Estimating local space-time properties of rainfall from a dense gauge network Gregoire Mariethoz (University of Lausanne), Lionel Benoit (University of Lausanne), Denis Allard (INRA Avignon)

Rainfall spatial variability The spatial and temporal distribution of rainfall is critical for hydrology, ecology, engineering, agriculture, etc. However we often don’t know what rainfall looks like at small scales! We can see it and experience it, but it is difficult to measure.

10 Km Spatial rainfall pattern: a main driver for floods Map from Brisbane City council Where will the flood hit a few hours later? RAIN

Temporal information is also critical Same intensity for each event. Which one will result in the highest flood? Milder floodSevere flood Time Rain What is measured

Satellite-based radars Terrestrial radars In-situ measurements How we measure rainfall Satellite radar (TRMM, GPM) Terrestrial weather radar (S- and C-Band) Rain gauges (e.g. Tipping Bucket) Terrestrial weather radar (X-Band) Disdrometers >10km >1km <1m ~1km

On week integrated rainfall measured by radar (1-7 January 2005). Measure influenced by radar locations. Topography causes missing areas. In many areas, no radar data are available. Radar measurements are imperfect Data Meteosuisse (1 km resolution)

Yilmaz et al. (2005) gauge measure (mm/6h) radar measure (mm/6h) Ground-based radar Radar-derived rain intensities are biased Ghiggi (2016) The radar measures reflectivity is non-uniquely related to rainfall.

Not really - numerical models don’t capture key small-scale processes. Can physics-based models help? Wmo.int  Need to resort to statistical models Cloud smaller than the grid cell? Wikipedia Analytic expression of clouds micro-physics? Scale issuesComplexity issues

Target spatial scale: within one radar pixel (~10-100m). Commensurate temporal scale: < 1 minute. Inferred uniquely from in-situ measurements. Goal: develop a stochastic model for the small scale Coherent properties

High-resolution data: Driptych Pluvimates acoustic rain gauges. Counts individual drops. Pluvimates 0.01mm, 30 sec resolution. Usual tipping bucket rain gauges: 0.254mm, 5min. Such precision and temporal resolution allow capturing the passage of small rain cells, invisible on the radar. Instrumentation

Rain rate time series Raw data and quantization effects (stratiform event, 6 Jan 2016) Pluvimate Tipping bucket Histograms Variograms

Rain rate time series After interpolation/smoothing on 3 a min window Pluvimate Tipping bucket Histograms Variograms Possible to identify nested structures (generally two)

Pluvimates in a network A network of 8 pluvimates set up on the campus of the University of Lausanne (~1km). Times series evidently correlated at this scale => possibility to track the movement of rain cells! 200m

New measurements call for a specific type of rain model Many stochastic models exist to represent rainfall properties. [Zhang and Switzer, 2007] [Barancourt and Creutin, 1992] [Schleiss et al, 2014] [Leblois and Creutin, 2013] [Lepioufle et al, 2012]

Latent field We choose to model rain as a truncated latent random field. MultiGaussian field + Values under a given threshold represent no-rain areas. Classically used to represent the very skewed distribution of rinfall and the presence of zeros. Advantages: A single field is generated: easy inference and conditioning. Ability to capture complex patterns. Convenient mathematical framework. Difficulty: we don’t observe the latent variable where there is no rain!

How to deal with space-time? © MeteoSuisse Advection Morphing t E N t E N t E N Symmetric space-time covariance: Rain patterns evolve in time No spatial shift “Frozen rain field”: Rain patterns constant Spatial shift only

Overall procedure 1.By time series cross-correlation, estimate storm velocity ν, and direction θ. 2.Project the time series in a Lagrangian reference frame. 3.Gaussian anamorphosis by normal score transform : y=G(z,t). (t based on the proportion of wet timesteps) 4.Compute space-time variogram. Eulerian reference (Earth fixed) Lagrangian reference (linked to the clouds) t

Observations are a truncated field, but we estimate a continuous latent field. When z=0, y The experimental variogram is computed by maximum likelihood. Limited spatial extent of the network => cannot observe all lags. Blue: time series only Red: space-time Experimental space-time variogram

Overall procedure 1.By time series cross-correlation, estimate storm velocity ν, and direction θ. 2.Project the time series in a Lagrangian reference frame. 3.Gaussian anamorphosis by normal score transform : y=G(z,t). 4.Compute space-time variogram. 5.Fit covariance model.

Two nested variograms models. Parameters: –Sill (2x) –Spatial range (2x) –Temporal range (2x) –Shape parameter (assumed identical) –Nugget A Metropolis algorithm is used to obtain samples of parameters combinations. Each sample is then used to generate a realization of the rain field. Experimental variogram Large scale structure (model 1) Local structure (model 2) Black: model fit Nested model

Overall procedure N t E t E N E Experimental variogram Large scale structure (model 1) Local structure (model 2) Black: model fit 1.By time series cross-correlation, estimate storm velocity ν, and direction θ. 2.Project the time series in a Lagrangian reference frame. 3.Gaussian anamorphosis by normal score transform : y=G(z). 4.Compute space-time variogram. 5.Fit covariance model. 6.Simulate latent Gaussian by FFT + conditioning by kriging and Gibbs sampling. 7.Inverse normal score transform. 8.Project back Eulerian/non-Gaussian reference z=G -1 (y,t).

Synthetic test Realistic synthetic reference field: v=5 m/s, θ =60°, Model 1: Gaussian model, spatial range: =3300 m, temporal range: ∞ (no temporal evolution, sill=0.5. Model 2: Matérn model, spatial range: =1650 m, temporal range: 1650 s, sill=0.5, shape parameter=0.5. Nugget=0 Inferred advection parameters: v=5.76 m/s, θ =59.4°. Same setting as our experimental gauge network. Domain size 1.4 km, 8 stations.

Spatial range model 1 Spatial range model 2 Temporal range model 2 Sill model 2 Shape parameter model 2 Nugget Results of model fit by metropolis For most parameters the reference is within the posterior (or close). The combination shape parameter/nugget is undetermined. Spatial ranges cannot be properly determined due to the lack of data at large spatial lags.

Reproduction of spatial dependence Reference Single realization Difference Small-scale variability slightly overestimated. General rainfall patterns correctly identified.

Reproduction of temporal dependence Small-scale variability correctly reproduced, although slightly overestimated. Increased variance away from the rain gauges.

Conclusions We implemented the first network of pluvimate rain gauges for direct observation of rainfall at high spatial and temporal scales. New measurements call for new models. Integration time matters! Compared to usual tipping bucket: –Different scales of variability, which are represented by nested correlation structures. –The advection of rain cells, allowing to infer the velocity and direction of the storm. A truncated latent field model reproduces these properties, although with inference difficulties. Better inference would require measurements at larger spatial lags or larger temporal lags. Solutions: larger network, moving rain gauge, infer some parameters from radar data.

Ensemble of simulations Least square adjustment Maximum likelihood Metropolis Geostatistical simulations