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Rainfall Measurement

Rainfall Measurement. 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)

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Rainfall Measurement

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  1. Rainfall Measurement

  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. Drop size distribution

  5. 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

  6. Basic relations, cont. Radar reflectivity: Observed radar reflectivity:

  7. Structure of model Data: [G|N(D),qG] [Z|N(D),qZ] Processes: [N|mN,qN] [D|xt,qD] log GARCH LN Temporal dynamics: [mN(t)|qm] AR(1) Model parameters: [qG,qZ,qN,qm,qD|qH] Hyperparameters: qH

  8. MCMC approach

  9. Observed and predicted rain rate

  10. Observed and calculated radar reflectivity

  11. Wave height prediction

  12. Misalignment in time and space

  13. 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

  14. 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:

  15. EOFs of wind forecasts

  16. Kalman filter forecast emulates forecast model

  17. The effect of satellite data

  18. Model assessment Difference from current forecast of Previous forecast Kalman filter Satellite data assimilated

  19. 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

  20. 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

  21. 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

  22. An example y=7+x+cos(2x) q=1, hT(x)=(1 x) n=5

  23. 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)

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