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

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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)