State space model of precipitation rate (Tamre Cardoso, PhD UW 2004) Updating wave height forecasts...
-
Upload
francine-bryant -
Category
Documents
-
view
213 -
download
0
Transcript of State space model of precipitation rate (Tamre Cardoso, PhD UW 2004) Updating wave height forecasts...
State space model of precipitation rate (Tamre Cardoso, PhD UW 2004)
Updating wave height forecasts using satellite data (Anders Malmberg, PhD U. Lund 2005)
Model emulators (O’Hagan and co-workers )
Rainfall measurementRain gauge (1 hr)
High wind, low rain rate (evaporation)Spatially localized, temporally moderate
Radar reflectivity (6 min)Attenuation, not ground measureSpatially integrated, temporally fine
Cloud top temp. (satellite, ca 12 hrs)Not directly related to precipitationSpatially integrated, temporally sparse
Distrometer (drop sizes, 1 min)Expensive measurementSpatially localized, temporally fine
Radar image
Drop size distribution
QuickTime™ and aPhoto - JPEG decompressor
are needed to see this picture.
Basic relations
Rainfall rate:
v(D) terminal velocity for drop size DN(t) number of drops at time tf(D) pdf for drop size distributionGauge data:
g(w) gauge type correction factorw(t) meteorological variables such as wind speed
€
R(t) = cRπ
6D3v(D)N(t)f(D)
0
∞
∫ dD
€
G(t)~ N g(w(t)) R(s)ds,σG2
t − Δ
t
∫ ⎛
⎝ ⎜
⎞
⎠ ⎟
Basic relations, cont.
Radar reflectivity:
Observed radar reflectivity:
€
ZD(t) = cZ D6v(D)N(t)f(D)dD0
∞
∫ ⎛
⎝ ⎜
⎞
⎠ ⎟
€
Z(t) ~ N(ZD (t),σZ2 )
Structure of model
Data: [G|N(D),G] [Z|N(D),Z]
Processes: [N|N,N] [D|t,D]
log GARCH LN
Temporal dynamics: [N(t)|]
AR(1)
Model parameters: [G,Z,N,,D|H]
Hyperparameters: H
MCMC approach
Observed and predicted rain rate
Observed and calculated radar reflectivity
Wave height prediction
Misalignment in time and space
The Kalman filter
Gauss (1795) least squaresKolmogorov (1941)-Wiener (1942)
dynamic predictionFollin (1955) Swerling (1958)Kalman (1960)
recursive formulationprediction depends onhow far current state isfrom average
Extensions
A state-space model
Write the forecast anomalies as a weighted average
of EOFs (computed from the empirical covariance) plus small-scale noise.
The average develops as a vector autoregressive model:
Y(s, t + τ) = ws (u)Y(u, t)du+∫ η(s, t+ τ)
Y(s, t) = ai (t)φi (s)∑
EOFs of wind forecasts
Kalman filter forecast emulates forecast model
The effect of satellite data
Model assessment
Difference from current forecast of
Previous forecast
Kalman filter
Satellite data assimilated
Statistical analysis of computer code output
Often the process model is expensive to run (in time, at least), especially if different runs needed for MCMC
Need to develop real-time approximation to process model
Kalman filter is a dynamic linear model approximation
SACCO is an alternative Bayesian approach
Basic framework
An emulator is a random (Gaussian) process η(x) approximating the process model for input x in Rm.
Prior mean m(x) = h(x)TPrior covariance
Run the model at n input values to get n output values, so
v(x1,x2 ) =σ2c(x1,x2 )
(d ,σ2 ) : N(H,σ2C)
(η g( ) ,σ2 ,d) : N(m∗, Σ∗)
The emulator
Integrating out and σ2 we get
where q = dim() and
where t(x)T = (c(x,x1),…,c(x,xn))
m** is the emulator, and we can also calculate its variance
η(x) − m∗∗(x)
σc∗∗(x,x)12
: tn−q
m∗∗(x) =h(x)T + t(x)TC−1(d−H)
An exampley=7+x+cos(2x)
q=1, hT(x)=(1 x) n=5
Conclusions
Model assessment constraints:• amount of data• data quality• ease of producing model runs• degree of misalignmentIdeally the model should have• similar first and second order properties to the data• similar peaks and troughs to data (or simulations based on the data)