Representing radar rainfall uncertainty with ensembles based on a time-variant geostatistical error modelling approach Bristol, 4 th March 2016 Francesca Cecinati
Background M.Eng. Environmental and Water Quality Eng. MIT M.Sc. Environmental and Energy Engineering Universitá di Genova Young Graduate Trainee European Space Agency PhD University of Bristol Started November 2014 (1 year and 4 months) Marie Curie ESR Project: QUICS (Quantifying Uncertainty in Integrated Catchment Studies) Seconded to TU Delft 4 months
Research topic: rainfall uncertainty Radar Uncertainty Radar Uncertainty Propagation 1) Radar Rainfall Ensembles Radar + Rain Gauge merged products Rain Gauge uncertainty Merged product uncertainty Merged Product uncertainty propagation
Radar Errors Some of the error sources are: Attenuation (a) Shielding and partial beam blocking (b/c) Ground clutter (d) Anomalous propagation (g) Different Z-R relationships for different types of precipitation (convective/ stratiform/drizzle/snow/hail (h) Beam overshooting (e) Bright band and vertical reflectivity profiles (f) Evaporation (i) Orographic lifting (l) d f g h e i a l c b Many of these errors can be partially corrected, but a residual uncertainty remains Villarini, G. & Krajewski, W.F., 2010. Review of the different sources of uncertainty in single polarization radar-based estimates of rainfall. Surveys in Geophysics, 31, pp.107 129.
1) Radar Rainfall Ensembles Estimate the overall radar rainfall error Statistically characterise the errors Synthetise N alternative error fields Estimate the overall radar rainfall uncertainty and represent it through N alternative possible rainfall fields Recombine with the radar to generate N ensemble members
1) Radar Error Model Radar error model: 10 log P = 10 log R + δ Radar error characterization: μ(δ) = E{δ} σ δ = E δ μ 2 Radar error observation: δ = 10 log P G 10 log(r) Spatial correlation: Covariance Matrix Variogram
Covariance Matrix approach The spatial correlation of error is characterised by the covariance matrix between the measured error in different positions ( i, j) C i,j = Cov ε i, ε j 1. Heavy computational and storage load for many rain gauges 2. Statistics done on time series temporally stationary 3. The covariance matrix needs to be decomposed unstable 4. The covariance matrix can condition only the measurement points, requiring interpolation Germann, U. et al., 2009. REAL Ensemble radar precipitation estimation for hydrology in mountainous region. Q. J. R. Meteorol. Soc, 135(February), pp.445 456.
Time-variant variogram definition Theoretical variogram: γ ε, d = 1 E δ(x) δ(x + d) 2 2 Experimental fitted variogram (exponential model): γ d = c 0 + (s c 0 ) 1 exp 3d r Time-variant calculation of variograms: The variogram is calculated at each time step with the minimum possible of a 3, 6, 9, or 12 hour window. If none of them is sufficient to define a variogram, all the data is used for average condition variogram Consider temporary phenomena like convective/stratiform rainfall, bright band, hail, snow, attenuation, etc.
1) Radar Rainfall Ensembles Estimate the overall radar rainfall error Statistically characterise the errors Synthetise N alternative error fields Recombine with the radar to generate N ensemble members
FFT-Moving Average 1. The covariance function can be derived from the variogram: C(t, d) = σ 2 γ(t, d) 2. The covariance function can be written as a convolution of a function g and its transpose g: C = g g where g(x) = g( x) 3. The convolution function is used to generate a Gaussian random field y with mean μ and covariance C, from an uncorrelated normal field z: δ = μ + z g 4. (Le Ravalec et al. 2000) demonstrate that the Fourier transform of g is obtained as: FFT(g) = dx FFT(C) 5. Hence, the product FFT(z) FFT(g) can be calculated and transformed in the space domain in z g, and the error components δ are generated with the equation at 3. Le Ravalec, M., Noetinger, B. & Hu, L.Y., 2000. The FFT moving average (FFT-MA) generator: An efficient numerical method for generating and conditioning Gaussian simulations. Mathematical Geology, 32(6), pp.701 723.
Ensemble for uncertainty propagation in models 4) Radar error simulation: δ = μ + z g (FFT-MA) 5) Radar ensemble generation: 10 log P i (t) = 10 log R(t) + δ i (t) 6) Radar ensemble correction: P new,i = σ G σ Porig P orig,i m Porig + m G
Mean and Variance inflation correction We use a logarithmic model Radar error model: 10 log P = 10 log R + δ Radar ensemble generation: 10 log P i (t) = 10 log R(t) + δ i (t) Positive deviations have a larger effect after the anti-log transformation than negative ones Radar ensemble correction: P new,i = σ G σ Porig P orig,i m Porig + m G
Ensemble for uncertainty propagation in models MODEL
Validation and test: case study Validation Model test
Validation Validation Rain Gauge 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0.59 0.78 0.77 0.81 0.8 0.78 0.85 0.88 0.81 0.8 0.72 0.59 0.74 0.75 0.8 0.86 0.83 0.85 0.22 % of the validation data covered by the ensemble
Hydrologic model test: PDM % of the validation data covered by the ensemble Ribble Lune Rawthey All seasons 0.37 0.24 0.32 Winter 0.44 0.29 0.38 Spring 0.30 0.19 0.30 Summer 0.38 0.24 0.28
Conclusions New geostatistical ensemble generator with the following advantages: Capture time variability of errors Produces fields that are already distributed (no interpolation) Corrects mean and variance inflation Fast and flexible also for large datasets Drawback: Spatial stationarity and isotropy of errors Validation proved good agreement with reference data (but RG are not perfect) Hydrologic test suggests that input uncertainty is around 70% of total uncertainty Winter results are, as expected, better than spring and summer ones Need to consider: Anisotropy Rain gauge errors Other model sources of uncertainty Representing radar rainfall uncertainty with ensembles based on a time-variant geostatistical error modelling approach Submitted to the Journal of Hydrology, F. Cecinati, M. A. Rico-Ramirez, D. Han, and G. B. M. Heuvelink
This project has received funding from the European Union s Seventh Framework Programme for research, technological development and demonstration under grant agreement no 607000. www.quics.eu Francesca.Cecinati@Bristol.ac.uk