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Present address: Meteoswiss, Zürich, Switzerland

Present address: Deutscher Wetterdienst, Offenbach, Germany

This article has been republished with minor changes. These changes do not impact the academic content of the article.

We present a method to estimate spatially and temporally variable uncertainty of areal precipitation data. The aim of the method is to merge measurements from different sources, remote sensing and in situ, into a combined precipitation product and to provide an associated dynamic uncertainty estimate. This estimate should provide an accurate representation of uncertainty both in time and space, an adjustment to additional observations merged into the product through data assimilation, and flow dependency. Such a detailed uncertainty description is important for example to generate precipitation ensembles for probabilistic hydrological modelling or to specify accurate error covariances when using precipitation observations for data assimilation into numerical weather prediction models. The presented method uses the Local Ensemble Transform Kalman Filter and an ensemble nowcasting model. The model provides information about the precipitation displacement over time and is continuously updated by assimilation of observations. In this way, the precipitation product and its uncertainty estimate provided by the nowcasting ensemble evolve consistently in time and become flow-dependent. The method is evaluated in a proof of concept study focusing on weather radar data of four precipitation events. The study demonstrates that the dynamic areal uncertainty estimate outperforms a constant benchmark uncertainty value in all cases for one of the evaluated scores, and in half the number of cases for the other score. Thus, the flow dependency introduced by the coupling of data assimilation and nowcasting enables a more accurate spatial and temporal distribution of uncertainty. The mixed results achieved in the second score point out the importance of a good probabilistic nowcasting scheme for the performance of the method.

Precipitation information with an accurate uncertainty estimate is highly relevant for research areas with special interest in areal precipitation information. The uncertainty information is needed for example to generate precipitation ensembles as input for probabilistic hydrological modelling (e.g. Krajewski and Ciach, 2006; Germann et al.,

To assess the reliability of any observation, it is crucial to know its uncertainty. For precipitation observations, accurate measurements as well as a reliable estimate of their uncertainty are an ongoing topic of research. Observations from weather radars are of particular interest because they provide valuable areal precipitation information with high spatial and temporal resolution, in contrast to in situ observations. However, radar measurements are affected by numerous sources of error that diminish the accuracy of the provided precipitation quantification, e.g. calibration errors, noise, interferences, clutter, and attenuation. Despite the use of correction algorithms and filters on radar measurements, residual error remains inherent to the data. In addition, the estimation of rainfall intensity from radar data requires a relation between measured reflectivity and rain rate which is empirical and uncertain. Therefore, accurate rainfall precipitation quantification is not possible with radar measurements only. An extensive review of uncertainty sources in single polarisation radar measurements is given in Villarini and Krajewski (

Improvements in the precipitation estimate are achieved by combining measurements from different sources, e.g. radar and rain gauge data, to take advantage of as much information as possible. Since no observation is error-free, they must be weighted with their respective uncertainty in order to obtain a statistically sound combined precipitation product. Many studies investigate the statistical combination of precipitation data. They rely on computing the optimal estimate of precipitation by minimising its error variance. The majority of the applied techniques focus on a static, i.e. time-independent, merging approach, where precipitation measurements are combined for every available time step independently. Most merging approaches use kriging or cokriging schemes to spatially merge radar and rain gauge data (e.g. Krajewski,

Some studies additionally take into account the temporal correlation of precipitation measurement errors while merging radar and gauge data. The temporal structure of the errors can be considered either by modelling spatio-temporal variograms in the kriging approach (e.g. Sideris et al.,

Very few studies include a forecast model that evolves the system’s state to take advantage of previous knowledge of the system. The coupling of a forecasting model can be done using Kalman filtering or variational data assimilation methods. In addition to the merging of observations, the forecasting component allows for the extrapolation of information to data-void regions. Advection is commonly used as an approximation for the evolution of precipitation systems. Zinevich et al. (

Considering radar precipitation data, uncertainty estimation is extensively studied outside the context of combined precipitation products. Numerous studies address the quantification of precipitation data uncertainty to overcome the deterministic view on radar precipitation measurements. The basis for the description of areal precipitation measurement errors mostly is an empirical, statistical study of radar and reference surface measurements. Some methods only provide static information (Krajewski and Ciach, 2006), while more recent approaches also allow for the description of the spatial and temporal structure of the errors through the description of the error covariance (e.g. Ciach et al.,

Here, we present a method that connects both aspects: precipitation data merging and probabilistic uncertainty assessment. The method allows for the combination of precipitation observations considering their respective uncertainty and takes advantage of the additional information provided by the temporal evolution of the system. At the same time, the method yields an areal uncertainty estimate for the resulting precipitation product. Because of the included temporal evolution, the uncertainty estimate is variable both in space and time and is flow-dependent. Thus, this method aims at providing both an accurate precipitation product and an improved areal and dynamical uncertainty estimate.

Our approach uses data assimilation as a tool to merge precipitation observations. This study considers radar data with different spatial resolution, but the method can be extended to incorporate any additional source of precipitation observation. Data assimilation techniques allow for statistically combining all available information considering its uncertainty within a temporal evolution model. The Local Ensemble Transform Kalman Filter (LETKF, Bishop et al.,

The added value of the presented method is demonstrated in four experiments with observations from an X-band radar network. The method’s functioning, including the LETKF and the coupled nowcasting scheme, is presented in

The combination of a data assimilation method with a nowcasting method introduces flow dependency to the areal uncertainty estimate, as known uncertainty is propagated in time and space. In this way, the method yields a situation-dependent uncertainty estimate whose structure follows the evolution of the system. The overall functioning of the method is outlined in the flowchart in

Flowchart illustrating the data assimilation cycle of the method for uncertainty estimation. Combined precipitation product and uncertainty estimate are obtained from the analysis ensemble mean and spread, respectively.

Data assimilation aims at combining different sources of information about an observed system, weighted by their respective uncertainty, in order to get the best possible estimate of the system’s true state. The data assimilation method used here is the LETKF that allows to work with an ensemble of moderate size through a localisation approach. It was first introduced by Hunt et al. (

The nowcasting scheme used for the proof of concept study is a simple approach based on template-matching between time-lagged precipitation images. The original algorithm, the Automated Precipitation Extrapolator (APEX), was developed by van Horne (

The algorithm considers two consecutive precipitation images and computes an estimate of the precipitation displacement from the first to the second image. A correlation is calculated between the second image and spatially shifted versions of the first image. The spatial shift is performed in every direction and with different amplitudes and the shift yielding the best correlation allows for estimating the displacement of the precipitation structure between both images. The resulting displacement vector field indicates the precipitation motion direction and speed, as demonstrated in

Example of input precipitation images for template-matching for the 3 July 2013. (a) 15:23:00 UTC and (b) 15:26:00 UTC and resulting displacement vector field, thinned out to every tenth pixel in each direction in this representation.

The complete displacement vector field (one vector per pixel in the precipitation field) is used to compute the precipitation forecast. Displacement vector lengths are scaled to the desired forecast time step and the future position of the precipitation field in the second image is predicted accordingly. In order to get a smooth, continuous precipitation field after the advection, a 3 × 3 window of pixels around each pixel is shifted to its new position. If more than one precipitation value is attributed to a pixel of the forecasted precipitation field, values are averaged, as illustrated in

Schematic representation of the forecasting step for three pixels, exemplary. 3 × 3 windows of pixels (initial position left) are shifted according to the local motion vectors calculated for the middle pixel. If these pixel windows overlap in the forecast (new position right), pixel values (precipitation intensity or motion velocity) are averaged to get the forecasted field. In this example, light grey pixels are averages over two values, dark grey over three values.

To create an ensemble forecast from the deterministic nowcasting scheme, we generate an ensemble of displacement vector fields from the deterministic field by randomly perturbing its

This study uses radar reflectivity data from a research radar network. Both the network and the data are thoroughly described in Lengfeld et al. (

Map of the radar network. X-band radar locations and maximum range in black, MRR locations in orange. X-band radars and associated MRRs are installed at BKM, HWT, MOD and QNS sites. Three MRRs are installed between X-band radars, at sites WST, MST, and OST, from west to east respectively, together with reference rain gauges.

The Cartesian composite radar data are used to initialise the nowcasting method presented in

Observations considered for assimilation in this study are taken from single radar measurements (X-band radar MOD in Lengfeld et al.,

Locations of thinned precipitation observations created for assimilation from data of the X-band radar at MOD site (Lengfeld et al.,

Observation errors on the thinned 5000 m × 5000 m grid are assumed to be uncorrelated and with constant error variance. This is a simplifying assumption, especially for radar data. The estimation of the full radar observation error statistics is a complex and ongoing topic of research (e.g. Xu et al., ^{2}.

We focus on reflectivity data to avoid additional uncertainty coming from the empirical conversion from reflectivity to rain rate. The logarithmic reflectivity unit dB^{–1}, these differences correspond to roughly a factor of 1.15 and 1.54, respectively. Following the example of Bick et al. (^{–1}, and all reflectivity data are thresholded accordingly throughout the study.

To assess the uncertainty estimate obtained by the method introduced in

X-band radar network composite reflectivity data (Lengfeld et al.,

As presented in ^{–1} and ^{–1} respectively. The speed of the precipitation cell is thus estimated to be 7 m s^{–1}. The forecast is performed for 50 ensemble members until 16:00:00 UTC in time steps of 2 min.

Observations used for assimilation are selected from single radar data and thinned to a 5000 m × 5000 m grid, as indicated in

The LETKF further requires the specification of an observation influence radius. During the analysis step, model grid points are affected by all observations within the influence radius. Other observations are ignored. We use an influence radius of 4000 m, which ensures a complete coverage of grid points within X-band radar MOD reach. Therefore, every grid point in this area is affected by the update during data assimilation which favours a smooth precipitation analysis. An analysis of the sensitivity of the system with respect to different observation influence radii can be found in Merker (

To illustrate the system’s behaviour, the performed data assimilation cycle is shown in

Reflectivity ensemble mean evolution (dark blue line) and uncertainty range (light blue envelope, ± one ensemble standard deviation) throughout the data assimilation cycle at observation grid point 0.041°E, – 0.053°N (indicated by a cross in

Spatial distribution of the reflectivity ensemble mean (with contour highlighting the region in which 80% of the ensemble members show precipitation above 5 dB

The spatial structure and temporal evolution of the uncertainty information is the main asset of the method presented here. The structure and evolution of the uncertainty is described by the ensemble spread at analysis time, which is shown together with the ensemble mean as an example for two time steps in

Verification is performed at locations selected on a 5000 m × 5000 m grid, similar to the assimilation locations. The verification grid is shifted by 2500 m in each direction with respect to the assimilation grid in order to select independent observation locations for assimilation and verification (

The overall root mean square error (RMSE) of the forecast ensemble mean for case 1 amounts to 4.49 dB and the bias, defined here as the mean error of the ensemble mean compared to the observations, to 1.54 dB. These forecast statistics can be improved slightly using the probabilistic ensemble information and applying a 80% precipitation probability threshold to the forecast data, i.e. setting the ensemble mean for grid points with precipitation probability lower than 80% to the ‘no rain’ value of 5 dB

To characterise the quality of the areal uncertainty estimate of the combined precipitation, we analyse the statistical relation between forecasted uncertainty, i.e. the ensemble standard deviation, and the actual error, i.e. the deviation between forecasted and observed reflectivity. In a so-called spread-skill diagram, the perfect relation between forecast uncertainty and forecast error is a one-to-one relation. The uncertainty forecast perfectly describes the uncertainty of the system if, statistically, the actual forecast error equals its forecasted uncertainty (Leutbecher and Palmer,

Comparison of absolute precipitation product (model ensemble mean) error and ensemble spread (model ensemble standard deviation) at the available 47 verification grid points and eight analysis time steps for (a) case 1, (b) case 2, (c) case 3 and (d) case 4. Ensemble spread values are divided into bins of 0.5 dB width, boxes indicate the median (blue line) and the first and third quartiles. Data distribution is shown in frequency histograms, the solid grey lines show the cumulative distribution function.

Most data points show ensemble spread and absolute product error below 0.5 dB. This category is dominated by grid points without precipitation, both in the forecast and the observations. Overall, there is a clear correlation between ensemble spread and absolute product error, especially for cases 1 and 3, hinting at a good uncertainty representation by the ensemble spread. In the range between 0 and 3 dB ensemble spread, the forecast model ensemble is overdispersive because ensemble spread is substantially larger than the median of the absolute product error distributions (median below the one-to-one line). This is due to non-zero model forecast values at grid points outside the observed (true) location of the precipitation cell. Most of the ensemble members correctly predict no precipitation for those grid points, but some have higher reflectivity values because of the differently predicted displacements of the cell. The ensemble standard deviation is more strongly impacted by these few high reflectivity values than the ensemble mean. Therefore, the ensemble standard deviation is larger than the deviation between ensemble mean and observation.

Two scores are defined to quantify the potential of the ensemble spread to correctly represent the uncertainty of the precipitation product in time and space. The aim is to prove that the spatial and temporal structure of the uncertainty estimate is not random, but actually provide valuable additional information. The first score indicates the amount of data points for which the absolute product error falls within the uncertainty range predicted by the ensemble spread—the percentage of ‘hits’. The ensemble spread is interpreted as the tolerated margin of error. The score
_{i}

To assess the areal uncertainty estimate for the combined precipitation product, above scores are computed with the spatially and temporally variable product uncertainty described by the ensemble spread after the update step at all available verification grid points. The reliability

The benchmark against which the spatially and temporally variable uncertainty estimate gained by the flow-dependent ensemble spread is evaluated is a constant uncertainty estimate valid for all grid points at all-time steps. Constant uncertainty information still is most common for precipitation data from radar or other sources (e.g. Tong and Xue, _{i}

Both scores are summarised in _{i}

Results for

The results of the

This article presents a method to estimate spatially and temporally variable uncertainty for a combined areal precipitation product. The method makes use of data assimilation to merge precipitation measurements from different sources and an ensemble nowcasting method to evolve the uncertainty estimate over time. The potential of the areal uncertainty estimate provided by the method is demonstrated in a proof of concept study.

Requirements for this improved uncertainty estimate are an accurate representation of the actual error of the initial product, an adjustment to additional observations merged into the product through data assimilation, and flow dependency. In this study, a research network of four X-band weather radar provides reflectivity data for four precipitation cases. Reflectivity can be converted to rain rate using empirical relations that are not part of the analysis.

An ensemble data assimilation method, the LETKF, is used to statistically merge precipitation observations and is coupled to an extrapolation-based nowcasting scheme. The implemented nowcasting scheme computes the cross-correlation between subsequent radar composite images and extrapolates the evolution of the precipitation field using the deduced displacement. The nowcast is started from composite data of the research radar network and a probabilistic forecast is generated using an ensemble technique. The ensemble is generated by stochastic perturbation of the computed precipitation displacement vectors.

Four data assimilation experiments are performed to test the presented method with emphasis on its potential to provide an improved spatial and temporal uncertainty estimation. We use an ensemble precipitation nowcast and continuously merge observations into the forecast using the LETKF, generating a combined precipitation product. The uncertainty of the precipitation product is estimated by the ensemble spread of the nowcast after each update step of the data assimilation cycle. Two scores are introduced for the assessment of the method. Both are based on the definition of a perfect uncertainty estimate, for which the actual observed error statistically corresponds to the predicted uncertainty. The first score describes the reliability,

The scores are computed at verification grid points selected on a regular grid and for all available analysis time steps. The potential of the obtained areal uncertainty estimate

The reliability of the uncertainty forecast,

The evaluation of both considered scores demonstrates that the provided areal uncertainty estimate outperforms constant benchmark uncertainty values, but that the method is sensitive to the quality of the probabilistic nowcasting. In subsequent work, the next logical step would therefore be the use of a more sophisticated nowcasting tool. Nevertheless, the proof of concept shows the potential of the developed method and establishes the groundwork for further studies. The possible applications of the method are numerous in hydrology, nowcasting or data assimilation. It provides the combination of areal and point measurements for comprehensive precipitation products and additionally yields probabilistic information allowing for the estimation of the product reliability and ensemble generation.

Gernot Geppert was supported by the Natural Environment Research Council (Agreement PR140015 between NERC and the National Centre for Earth Observation). We would also like to acknowledge the contributions of the reviewers which led to a significant improvement of the manuscript.

No potential conflict of interest was reported by the authors.