State space model of precipitation rate Tamre Cardoso, PhD UW 2004

1

• 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 (OHagan and co-workers )

2
Rainfall measurement

• Rain gauge (1 hr)
• High wind, low rain rate (evaporation)
• Spatially localized, temporally moderate
• Radar reflectivity (6 min)
• Attenuation, not ground measure
• Spatially integrated, temporally fine
• Cloud top temp. (satellite, ca 12 hrs)
• Not directly related to precipitation
• Spatially integrated, temporally sparse
• Distrometer (drop sizes, 1 min)
• Expensive measurement
• Spatially localized, temporally fine

3
Radar image
4
(No Transcript)
5
Drop size distribution
6
Basic relations

• Rainfall rate
• v(D) terminal velocity for drop size D
• N(t) number of drops at time t
• f(D) pdf for drop size distribution
• Gauge data
• g(w) gauge type correction factor
• w(t) meteorological variables such as wind speed

7
Basic relations, cont.

• Radar reflectivity
• Observed radar reflectivity

8
Structure of model

• Data GN(D),qG ZN(D),qZ
• Processes NmN,qN Dxt,qD
• log GARCH LN
• Temporal dynamics mN(t)qm
• AR(1)
• Model parameters qG,qZ,qN,qm,qDqH
• Hyperparameters qH

9
MCMC approach
10
Observed and predicted rain rate
11
Observed and calculated radar reflectivity
12
Wave height prediction
13
Misalignment in time and space
14
The Kalman filter

• Gauss (1795) least squares
• Kolmogorov (1941)-Wiener (1942)
• dynamic prediction
• Follin (1955) Swerling (1958)
• Kalman (1960)
• recursive formulation
• prediction depends on
• how far current state is
• from average
• Extensions

15
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

16
EOFs of wind forecasts
17
Kalman filter forecast emulates forecast model
18
The effect of satellite data
19
Model assessment

• Difference from current forecast of
• Previous forecast
• Kalman filter
• Satellite data assimilated

20
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

21
Basic framework

• An emulator is a random (Gaussian) process ?(x)
  approximating the process model for input x in
  Rm.
• Prior mean m(x) h(x)T?
• Prior covariance
• Run the model at n input values to get n output
  values, so

22
The emulator

• Integrating out ? and ?2 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

23
An example

• y7xcos(2x)
• q1, hT(x)(1 x) n5

24
Conclusions

• Model assessment constraints
• amount of data
• data quality
• ease of producing model runs
• degree of misalignment
• Ideally 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)