Uncertainty in merged radar - rain gauge rainfall products Francesca Cecinati University of Bristol francesca.cecinati@bristol.ac.uk Supervisor: Miguel A. Rico-Ramirez 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
Who I am Environmental Engineer (M.Sc. Univ. Genova / M.Eng. MIT) Young Graduate Trainee ESA PhD University of Bristol Started November 2014 (2 year) Marie Curie Early Stage Researcher Project: QUICS (Quantifying Uncertainty in Integrated Catchment Studies) Seconded to TU Delft 4 months (2015) Seconded to EAWAG 3 months (summer 2016) Seconded to W+B 2 months (soon)
Quantifying Uncertainty in Integrated Catchment Studies Rainfall Water Framework Directive (2000/60/EC) Image credits: Melbourne Water
PhD topic: rainfall uncertainty Radar Uncertainty Radar Uncertainty Propagation Rain Gauge uncertainty Merged product uncertainty Radar + Rain Gauge merged products Merged Product uncertainty propagation
Merging Radar Rain Gauge rainfall products Trying to keep the advantages of both: areal estimates at high spatio-temporal resolution high accuracy Many methods, two main families: Bias corrections Kriging based
How to estimate uncertainty Radar: Comparison with rain gauges Modelling source by source Rain gauges: Test studies Theoretical models Merged radar rain gauge : It depends on the method!
Kriging with External Drift KED is one of the best and most efficients merging methods The estimate is based on the kriging interpolation of rain gauges The mean of the process is modelled as a linear function of the radar (external drift) The process is assumed to be Gaussian It also estimates the uncertainty associated with the prediction (kriging variance)
KED uncertainty modelling 1) Interpolation uncertainty Kriging variance 2) Rain gauge errors Kriging for uncertain data (KUD) 3) Radar residual biases Variation of KED including external drift for SD too 4) Gaussian approximation TransGaussian Kriging, etc
KED Z x = m x + Y x m x = a R x + b n Z x 0 = λ i i=1 G x i n i=1 λ i = 1 n j=1 λ i γ ij + μ 1 + μ 2 R x i = γ i0 i = 1,2,, n n λ i i=1 R x i = R x 0
1) Kriging variance Z x 0 σ 2 x 0 = W T G = c W T D C = W = C 1 D The kriging variance mainly describes the interpolation uncertainty C 10 C 11 C 1n 1 R 1 C n1 C nn 1 R n 1 1 0 0 R 1 R n 0 0 D = C n0 1 R 0 W = λ 1 λ n μ 1 μ 2
KED uncertainty modelling 1) Interpolation uncertainty Kriging variance 2) Rain gauge errors Kriging for uncertain data (KUD) 3) Radar residual biases Variation of KED including external drift for SD too 4) Gaussian approximation TransGaussian Kriging, etc
2) Kriging for Uncertain Data (KUD) The nugget effect is a jump in the covariance function that describes the variance at infinitesimal distance C d = c + c 0 d = 0 c exp 3d r d > 0 If the measurement errors are different for the different rain gauges we need a specific c 0i for the i th rain gauge: C = c + c 01 C 1n 1 R 1 C n1 c + c 0n 1 R n 1 1 0 0 R 1 R n 0 0
mm/h Findings Predictions 1. Applying KUD to KED usually improves the estimates Variance 2. It is not always true for ordinary kriging mm 2 /h 2 3. KUD has a smoothing effect on the less reliable rain gauges 4. The improvement (or not) is a function of many factors, including the geometry and the characteristics of the rainfall event
KED uncertainty modelling 1) Interpolation uncertainty Kriging variance 2) Rain gauge errors Kriging for uncertain data (KUD) 3) Radar residual biases Variation of KED including external drift for SD too 4) Gaussian approximation TransGaussian Kriging, etc
3) Radar residual biases The absolute value of radar estimates does not affect the predictions of KED, because a linear function of it is used as trend. If radar bias was spatially homogeneous, it wouldn t affect the KED prediction. Nevertheless, radar uncertainty is not uniform in space, and this needs to be modelled
Modification of KED: Regular KED: Z x = m x + Y(x) p m x = α i f i (x) i=1 Radar estimates Modified KED: Z x = m x + σ x ε(x) p m x = α i f i (x) i=1 p σ x = β j g j (x) j=1 Y x zero-mean, second order stationary process ε x zero-mean, unit variance, second order stationary process Elevation Clutter map Bright-band area
KED uncertainty modelling 1) Interpolation uncertainty Kriging variance 2) Rain gauge errors Kriging for uncertain data (KUD) 3) Radar residual biases Variation of KED including external drift for SD too 4) Gaussian approximation TransGaussian Kriging, etc
4) Gaussian approximations Kriging methods are based on the assumption of secondorder stationary Gaussian processes. Rainfall distribution is not Gaussian, it is skewed, only positive, with a discontinuity in zero. Partial solutions: TransGaussian Kriging Indicator Kriging Disjunctive Kriging Singularity analysis. Controversial Not easily adaptable to KED
TransGaussian kriging Rain Gauge data Radar data transformation transformation KED Back transformation (Box-Cox, normal quantiles, etc ) KED merged rainfall estimate Box-Cox Transformations: y = λ = 0.5 Square root λ = 0.25 Square root Square root λ = 0 Logarithm log x if λ = 0 x λ 1 λ if λ 0 Normal quantiles (Normal Scores): empirical transformation
Singularity analysis Born in fractal theories, adapted to Bayesian merging by Wang et al. 2015, Imperial College Local Singularities: structures in which the areal average varies with the considered area with a power function They are not fully captured by second-order tools, like the variogram used in kriging, and therefore cannot be maintained in kriging merging, due to the Gaussianity assumption. Radar Rain Gauges Remove singularities KED Recover Singularities Result
Evaluation Rain Gauge data Radar data transformation transformation 1 How effective are the transformation methods? How gaussian are the transformed variables? KED Back transformation How well are the results backtransformed? Is the original PDF reproduced? 2 KED merged rainfall estimate What is the quality and the reliability of the final rainfall product? 3 1) Kurtosis 2) Skewness 3) Approx. Negentropy R 2 in QQ-plot regression RG validation: 1) MRTE 2) BIAS 3) Hanssen Kuipers d.
1) Gaussianity Test
2) QQ-plots
3) Validation with Rain Gauges Box-Cox with λ = 0.1 introduces a high bias Singularity analysis not suitable for KED Merging usually improves the results Transformations are helpful, but a simple square root does already the job
Event on 2009/06/17 20:00 (event 2) Black-border dots: rain gauges used for merging Green-border dots: rain gauges used for validation Notice the radar artefact attenuation
Some work done 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 a ) b ) c ) Integration of rain gauge measurement errors with the overall rainfall uncertainty estimation using kriging methods Presented at EGU 2016 To be submitted soon to a journal Evaluation and correction of uncertainty due to Gaussian approximation in radar rain gauge merging using kriging with external drift Abstract submitted for AGU 2016 To be submitted soon to a journal
To conclude Understanding and modelling uncertainty in merged radar-rain gauge products is complex Rain gauge uncertainty: you need to know your gauges. Can you help me with the UK? Radar uncertainty: what spatially variant covariates can affect NIMROD radar uncertainty magnitude? Transformations are good, but advanced methods only introduce instability/approximations What do you think affects KED uncertainty? Thank you!!! This project has received funding from the European Union s Seventh Framework Programme for research, technological development and demonstration under grant agreement no 607000.
Some references Berndt, C., Rabiei, E. & Haberlandt, U., 2014. Geostatistical merging of rain gauge and radar data for high temporal resolutions and various station density scenarios. Journal of Hydrology, 508, pp.88 101. Clark, I., 2010. Statistics or geostatistics? Sampling error or nugget effect? Journal of the Southern African Institute of Mining and Metallurgy, 110(6), pp.307 312. Mazzetti, C. & Todini, E., 2009. Combining Weather Radar and Raingauge Data for Hydrologic Applications. In Flood Risk Management: Research and Practice. London: Taylor & Francis Group Erdin, R., Frei, C. & Künsch, H.R., 2012. Data Transformation and Uncertainty in Geostatistical Combination of Radar and Rain Gauges. Journal of Hydrometeorology, 13(1987), pp.1332 1346 Wang, L.-P. et al., 2015. Singularity-sensitive gauge-based radar rainfall adjustment methods for urban hydrological applications. Hydrology and Earth System Sciences, 19(9), pp.4001 4021.