Data
Before assimilating radar data, the
algorithms have been validated using
simulated data.
– SWAN is used to generate a ‘true’ state.
– Model errors are assumed to be uniform over
the grid and equal to background uncertainty
estimated from buoy data.
– Radar measurements are assumed to be
available at all sites and errors also uniform.
Results for Simulated Case - 3DVAR
Results for Simulated Case - ENKF
Performance Error Statistics
Scheme
MSE Hs
No Assimilation 5.26 * 10-3
MSE Tm
6.57 * 10-2
3DVAR
2.679 * 10-3
2.209 * 10-2
Ens_OI
2.709 * 10-3
2.267 * 10-2
ENKF-16
1.05 * 10-3
2.45 * 10-2
Assimilation with Real Data ENS-OI
Assimilation with Real Data EnKF
Assimilation of Band Parameters
• Assimilation of Hs and Tm in the following
frequency bands.
Band 1 = 0.03Hz
Band 2 = 0.1Hz
Band 3 = 0.2Hz
Band 4 = 0.3Hz
- 0.1Hz
- 0.2Hz
- 0.3Hz
- 0.4Hz
Performance Error Statistics for Test
Case Assimilation of Band Parameters
Scheme
B1 Hs
*10-3
B1 Tm B2 Hs
*10-3
*10-3
B2 Tm
*10-3
B3 Hs
*10-3
B3 Tm
*10-3
B4 Hs
*10-3
B4 Tm
*10-3
No
Assimilation
0.16
0.18
11.9
12.0
5.8
4.5
16.3
0.16
3DVAR
0.16
0.13
6.8
6.7
3.6
2.5
8.2
0.17
Ens-OI
0.16
0.14
6.7
6.8
3.7
2.6
8.7
0.16
ENKF-16
0.02
0.15
0.7
3.5
3.1
1.8
3.6
0.29
Assimilation with Real Data 3DVAR
Assimilation with Real Data EnKF
Future Work
• Re-estimate model and radar errors to use
in the assimilation schemes.
• Perform a probability sensitivity analysis
on the model to find model sensitivities.
• Extend range of assimilated parameters
e.g. by using partitioned directional
spectra
Energy spectrum
m2/Hz
OSCR
Mean direction
spectrum
Directional frequency
spectrum m2/Hz/rad
buoy
OSCR wind
direction