## 1. Introduction

Accurate hydrological modeling requires the combined use of distributed rainfall-runoff models and appropriate spatial resolution rainfall fields. Meteorological radars provide detailed precipitation fields with high spatial and temporal resolution to couple with hydrological models as shown in Greene and Flanders (1976). The size (small or medium), response time (flash flood prone basin), and type of catchment (e.g., urban or rural) change the requirements on the spatial and temporal resolution of the radar data (Krajewski et al. 1991; Yang et al. 2000; Atencia et al. 2010).

The use of rainfall forecasts in these kinds of hydrometeorological systems results in longer hydrologic forecast times (i.e., the difference between the time when the forecast event occurs and the time when the forecast is issued). Different types of quantitative precipitation forecasts (QPFs), such as radar-based nowcasts (Mecklenburg et al. 2002), numerical weather prediction (NWP) rainfall forecasts (Bartholmes et al. 2005), or blended QPFs (Atencia 2010), have showed benefits in the real-time flood forecasting and warning systems. However, most of the uncertainty in rainfall-runoff forecasting schemes over small and medium catchments comes from the uncertainty in predicting rainfall at high temporal and spatial resolutions (Zappa et al. 2010).

A meteorological radar can solve this problem by providing quantitative precipitation estimates (QPEs) at the required scales. Nowcasting refers to short-term forecasts with forecast length between 0 and 6 h and with high spatial (â‰¤5 km) and temporal (<1 h) resolutions (Wilson et al. 1998). Therefore, radar-based heuristic nowcasting techniques have become a key component in hydrometeorological forecasting (Harrison et al. 2012).

Quantitative precipitation nowcasts are mainly based on the advection of rainfall fields observed by the meteorological radar. These advected rainfall fields (also called Lagrangian extrapolation nowcasts or persistence) are characterized by a high initial skill that rapidly decreases with the forecast lead time as shown by Golding (1998). The two main sources of error in Lagrangian persistence forecasts are the growth and decay of precipitation and the temporal evolution of the advection field (Tsonis and Austin 1981; Radhakrishna et al. 2012). Germann et al. (2006) showed that the importance of these two factors is case dependent.

Lagrangian extrapolation produces advection of the latest rainfall field. Consequently, these two processes (growth and decay and evolution of advection field) are not taken into account. A first attempt to introduce these processes as stochastic factors in nowcasting was carried out by Andersson and Ivarsson (1991), producing probability of precipitation forecasts based on the fraction of precipitating pixels in the area around the point of interest. The following studies (Schmid et al. 2000; Germann and Zawadzki 2004) researched techniques for producing future probability distribution function (pdf) of rain rate by different methods. These probability QPF schemes have been unable to provide a sufficiently dense description of the spatial and temporal structure of precipitation. Consequently, they are unsuitable for hydrological modeling where these spatial factors play a key role (Zappa et al. 2010).

Several authors (Lovejoy and Schertzer 1990; Perica and Foufoula-Georgiou 1996; Venugopal et al. 1999, among others) have provided evidence that rainfall patterns show a scaling behavior based on multifractal models. These scaling properties have been used in the formulation of spatiotemporal models of rainfall fields, such as that developed by Mackay et al. (2001) or the String of Beads Model for Nowcast (SBMcast; Pegram and Clothier 2001). This model has been recently used by Berenguer et al. (2011) to generate a number of realistic future rainfall scenarios (called members) compatible with observations in a nowcasting framework (SBMcast). Some works (Venugopal et al. 1999; Germann and Zawadzki 2002, hereafter GZ02; Surcel et al. 2014, manuscript submitted to *J. Atmos. Sci.*) have shown that the predictability of rainfall structures has a scale dependence based on dynamic scaling processes. Bowler et al. (2006) developed a stochastic precipitation nowcasting system called Short-Term Ensemble Prediction System (STEPS) that combines an extrapolation nowcast, a downscaled NWP model, and a small-scale noise field. The Spectral Prognosis (S-PROG) model developed by Seed (2003) has been used in STEPS to simulate the uncertainties in the evolution of precipitation patterns. Fourier filters are used to decompose the rainfall field into a multiplicative spectral cascade. This separation allows treating the different scales components of the precipitation pattern independently and attributing different weights to different scales.

In this article, a new probabilistic methodology to create realistic rainfall nowcasts is developed. Section 2 presents the data used in this study and the case studies. Details for implementing the new methodology are presented in section 3 where both mathematical description and illustrative flow charts are used for this purpose. The verification results are shown in section 4 and the main findings of this study are discussed in section 5.

## 2. Dataset and case studies

The radar data are produced by Weather Decision Technologies (WDT) and use radar data from the entire Weather Surveillance Radar-1988 Doppler (WSR-88D) network in the continental United States (CONUS). This allows WDT to apply their most up-to-date, technologically advanced algorithms to provide superior quality radar data through the removal of false echoes and through the blending of multiple radars. WDT creates seamless radar mosaics with a high spatial resolution (1 km) and temporal resolution (5 min). Our database covers the period from January 2004 to April 2011.

A new grid is defined for the dataset used in this study (Fig. 1). The new 512 Ã— 512 points grid has a 4-km resolution. Each reflectivity map at the original resolution, *Z*^{1km}, is converted to rainfall rate map, *R*^{1km}, by the relation, *R*^{1km}) are interpolated to the new grid (*R*^{4km}). After the upscaling process in rainfall units, the obtained field is converted to reflectivity *Z*). The temporal resolution of the new dataset is 15 min. The selected domain avoids the Rocky Mountains because of their orographic effects on rainfall fields and the blockage they produce in the radar rainfall images.

Four case studies have been selected to test the probabilistic nowcasting technique. Two of them are high-predictable cases according to the lifetime definition of GZ02. The other two events are low predictable cases consisting of mesoscale systems. In GZ02 the threshold 1/*e* for the temporal correlation between Lagrangian extrapolation and observed reflectivity field at the same time in the future is determined as the lower limit to attribute predictability to a forecast. Consequently, lifetime is defined as the time when the correlation reaches this threshold. Figure 2 depicts one representative image of each system. Table 1 lists some properties of the selected events, such as duration, lifetime, spatial extent, and percentage of convective precipitation. It is important to mention that diurnal heating was the main trigger effect in one of the events selected.

A snapshot for each of the four 2008 precipitation events used in this study: (a) 2115 UTC 22 Jul, (b) 1000 UTC 6 May, (c) 0345 UTC 18 Apr, and (d) 0530 UTC 6 Jun. The plots depict the reflectivity field from a minimum threshold of 15 dB*Z*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

A snapshot for each of the four 2008 precipitation events used in this study: (a) 2115 UTC 22 Jul, (b) 1000 UTC 6 May, (c) 0345 UTC 18 Apr, and (d) 0530 UTC 6 Jun. The plots depict the reflectivity field from a minimum threshold of 15 dB*Z*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

A snapshot for each of the four 2008 precipitation events used in this study: (a) 2115 UTC 22 Jul, (b) 1000 UTC 6 May, (c) 0345 UTC 18 Apr, and (d) 0530 UTC 6 Jun. The plots depict the reflectivity field from a minimum threshold of 15 dB*Z*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Properties and statistics of the four precipitation events used in this study. The extent is defined as the area with a reflectivity >15 dB*Z*. The lifetime of the event is determined by the decorrelation time in Lagrangian coordinates. The fraction of that area having reflectivities >35 dB*Z* defines the percentage of convective precipitation (C. fraction). The values are averages over the entire period.

## 3. Methodology

Despite the significant advances in NWP models and data assimilation, the resulting forecasts are not up to the desired level of accuracy at meso-*Î²* and meso-*Î³* scales (Lin et al. 2005; Wilson et al. 2010). Such studies have reported that the radar-based extrapolation forecasts (nowcasting) have better skill than those from the NWP models for forecast hours from 0 to 4 h (Berenguer et al. 2012). Consequently, Lagrangian extrapolation forecast is the preferred technique for nowcasting purposes. The extrapolation algorithm used in this study is introduced in section 3a.

Germann et al. (2006) showed that there are two sources of forecast uncertainty when using Lagrangian extrapolation of reflectivity fields: the nonstationarity of the storm motion field and the growth and decay of precipitation. The uncertainty in the Lagrangian forecast due to these effects and the importance of these two limiting factors are case dependent. It is also dependent on the geographic region, with the longest lifetime in the central United States.

A probabilistic nowcasting method is presented herein. The approach takes advantage of the predictability of large-scale features existing in the latest radar image to produce a new reflectivity field that maintains the observed spatial and temporal structure of rainfall fields. This technique is developed in Lagrangian coordinates assuming that the advection of the rainfall field is a known process but introducing, as a redistribution of power in the small scales, the growth and decay of the rainfall field as a stochastic process. The methodology to reproduce this growth-and-decay uncertainty in the Lagrangian extrapolation is detailed in section 3b.

### a. Lagrangian extrapolation

The Lagrangian extrapolation of the radar reflectivity field is carried out in two steps. First, the motion field is estimated based on the Continuity Tracking Radar Echoes by Correlation (COTREC) algorithm developed by Li et al. (1995). This algorithm imposes the continuity equation on the vectors obtained by maximizing the cross-correlation Tracking Radar Echoes by Correlation (TREC) technique by Rinehart and Garvey (1978) to avoid null vectors due to nonprecipitation areas and to smooth the motion vector field. Second, the last observed reflectivity field is advected according to the previously estimated motion field. The advection is applied using a semi-Lagrangian backward scheme (Staniforth and CÃ´tÃ© 1991). In this scheme, a given pixelâ€™s reflectivity value is obtained by choosing the observed reflectivity value at the location obtained by integrating the motion field backward along the forecasting period. This scheme is selected because, as showed in GZ02, it allows rotation, it almost conserves the mass, and it limits the loss of power at small scales that would results from interpolation. The motion field is kept stationary in time during the forecast period.

### b. Ensemble generation

To generate a realistic ensemble of forecast reflectivity fields from the Lagrangian extrapolation, a new methodology has been developed in this article. This methodology, which is based on the Fourier space, is divided into two main steps that will ensure that the phase and amplitude of the generated ensemble are compatible with the latest observation. Here is a summary of the two steps. An in-depth explanation follows.

Step 1, phase, illustrated in Fig. 3: several substeps are needed to construct a realistic rainfall mask, which holds that part of the phase that exhibits persistence. These substeps are the following:

Determine the cutoff scale for constructing a low-pass-filtered reflectivity field (

*Z*_{L}) for a given lead time as a function of the temporal autocorrelation.Generate a spatialâ€“temporal correlated noise field from a conditional autoregressive model of second-order [CAR(2)] with a zero mean and one standard deviation (first row of Fig. 3).

Creation of a contour

*Z*_{L}level map (first field of the second row in Fig. 3) illustrating the probabilities of no rain for a given*Z*_{L}value (Fig. 6).Compute the threshold value of the previously obtained correlated noise field (third field of the first row in Fig. 3) that reproduces the no-rain probability map (first field of the second row in Fig. 3). (The threshold map obtained is similar to the second field of the second row of Fig. 3).

Compute the factors that normalize the different thresholds.

Smooth the factor field (third field of the second row of Fig. 3) and multiply it by the correlated noise field.

Pixel values superior to the normalized threshold are selected to become the rain area of the rainfall mask. An example is shown in Fig. 3.

Flowchart of the methodology used to generate the rainfall mask. (top) Representation of a CAR(2) used to reproduce a temporal correlated noise field. (bottom) Information about the steps needed to generate the rainfall mask. From the noise field and the factor field, the rainfall mask field (extreme right map) is obtained.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Flowchart of the methodology used to generate the rainfall mask. (top) Representation of a CAR(2) used to reproduce a temporal correlated noise field. (bottom) Information about the steps needed to generate the rainfall mask. From the noise field and the factor field, the rainfall mask field (extreme right map) is obtained.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Flowchart of the methodology used to generate the rainfall mask. (top) Representation of a CAR(2) used to reproduce a temporal correlated noise field. (bottom) Information about the steps needed to generate the rainfall mask. From the noise field and the factor field, the rainfall mask field (extreme right map) is obtained.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Step 2, amplitude, illustrated in Fig. 4: to reconstruct the power spectrum, which is related to the square of the amplitude, an iterative process divided into three substeps is carried out:

Creation of a generator (

*g*) to reproduce the correlation of the last observed reflectivity field (*Z*).Convolution of a field of white noise with the generator field to obtain a spatially correlated noise field (left arrows in Fig. 4).

Masking the image with the previous obtained rainfall mask (right arrows in Fig. 4).

Example of the iterative process used to reproduce the power-spectrum slope from a Gaussian white noise field. A generator *g* is applied to resemble the power-spectrum slope [gray shaded line in the power spectrum figures (second column)] and the same rainfall mask is applied afterward [white solid contours in noise field (third column)]. This process is repeated *n* times until there is little variation in the square mean and in the power-spectrum slope. (first column) The distribution of observed reflectivity values (dashed histograms) and noise field values (solid histograms).

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Example of the iterative process used to reproduce the power-spectrum slope from a Gaussian white noise field. A generator *g* is applied to resemble the power-spectrum slope [gray shaded line in the power spectrum figures (second column)] and the same rainfall mask is applied afterward [white solid contours in noise field (third column)]. This process is repeated *n* times until there is little variation in the square mean and in the power-spectrum slope. (first column) The distribution of observed reflectivity values (dashed histograms) and noise field values (solid histograms).

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Example of the iterative process used to reproduce the power-spectrum slope from a Gaussian white noise field. A generator *g* is applied to resemble the power-spectrum slope [gray shaded line in the power spectrum figures (second column)] and the same rainfall mask is applied afterward [white solid contours in noise field (third column)]. This process is repeated *n* times until there is little variation in the square mean and in the power-spectrum slope. (first column) The distribution of observed reflectivity values (dashed histograms) and noise field values (solid histograms).

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

The substeps (ii) and (iii) are repeated in an iterative process until both the slope of the power spectrum and the mean of masked correlated field vary by less than 0.1% between iterations.

*Î¼*stands for the mean of the reflectivity field

*Z*and

*Ïƒ*for the standard deviation.

*X*) and the original reflectivity field (

*Z*):

*Ï†*

_{Z}is the phase. The change in the power spectrum by means of the normalization of the reflectivity field depends on the phase as can be seen in last term of the Eq. (3).

It can be observed in Fig. 5a the phase changes quickly for scales smaller than 100 km. Consequently, the use of previous reflectivity fieldâ€™s phase information does not seem useful. On the other hand, it can be observed (Fig. 5b) that the phase information is mostly in the rain area. Because of that, the phase information is introduced in our methodology by the construction of a possible future rainfall mask.

The 2D field of phase difference in the *x-* and *y*-direction wavenumbers between: (a) two reflectivity fields that are 5 min apart and (b) rainfall mask and original reflectivity field. The circles represent the isoline wavelength in our domain.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

The 2D field of phase difference in the *x-* and *y*-direction wavenumbers between: (a) two reflectivity fields that are 5 min apart and (b) rainfall mask and original reflectivity field. The circles represent the isoline wavelength in our domain.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

The 2D field of phase difference in the *x-* and *y*-direction wavenumbers between: (a) two reflectivity fields that are 5 min apart and (b) rainfall mask and original reflectivity field. The circles represent the isoline wavelength in our domain.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

To create a realistic rainfall mask, we take advantage of the very good relationship between the probability of a dry pixel and the reflectivity value of the low-pass-filtered field (*Z*_{L}) for all lead times (Fig. 6). This will be called hereinafter no-rain probability.

Probability of a dry pixel (no rain) as function of the reflectivity of the *Z*_{L} field at 64-km cutoff scale for a period of 24 h (different color lines stands for different times of the day) of (a) a highly and (b) a lowly predictable event. The no-rain probability is computed as the ratio of dry pixels and total of pixels in the area covered by a given *Z*_{L} value.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Probability of a dry pixel (no rain) as function of the reflectivity of the *Z*_{L} field at 64-km cutoff scale for a period of 24 h (different color lines stands for different times of the day) of (a) a highly and (b) a lowly predictable event. The no-rain probability is computed as the ratio of dry pixels and total of pixels in the area covered by a given *Z*_{L} value.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Probability of a dry pixel (no rain) as function of the reflectivity of the *Z*_{L} field at 64-km cutoff scale for a period of 24 h (different color lines stands for different times of the day) of (a) a highly and (b) a lowly predictable event. The no-rain probability is computed as the ratio of dry pixels and total of pixels in the area covered by a given *Z*_{L} value.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

To determine the cutoff scale for low-pass filtering of the reflectivity field, the ratio of pixels for a future observed reflectivity field that are contained in the area inside of the contour of 2 dB*Z* of the *Z*_{L} field (Fig. 7) is computed. The smallest scale with a ratio higher than a 95% is the selected cutoff scale. This ratio is highly correlated to the temporal autocorrelation of both the *Z*_{L} field and the reflectivity field. Consequently, the scale selected as the cutoff scale for low-pass filtering of the reflectivity field is related to its temporal autocorrelation at a given time. An example of the averaged correspondence between temporal autocorrelation and cutoff scale is shown in Fig. 8. This relationship is different for each event.

Example of the computation of the cutoff scale. The ratio is computed as the total number of pixels (colored shaded field) contained within the area inside the 2-dB*Z* contour of the low-pass-filtered reflectivity field *Z*_{L} (gray contour). The smallest scale with a ratio â‰¥0.95 is the selected scale (72 km in this example).

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Example of the computation of the cutoff scale. The ratio is computed as the total number of pixels (colored shaded field) contained within the area inside the 2-dB*Z* contour of the low-pass-filtered reflectivity field *Z*_{L} (gray contour). The smallest scale with a ratio â‰¥0.95 is the selected scale (72 km in this example).

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Example of the computation of the cutoff scale. The ratio is computed as the total number of pixels (colored shaded field) contained within the area inside the 2-dB*Z* contour of the low-pass-filtered reflectivity field *Z*_{L} (gray contour). The smallest scale with a ratio â‰¥0.95 is the selected scale (72 km in this example).

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Relation between the cutoff scale for the low-pass filter *Z*_{L} and the temporal autocorrelation used in the modified AR(2) model applied in Fig. 10. The horizontal dashed line indicates a correlation of 1/*e*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Relation between the cutoff scale for the low-pass filter *Z*_{L} and the temporal autocorrelation used in the modified AR(2) model applied in Fig. 10. The horizontal dashed line indicates a correlation of 1/*e*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Relation between the cutoff scale for the low-pass filter *Z*_{L} and the temporal autocorrelation used in the modified AR(2) model applied in Fig. 10. The horizontal dashed line indicates a correlation of 1/*e*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Once the scale is determined for a given lead time, we obtain a no-rain probability map (bottom-left map in Fig. 3). This map contains similar information as the *Z*_{L} field. Assuming the decorrelation time is a measure of predictability (as was proposed in GZ02), it can be observed in Fig. 9 that the larger the scale of the system, the more predictable is the reflectivity field. Consequently, these probability maps for different lead times take advantage of this predictability.

Temporal autocorrelation of the reflectivity field at different scales (low-pass filter) from 4 to 512 km for the 18 Apr 2008 event. The horizontal dashed line indicates the decorrelation time or lifetime for 1/*e*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Temporal autocorrelation of the reflectivity field at different scales (low-pass filter) from 4 to 512 km for the 18 Apr 2008 event. The horizontal dashed line indicates the decorrelation time or lifetime for 1/*e*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Temporal autocorrelation of the reflectivity field at different scales (low-pass filter) from 4 to 512 km for the 18 Apr 2008 event. The horizontal dashed line indicates the decorrelation time or lifetime for 1/*e*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

The latest observations have spatial and temporal correlations. To create a set of fields with these properties an autoregressive modeling can be used. In previous works (Pegram and Clothier 2001; Berenguer et al. 2011), a second-order autoregressive model AR(2) (Priestley 1981) has been used to reproduce the temporal correlation at pixel scale. Only two parameters are needed to determine AR(2): the time lag-1 (*r*_{1}) and lag-2 (*r*_{2}) error correlations. It has been showed in Germann et al. (2009), and it is verified in this study (Fig. 10), that the AR(2) model reproduces well the temporal correlation for about 2â€“3 h but it is not able to properly model the tail (from 5 to 10 h) in the temporal correlation function. The lead time selected in this study is 10 h. Consequently, a simple AR(2) model cannot be used to generate future realizations for a 10-h lead time.

Temporal autocorrelation of the original reflectivity field (black solid line), the AR(2) model filtered field (red dashâ€“dotted line), and the CAR(2) model filtered line (blue dashâ€“dotted line). The maximum and minimum values are plotted as light color shaded areas and the interquartile range is the darker colored shaded area. The horizontal dashed line indicates the decorrelation time or lifetime for 1/*e*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Temporal autocorrelation of the original reflectivity field (black solid line), the AR(2) model filtered field (red dashâ€“dotted line), and the CAR(2) model filtered line (blue dashâ€“dotted line). The maximum and minimum values are plotted as light color shaded areas and the interquartile range is the darker colored shaded area. The horizontal dashed line indicates the decorrelation time or lifetime for 1/*e*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Temporal autocorrelation of the original reflectivity field (black solid line), the AR(2) model filtered field (red dashâ€“dotted line), and the CAR(2) model filtered line (blue dashâ€“dotted line). The maximum and minimum values are plotted as light color shaded areas and the interquartile range is the darker colored shaded area. The horizontal dashed line indicates the decorrelation time or lifetime for 1/*e*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

*X*

_{t},

*X*

_{(tâˆ’1)}, and

*X*

_{(tâˆ’2)}stand for the normalized reflectivity field at time

*t*and one and two time steps before, respectively. The quantity

*W*

_{D}is a normalized matrix that provides the rain probability. This matrix is multiplied by a spatially correlated Gaussian distributed noise field (

*Îµ*

_{t}) obtaining a field similar as seen in the middle field in the top row of Fig. 3. The result of this equation is the field plotted on the right side of the first row of Fig. 3.

From this temporal and spatial correlated noise field and the no-rain probability field, a possible rainfall mask can be obtained by the following steps. First compute the noise threshold to reproduce the no-rain probability field from the correlated noise field (second field of the second row in Fig. 3). Second, obtain the factors that reduce all the thresholds to a single value. Map these factors and smooth it spatially in order to avoid discontinuities in this field (third field of the second row in Fig. 3). This normalization factor field is multiplied by the correlated noise field that makes up the rain pixels of the rainfall mask.

The next step is to reproduce the power spectrum (square of the amplitude) of the last observed reflectivity image. Many Fourier analysis books, such as Bochner and Chandrasekharan (1949), report that the correlation and the power spectrum functions provide the same information, but in different spaces.

*C*

_{Z}) or power spectrum

*Z*by means of a generator (

*g*). The convolution of this functionâ€“generator and its transpose (

*g*

^{T}) returns

*C*

_{Z}. A correlated Gaussian random field (

*Y*) can be generated by the following operation:

*N*is a

*M*Ã—

*M*size uncorrelated field (also known as white noise) and â˜… stands for the convolution.

*g*using the following relation:

*Y*field. If the rainfall field is not symmetric, the result of this operation

*g*has been obtained by the inverse Fourier transform, the Gaussian white noise

*N*is converted to a correlated Gaussian random field

*Y*. According to Eulerâ€™s formulation of the Fourier transform, this new correlated Gaussian noise just changes its amplitude spectrum:

*Ï•*

_{N}(

*f*) is the phase of the Fourier transform of Gaussian white noise field

*N*.

To introduce the phase information provided by the rainfall mask, an iterative process is carried out. An example of the process is depicted in Fig. 4. The mask is multiplied by the correlated random field *Y* obtaining a masked correlated random field (*Y*_{m}). This new field does not have the same correlation as the previous one but it has similar phases as the rainfall mask. For this reason, a new generator is created (*g*_{m}) to get a new correlated random field *Y* and the whole process is repeated again. This iterative process is finished when little variations are introduced in the power spectrum slope and in the square mean of the masked correlated field. The number of iterations depends on the lead time and it is a process that is not always convergent to a realistic solution.

The obtained field (*F*_{j}) is one member of the ensemble for a given lead time. The phase and amplitude information provides different properties to this member. The phase information reproduces the temporal correlation of the observation by using the CAR(2) model. Besides, the coverage of the rainfall mask (phase information) is similar to the latest observation. Finally, the shape of the rainfall mask, which is related to the large-scale pattern given by the low-pass-filtered field *Z*_{L}, introduces anisotropy in the fields. The iterative process ensures that the ensemble member has the same power-spectrum slope. Furthermore, the obtained field has the same mean as the Lagrangian extrapolation. Summarizing, the ensemble members are forced to have not only the same power-spectrum slope, but also a similar anisotropy as the Lagrangian forecast. A similar reflectivity probability distribution function is reproduced in the ensemble members. And the observed temporal decorrelation is kept for different lead times in a given ensemble member.

There are conceptual similarities between the introduced technique and two other ensemble generation techniques developed previously (STEPS and SBMcast). The first technique (STEPS) introduces a normal noise field that result in log-normally distributed rain rates. This problem is solved by the introduction of the mask in the iterative method developed in our technique. Furthermore, the authors of STEPS claim that the filtering applied to the data is isotropic and tends to generate noise circular in nature. The creation of a realistic rainfall mask from the no-rain probability associated with the low-pass-filtered rainfall field (*Z*_{L}) helps to create anisotropic fields. The second technique (SBMcast) uses an autoregressive model of second order to reproduce the temporal autocorrelation. Their lead time is 2 h, where the AR(2) model works well; however, our lead time is 10 h and the AR(2) does not reproduce properly the temporal autocorrelation after the third hour. In addition, the transformations applied to the normalized field modify the power spectrum. This effect is not important for the first hours of forecast but it plays a key role in our 10-h lead-time forecast. Both correlation effects are corrected in our technique.

*N*

_{p}is the total number of points of the domain,

*N*

_{m}is the number of ensemble members,

*f*

_{ij}stand for

*i*pixel of the

*j*th ensemble member (

*F*

_{j}), and

*i*.

*N*

_{p}is the total number of points of the domain,

*i*, and

*z*

_{i}is the

*i*pixel value of the observed radar field

*Z*.

## 4. Results

The Lagrangian ensemble generation technique must respect the statistical properties of three radar images in order for the resulting reflectivity field to be appropriate. The first requirement is that each member conserves the slope of the power spectrum in space, which is introduced by the iterative process. This is fulfilled as it is showed in Fig. 11 where the evolution of the observation power-spectrum slope is plotted for the highly predictable 18 April event against the power-spectrum slope for the Lagrangian ensemble forecast. It is observed that the evolution of the slope varies little along the whole episode. Thus, the hypothesis that the slope is maintained is sound. The same analysis was carried out in the low predictable events and similar results were obtained (Fig. 11). Actually, it has a small change at the beginning due to the iterative process. This change is small (<10%) in comparison with the climatological area of slopes that have been computed with the whole dataset of reflectivity fields. This variation is due to the changes in the small scales when masking the noise field. It can be observed in Fig. 12 that the 1D radially averaged power spectrum for the Lagrangian ensemble is increasing the variance for the small scales producing the effect in the slope observed in Fig. 11.

The 1D radially averaged power-spectrum slope as function of the lead time. The black and gray solid lines represent the slope of the actual radar data for the high predictable event of 18 Apr 2008 and the low one of 6 May 2008, respectively. The red shaded lines are obtained by means of the ensemble generation technique developed in this study.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

The 1D radially averaged power-spectrum slope as function of the lead time. The black and gray solid lines represent the slope of the actual radar data for the high predictable event of 18 Apr 2008 and the low one of 6 May 2008, respectively. The red shaded lines are obtained by means of the ensemble generation technique developed in this study.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

The 1D radially averaged power-spectrum slope as function of the lead time. The black and gray solid lines represent the slope of the actual radar data for the high predictable event of 18 Apr 2008 and the low one of 6 May 2008, respectively. The red shaded lines are obtained by means of the ensemble generation technique developed in this study.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

The 1D radially averaged power spectrum as function of wavelength. The black lines represent the power spectrum of the observed radar data for the high predictable event of 18 Apr 2008 (shaded line) and the low one of 6 May 2008 (solid line). The power spectrums of the observed future radar fields are plotted in gray. The power spectrums of the members (for all the lead times) of the Lagrangian ensembles are plotted in blue.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

The 1D radially averaged power spectrum as function of wavelength. The black lines represent the power spectrum of the observed radar data for the high predictable event of 18 Apr 2008 (shaded line) and the low one of 6 May 2008 (solid line). The power spectrums of the observed future radar fields are plotted in gray. The power spectrums of the members (for all the lead times) of the Lagrangian ensembles are plotted in blue.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

The 1D radially averaged power spectrum as function of wavelength. The black lines represent the power spectrum of the observed radar data for the high predictable event of 18 Apr 2008 (shaded line) and the low one of 6 May 2008 (solid line). The power spectrums of the observed future radar fields are plotted in gray. The power spectrums of the members (for all the lead times) of the Lagrangian ensembles are plotted in blue.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Fields with similar power-spectrum slopes are expected to have similar variability (power) distribution among scales. Consequently, the field should have the same number of storms and mesoscale systems, etc. However, the 1D power spectrum is computed by averaging the 2D power spectrum radially, thus losing the anisotropy existent in the field. This anisotropy could be maintained by using information from the predictable low-frequency *Z*_{L} field. In fact, the creation of a rainfall mask from the no-rain probability introduces part of the observed anisotropy. This can be observed in Fig. 13, where the anisotropy is computed as a function of the eccentricity of the correlation ellipse. By construction, the evolution of the anisotropy cannot be reproduced by the Lagrangian forecasts. However, this property of the rainfall field is well reproduced for the highly predictable cases. This is because the low-pass field *Z*_{L} contains more information of the anisotropy in these events. On the other hand, the anisotropy of low predictable events varies rapidly in time and the low-pass field constraint from the Lagrangian extrapolation does not permit the same rate of anisotropy variation in the the perturbed field.

Evolution of the eccentricity of the correlation ellipse for the highly predictable event (14 Apr 2008) and the low one (6 May 2008). Plotted are the actual radar image evolution (solid line), the Lagrangian forecast eccentricity (dashed line), and the ensembles eccentricities (shaded areas). Eccentricity close to 1 indicates high anisotropy, whereas eccentricity near 0 indicates circular-shaped correlations.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Evolution of the eccentricity of the correlation ellipse for the highly predictable event (14 Apr 2008) and the low one (6 May 2008). Plotted are the actual radar image evolution (solid line), the Lagrangian forecast eccentricity (dashed line), and the ensembles eccentricities (shaded areas). Eccentricity close to 1 indicates high anisotropy, whereas eccentricity near 0 indicates circular-shaped correlations.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Evolution of the eccentricity of the correlation ellipse for the highly predictable event (14 Apr 2008) and the low one (6 May 2008). Plotted are the actual radar image evolution (solid line), the Lagrangian forecast eccentricity (dashed line), and the ensembles eccentricities (shaded areas). Eccentricity close to 1 indicates high anisotropy, whereas eccentricity near 0 indicates circular-shaped correlations.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

The 2D spatial decorrelation distance (i.e., the distance where the spatial autocorrelation values is equal to 1/*e*) is plotted in Fig. 14. It can be observed as the anisotropy is mainly maintained in the highly predictable event (Fig. 14a) and the decorrelation distance is reduced for the latest hours of forecast (around 8 h of lead time). On the other hand, the less predictable event (Fig. 14b) does not maintain either the eccentricity or the decorrelation distance from the fourth hour of lead time. This is because the cutoff scale is smaller for the same lead time that in the less predictable event and consequently the mask created from the no-rain probability keeps less information from the large scales. Consequently, the noise is just a correlated field by a filter of a 1D power spectrum slope isotropic function.

Spatial autocorrelation of the reflectivity fields as a function of the displacement. The observed rainfall field autocorrelation is plotted as a solid black contour for the value of 1/*e*. The 2D decorrelation distances for all the ensembles members are plotted as dashed lines. The different colors indicate the lead time of the Lagrangian ensembles.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Spatial autocorrelation of the reflectivity fields as a function of the displacement. The observed rainfall field autocorrelation is plotted as a solid black contour for the value of 1/*e*. The 2D decorrelation distances for all the ensembles members are plotted as dashed lines. The different colors indicate the lead time of the Lagrangian ensembles.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Spatial autocorrelation of the reflectivity fields as a function of the displacement. The observed rainfall field autocorrelation is plotted as a solid black contour for the value of 1/*e*. The 2D decorrelation distances for all the ensembles members are plotted as dashed lines. The different colors indicate the lead time of the Lagrangian ensembles.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

The second requirement is reproducing the observed temporal decorrelation, which is generated by CAR(2) where the spatial probability of rainy areas is introduced. Figure 15 shows the temporal decorrelation of the observations and ensemble members for the highly predictable event (18 April 2008; Fig. 15a) and the less predictable event (6 May 2008; Fig. 15b). The temporal decorrelation spread increases as a function of the lead time.

Temporal autocorrelation of the original reflectivity field (black solid line) and every member of the Lagrangian ensemble forecasted reflectivity field (blue dashed lines) for (a) a highly and (b) a lowly predictable event. The horizontal red dashed line indicates the decorrelation time or lifetime for 1/*e*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Temporal autocorrelation of the original reflectivity field (black solid line) and every member of the Lagrangian ensemble forecasted reflectivity field (blue dashed lines) for (a) a highly and (b) a lowly predictable event. The horizontal red dashed line indicates the decorrelation time or lifetime for 1/*e*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Temporal autocorrelation of the original reflectivity field (black solid line) and every member of the Lagrangian ensemble forecasted reflectivity field (blue dashed lines) for (a) a highly and (b) a lowly predictable event. The horizontal red dashed line indicates the decorrelation time or lifetime for 1/*e*.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

The last property to be maintained is the probability distribution of reflectivity values. Here, the following three metrics are proposed to characterize the probability distribution: the mean of the reflectivity values, the standard deviation, and the ratio of rainfall pixels (fractional coverage). The evolution of these parameters is plotted in Fig. 16 as a function of time for the two cases: one highly predictable event and the other one less predictable one. The temporal evolution of these values for the set of Lagrangian ensembles shows little variation along the forecasting period. Consequently, it can be observed in Fig. 16a that these results are close enough to the actual probability distribution of reflectivity values for the high predictable event. However, the spread of these parameters is larger in the highly predictable event, probably because of the backward semi-Lagrangian scheme applied. Notwithstanding, the actual values fall within the range of values of ensemble Lagrangian forecasts. On the other hand, Fig. 16b shows these parameters for a less predictable event. The variability is lower at the beginning of the event, but it increases drastically at the end of the event due to diurnal heating. Despite the fact that variability is lower, the Lagrangian forecasts do not reproduce this variability and, consequently, they do not fall within the range of values of the ensemble of Lagrangian forecasts.

Temporal evolution of the mean (black), standard deviation (red), and total coverage (blue) of the original reflectivity field (dark solid lines) and ensemble forecast (light solid lines) for (a) a highly and (b) a lowly predictable event.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Temporal evolution of the mean (black), standard deviation (red), and total coverage (blue) of the original reflectivity field (dark solid lines) and ensemble forecast (light solid lines) for (a) a highly and (b) a lowly predictable event.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Temporal evolution of the mean (black), standard deviation (red), and total coverage (blue) of the original reflectivity field (dark solid lines) and ensemble forecast (light solid lines) for (a) a highly and (b) a lowly predictable event.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Once the three properties required to generate the ensemble have been fulfilled, the next step is to verify if the uncertainty introduced is a correct representation of the forecast uncertainty. As mentioned previously, the best way to do this verification is to compare the ensemble spread agains the RMSE of ensemble mean. This comparison is showed in function of the lead time in Fig. 17. It can be observed that during the whole forecasting period both measurements are similar. The maximum separation is around a 10% and this separation is increasing with the lead time. Consequently, it can be said that the uncertainty is well reproduced for the first 4â€“6 h (depending on the event) and it is underestimated from the sixth hour for both less predictable events. This underestimation can be caused by the introduction of the no-rain probability from the *Z*_{L} field. In the 22 July event, the last separation could be caused by the presence of the diurnal cycle at the end of the forecast period. This effect increases the rainfall in the whole domain. However, the introduced technique for generating nowcasting ensembles does not modify the mean (among other properties) from the latest observation. Consequently, this effect is not taken into account as an uncertainty and the ensemble spread is underestimated.

Ensemble spread and RMSE of the ensemble mean in function of the lead time for the four events listed.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Ensemble spread and RMSE of the ensemble mean in function of the lead time for the four events listed.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Ensemble spread and RMSE of the ensemble mean in function of the lead time for the four events listed.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Finally, an example of three members of the ensemble nowcast is shown in Fig. 18. It can be observed that the shape of the front (consequently, the anisotropy of the correlation function) is kept for the whole forecasting period. Besides, the internal structure, even though is becoming noisy as a function of the lead time, seems realistic (as the power-spectrum function has shown).

Examples of Lagrangian ensemble forecasts for (left to right) 1-, 4-, and 9-h lead times. (top) The observed reflectivity field and (second, third, and bottom) the different ensemble members.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Examples of Lagrangian ensemble forecasts for (left to right) 1-, 4-, and 9-h lead times. (top) The observed reflectivity field and (second, third, and bottom) the different ensemble members.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

Examples of Lagrangian ensemble forecasts for (left to right) 1-, 4-, and 9-h lead times. (top) The observed reflectivity field and (second, third, and bottom) the different ensemble members.

Citation: Monthly Weather Review 142, 11; 10.1175/MWR-D-13-00117.1

## 5. Conclusions

An ensemble-generation nowcasting technique has been introduced and tested. It is based on the stochastic perturbation of Lagrangian extrapolation of the last observed rainfall field to introduce the uncertainty associated with the growth and decay of the precipitation field.

This study demonstrates that the Lagrangian-based technique, where an iterative process to reproduce the last observation power spectrum slope is applied, is able to reproduce the spatial structure of the rainfall field. Besides, the introduction of a modification in an autoregressive model of order two procreates the temporal autocorrelation of the observed rainfall field. Consequently, the space and time structures of the rainfall field are preserved by the future realizations obtained by adding the uncertainty to the deterministic Lagrangian extrapolation. This is a prerequisite that had to be fulfilled for the members to be suitable to be coupled with distributed hydrologic models.

However, other properties that vary during the episode, such as the anisotropy or the coverage of the field, are introduced but not well reproduced. This is because these properties are not forecast either from an empirical model or from a physical model. Consequently, they evolve with a determined uncertainty (in the case of the anisotropy) or they are kept constant (small variations are observed) during the forecast period for quantities such as coverage or mean value of the rainfall field.

There are conceptual similarities between the introduced technique and two other ensemble generation techniques developed previously (STEPS and SBMcast). These similarities are related to the properties maintained in the creation of the ensemble members such as the power-spectrum slope and the temporal correlation. However, the previous methodologies keep some of these properties only for the first hours of forecast. The present methodology introduces changes to these techniques to preserve the mentioned properties for the 10-h forecast. Simultaneously, our technique reproduces other properties, such as the anisotropy of the rainfall fields. On the other hand, STEPS and SBMcast techniques have benefits that are omitted in our methodology. For instance, SBMcast model introduces the forecast of quantities that are kept constant in our methodology. Consequently, features from the other two methodologies could be introduced in future work.

It can be concluded that the developed technique reproduce realistic ensemble members. Besides, these forecasted realistic rainfall realizations properly represent the forecast uncertainty. It had been observed that for the highly predictable events the ensemble mean RMSE grows in a similar way as the ensemble spread. Regarding the less predictable events, the uncertainty is well reproduced for the first 6 h and it is underestimated afterward.

## Acknowledgments

The authors express their gratitude to the Global Hydrology and Climate Center (GHRC) for providing access to the WSI radar composites data. Special thanks go to all our group members for their fruitful discussions during the group meetings and especially to Dr. Radhakrishna Basivi, Madalina Surcel, and Georgina Paull for their corrections and suggestions to improve the manuscript. The comments of two anonymous reviewers and Prof. Geoff Pegram also substantially improved our presentation.

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