1. Introduction
As shown by Lorenz (1963), two solutions of a system of nonlinear differential equations that differ only slightly in the specification of the initial conditions will diverge with time until they become as similar as two random states. This is an intrinsic property of nonlinear deterministic systems and leads to deterministic chaos. As the atmospheric processes are considered nonlinear deterministic processes, the results of Lorenz (1963) impose a finite limit on the intrinsic predictability of the atmosphere, even if a perfect model for prediction existed and if the initial condition errors were much smaller than those currently used in atmospheric models (Lorenz 1996). Furthermore, this finite limit is highly dependent on the initial conditions, and an excellent illustration of this behavior for the Lorenz system is provided in Fig. 4 of Palmer (1993). In practice, the atmospheric state is predicted using an imperfect model initialized with initial conditions that contain considerable errors. Therefore, the practical predictability of the atmosphere (defined as the extent to which prediction is possible given current forecasting methods; Lorenz 1996) is affected by, but not equivalent to, the intrinsic predictability of the atmosphere. Moreover, estimates of the practical atmospheric predictability are highly dependent on the model used for prediction and on the initial state (just as for the Lorenz system). These considerations have made evident the need to assess the uncertainty associated with a given forecast, which can be done through ensemble forecasting. Ensemble forecasting techniques for medium- to long-range weather prediction in midlatitudes, which is mostly affected by synoptic-scale baroclinic instabilities, are well established by now (Kalnay 2003, section 6.5). On the other hand, the mesoscale details of weather, such as the evolution of precipitation systems, are affected by moist convective processes. Since computational resources have become available in the early 2000s to allow the simulation of mesoscale phenomena with high horizontal grid spacing, significant effort has been devoted to understanding the processes responsible for error growth in such models. The various studies on error growth at the mesoscale (e.g., Walser et al. 2004; Zhang et al. 2002, 2003, 2006, 2007; Hohenegger and Schar 2007a,b; Bei and Zhang 2007; Melhauser and Zhang 2012; Wu et al. 2013) indicated that moist convection is the primary mechanism promoting the growth of small initial-condition errors. Moreover, it has been shown that small errors saturate at the convective scale and then grow upscale through geostrophic adjustment or cold pool dynamics to limit the predictability at the mesoscale within the time of interest of such forecasts (about 24 h), resulting in error growth rates for convection-allowing models being much higher than for large-scale models (Hohenegger and Schar 2007a). However, just as for the very simple Lorenz system, the exact limit of predictability is highly case dependent (Done et al. 2012; Hohenegger et al. 2006; Walser et al. 2004). Moreover, Hohenegger et al. (2006) showed that even cases with apparently similar intensity of moist convection might exhibit different predictability depending on the relation between the moist convection and the larger-scale flow.
Given these considerations, the formulation of proper ensemble techniques at the storm scale remains very difficult (Johnson et al. 2014). A significant effort in developing ensemble forecasting strategies at convection-allowing resolutions is represented by the National Oceanic and Atmospheric Administration (NOAA) Hazardous Weather Testbed (HWT) Spring Experiments, which have been taking place every spring since 20071 (http://hwt.nssl.noaa.gov/spring_experiment/). As part of this experiment, ensemble forecasts were produced for 30–48 h at very high resolution (dx = 4 km) using the storm-scale ensemble forecasting (SSEF) system developed at the Center for the Analysis and Prediction of Storms (CAPS; Xue et al. 2008; Kong et al. 2008). While the ensemble configuration has changed throughout the years, each year a set of members with perturbed initial conditions (IC) and lateral boundary conditions (LBC), different model physics, and mesoscale data assimilation (DA; including radar) was produced. The ICs and LBCs for these members were derived from the operational short-range ensemble forecast (SREF) system run at the National Centers for Environmental Prediction (NCEP; Du et al. 2009). SREF forecasts are produced with grid spacing of 32–45 km (depending on the member), and thus, the IC–LBC perturbations do not include information at very small scales. A recent study by Johnson et al. (2014) compared the effect of this type of perturbation to smaller-scale IC–LBC perturbations. It was found that the relative importance of the two types of errors is case dependent, but that on average, the small-scale perturbations are less important than the larger-scale errors for precipitation forecasts at medium and large scales (64–4096 km in their study). Furthermore, they concluded that the current CAPS SSEF configuration samples the primary sources of error. However, the evaluation by Clark et al. (2011) of ensemble quantitative precipitation forecasts (QPFs) from the 2009 Spring Experiment showed that the same type of ensemble is underdispersive for lead times between 6 and 18 h. On the other hand, by investigating the filtering effect of ensemble averaging for precipitation forecasts from the 2008 Spring Experiment, Surcel et al. (2014) indicated that QPFs from the ensemble members become fully decorrelated at larger scales with increasing lead time.
The objective of this paper is to further investigate the scale dependence of precipitation predictability by the 2008 CAPS SSEF by extending the analysis of Surcel et al. (2014) to determine how the range of scales over which the ensemble QPFs are fully decorrelated evolves with forecast lead time for 22 cases during spring 2008. As mentioned by Surcel et al. (2014), the complete decorrelation of the ensemble forecasts can be regarded as a lack of predictability of precipitation patterns by the ensemble at those scales. As the purpose of ensemble forecasts is to provide information on the uncertainty in the forecast, it is desirable to compare the predictability by the ensemble to the actual ability of the ensemble members to forecast precipitation at those scales as quantified by comparison to observations.
Therefore, in this paper we will provide quantitative estimates of the loss of precipitation predictability with spatial scale and forecast lead time by particular NWP model configurations. These estimates will be obtained in two ways: (i) by analyzing the difference between forecasts from the CAPS ensemble and (ii) by comparing forecasts from the CAPS SSEF to observations. As the CAPS SSEF has both IC–LBC perturbations derived from a regional-scale ensemble and varied model physics, both estimates correspond to the loss of the “practical predictability” of precipitation, as defined by Lorenz (1996) and discussed by Zhang et al. (2006). As explained in previous work by Zhang et al. (2006), Bei and Zhang (2007), and Melhauser and Zhang (2012), the practical predictability is influenced by the intrinsic predictability of the atmosphere, that is, to the growth of small IC errors due to the chaotic nature of the atmosphere. While understanding how the intrinsic predictability of precipitation for our dataset affects the estimates of predictability loss that we obtain is very important, it is outside the scope of the current study and is left for future work. The objective of this paper is to present quantitative estimates of the loss of practical predictability and to intercompare the estimates obtained through the two methods discussed above for a dataset consisting of 22 precipitation cases during spring 2008. Therefore, to facilitate the discussion of the comparison presented in this paper, we will refer to the loss of practical predictability as estimated from the differences among forecasts from the ensemble as the loss of “the predictability of the model state.” This depends on the numerical weather prediction (NWP) model, the method of generating the ensemble forecasts, and the metric used to quantify the variability within the members. Herein, the predictability of the model state will be estimated by the decorrelation scale corresponding to QPFs from the CAPS SSEF. On the other hand, when estimates of the loss of practical predictability are obtained by comparing the outputs of NWP models or of other forecasting methods to observations, we will refer to them as estimates of “the model predictability of the atmospheric state” (in this paper, we assume that observations closely describe the atmospheric state, although in section 4d we provide an assessment of the effect of this assumption on our results). These estimates of predictability depend both on the prediction model and on the particular metric of model–observation comparison. The model predictability of the atmospheric state will be quantified by the decorrelation scale between precipitation forecasts and precipitation observations. Previous verification studies of precipitation forecasts have reported that forecasting skill shows scale dependence (Casati et al. 2004; Gilleland et al. 2009; Roberts and Lean 2008; Surcel et al. 2014), with a loss of useful skill at larger scales with increasing forecast lead time (Germann et al. 2006; Roberts 2008). Therefore, we aim to determine how the range of scales with a complete lack of skill (no model predictability of the atmospheric state) compares to the range of scales lacking predictability of the modeled precipitation for the 2008 CAPS SSEF. Ideally, the predictability of the model state and the model predictability of the atmospheric state should be consistent with each other when estimated over sufficiently large datasets. That is, it is desirable, on average, for small ensemble spread to correspond to good forecast skill, and vice versa. Therefore, our results could provide insight on whether the perturbations currently employed in the CAPS SSEF are sufficient to represent uncertainty as a function of scale.
In addition to dynamical models for precipitation forecasting, statistical models are still often employed for the very-short-term prediction of precipitation (nowcasting). The simplest one is Eulerian persistence (EP), which simply assumes no evolution of the current state (in this case, of the precipitation field). Evidently, this is a very poor model, as a cursory examination of radar imagery demonstrates that rainfall is neither stationary nor steady. However, the EP model remains a baseline for validating predictions from more complex models.
Another more appropriate statistical method for short-term precipitation forecasting is Lagrangian persistence (LP), which assumes persistence in the reference frame of the moving precipitation system. Therefore, to obtain a forecast, it suffices to characterize the motion of the precipitation patterns in the immediate past and to extrapolate the current precipitation field using this motion field. This method is known to work well for very-short-term (0–6 h) precipitation forecasting when used in radar-based extrapolation algorithms (Berenguer et al. 2012; Lin et al. 2005; Turner et al. 2004). Furthermore, LP precipitation forecasts were found to outperform the deterministic forecasts from the CAPS SSEF radar data–assimilating members for about 3 h (Berenguer et al. 2012). To complement the results of their analysis, we will also investigate here how the scale dependence of the lack of predictability of rainfall compares between the radar data–assimilating members and LP forecasting.
Our study will show the potential of using the decorrelation scale introduced by Surcel et al. (2014) as an evaluation metric for quantifying the scale dependence of the predictability of precipitation patterns. Furthermore, it will offer a quantitative estimate of the loss of predictability of precipitation by both dynamical and statistical forecasting methods, as a function of scale and forecast lead time for a set of 22 cases during spring 2008. By analyzing the scale dependence of precipitation predictability for a set of cases, rather than adopting a case study approach, we attempt to generalize and thus complement the results obtained by previous predictability studies. The results of our study are also relevant to forecasting applications consisting of postprocessing model output, such as some ensemble averaging and blending applications (Atencia et al. 2010; Ebert 2001; Kober et al. 2012), as it indicates which components of the very detailed two-dimensional picture provided by the model do not contain useful information and therefore should not be used in such applications.
The paper is organized as follows. Section 2 describes the precipitation forecasts used in the analysis. Section 3 explains the methodology used to derive the decorrelation scale. Section 4 presents the results. Section 5 offers a discussion on the predictability of precipitation at the mesoscale and suggestions for future work, and section 6 presents the conclusion.
2. Data
Both precipitation forecasts and precipitation observations are used in the study, and they are described next. All the forecasts and observations have been remapped on a common grid using a nearest-neighbor interpolation method, and the analysis is performed on a domain covering most of the central and eastern United States, extending from 32° to 45°N and from 103° to 78°W, as illustrated in Fig. 1. The dataset consists of 22 precipitation cases from 18 April to 6 June 2008. Both hourly rainfall accumulations fields and instantaneous reflectivity fields (simulated or observed) were available for each dataset, and the entire analysis has been performed on both types of fields, with consistent results. However, in this paper, only the results corresponding to hourly rainfall accumulation fields are presented.
Analysis domain. The red contours represent the coverage of the 2.5-km constant-altitude plan position indicator (CAPPI) maps, while the rectangle ranging from 32° to 45°N and from 103° to 78°W corresponds to the analysis domain. The precipitation field presented in this figure is typical for spring 2008.
Citation: Journal of the Atmospheric Sciences 72, 1; 10.1175/JAS-D-14-0071.1
a. Precipitation forecasts
1) CAPS SSEF forecasts
The SSEF system was developed at CAPS, and it was run during the 2008 NOAA HWT Spring Experiment (Xue et al. 2008; Kong et al. 2008). It uses the Advanced Research version of the Weather Research and Forecasting (WRF-ARW) Model (Skamarock et al. 2008), version 2.2, and consists of 10 members with different physical schemes, mesoscale data assimilation including radar in 9 out of the 10 members, and perturbed ICs and LBCs. The background ICs are interpolated from the North American Model (NAM; Janjić 2003) 12-km analysis, and the IC–LBC perturbations are directly obtained from the SREF system run operationally at NCEP (Du et al. 2009). The SREF members are based on different dynamic cores [Eta, WRF Nonhydrostatic Mesoscale Model (WRF-NMM), and WRF-ARW] and are run with grid spacing of 32 or 45 km. Therefore, the IC–LBC perturbations do not have variability at the scale at which the 4-km members are run. In addition to IC–LBC perturbations, the ensemble members have different microphysical schemes varying among Thompson (Thompson et al. 2004), WRF single-moment 6-class (WSM6; Hong and Lim 2006), and Ferrier (Ferrier et al. 2002); different planetary boundary layer (PBL) schemes varying between Mellor–Yamada–Janjić (Mellor and Yamada 1982; Janjić 2001) and Yonsei University (YSU; Noh et al. 2003); and different shortwave radiation schemes varying between Goddard (Tao et al. 2003) and Dudhia (1989). Thirty-hour forecasts on a 4-km grid were performed almost daily in April–June 2008. Two of the members (control members C0 and CN) do not have SREF-based IC–LBC perturbations and have identical model configurations. However, convective-scale observations from radar [from the Weather Surveillance Radar-1988 Doppler (WSR-88D) network] and surface stations are assimilated only within CN. The assimilation of mesoscale observations was performed using the Advanced Regional Prediction System (ARPS) three-dimensional variational data assimilation (3DVAR) and cloud analysis package (Gao et al. 2004; Hu et al. 2006a,b; Xue et al. 2003). Radar reflectivity, surface data, and visible and 10.5-μm infrared data from the Geostationary Operational Environmental Satellite (GOES) were processed by the cloud analysis scheme to retrieve hydrometeor information. Radar radial velocity data and data from the Oklahoma Mesonet, METAR, and wind profiler networks were assimilated with the ARPS 3DVAR (Johnson et al. 2014).
2) MAPLE LP forecasts
The LP forecasts analyzed in this paper were produced with the McGill Algorithm of Precipitation Forecasting by Lagrangian Extrapolation (MAPLE; Germann and Zawadzki 2002). These are very-short-term precipitation forecasts produced using an extrapolation-based technique that employs the variational echo tracking (VET) algorithm (Laroche and Zawadzki 1995) to estimate the motion field of precipitation and a modified semi-Lagrangian backward scheme for advection. MAPLE was run using the National Severe Storms Laboratory (NSSL) 2.5-km height rainfall maps described below to generate 8-h forecasts initialized every hour with a temporal resolution of 15 min. For the analysis of hourly rainfall accumulations, maps of radar reflectivity Z are converted into rain rate R according to Z = 300R1.5, and then instantaneous reflectivity maps every 15 min are averaged to obtain radar-derived hourly rainfall accumulations.
b. Precipitation observations
The precipitation observations used in this study are U.S. radar reflectivity mosaics at 2.5-km altitude generated by NSSL (Zhang et al. 2005) every 5 min and mapped with a spatial resolution of 1 km. For the analysis of hourly rainfall accumulations, observed reflectivity maps every 15 min have been processed to obtain maps of hourly rainfall accumulations as in the case of MAPLE.
3. Methodology
(a) The power ratio for hourly accumulations from the nine-member ensemble at 1000 UTC 24 Apr 2008 (10-h lead time). The red filled circle indicates the scale (76 km). (b) Decorrelation scale as a function of forecast lead time for 24 Apr 2008.
Citation: Journal of the Atmospheric Sciences 72, 1; 10.1175/JAS-D-14-0071.1
This methodology can be applied for any number of precipitation fields, even though the smaller the number of precipitation fields, the noisier the 
The decorrelation scale 
The decorrelation scale could similarly be obtained by investigating the range of scales over which two forecasts are decorrelated. However, that would involve decomposing the precipitation fields in different scale components using decomposition methods such as DCT or FFT or the Haar wavelet transform (Casati et al. 2004; Germann et al. 2006; Johnson et al. 2014). DCT or FFT bandpass-filtered rainfall fields are strongly affected by the Gibbs effect and thus would make such computations impossible. Haar filtering is less prone to Gibbs effects (Turner et al. 2004), but the Haar transform imposes a coarse sampling in scale. Predictability estimates for LP precipitation forecasts were obtained in this way by Germann et al. (2006), and we will discuss the comparison to their results in section 4c.
4. Results
This section presents the results obtained by applying the above methodology to precipitation forecasts and observations from 22 cases during spring 2008. This set of cases has been previously analyzed by Surcel et al. (2010) and Berenguer et al. (2012), who have shown that the period was dominated by large-scale precipitation systems that nonetheless exhibited a marked diurnal cycle. Figures 3 and 4 illustrate the 22 cases in terms of the evolution of the power spectra of hourly rainfall fields derived from radar and in terms of radar-derived total accumulations for the entire time period (30 h). As shown by the power spectra (left panels), most cases show a clear diurnal cycle in the evolution of the statistical properties of the precipitation fields, with variance decreasing at all scales with time from 0000 UTC, reaching a minimum during early afternoon, and then beginning to increase again through the evening. This diurnal signal can affect evaluation metrics, as evident in Johnson et al. (2014, their Figs. 6, 8, and 11), and can make it difficult to analyze the evolution of skill with lead time. This problem is avoided in our analysis, as the decorrelation scale is computed from a power ratio, thus removing the effect of the large changes in the variance of a precipitation field with the time of day. The results presented in this section are usually averaged over the entire dataset, but the case-to-case variability is also addressed wherever relevant.
Temporal evolution of the power spectrum of (left) observed hourly rainfall accumulations and (right) total rainfall accumulations for each case between 23 Apr and 21 May 2008. All the evaluations described in the text are performed on the domain illustrated in this figure.
Citation: Journal of the Atmospheric Sciences 72, 1; 10.1175/JAS-D-14-0071.1
As in Fig. 3, except from 22 May to 6 Jun 2008.
Citation: Journal of the Atmospheric Sciences 72, 1; 10.1175/JAS-D-14-0071.1
a. Predictability of the model state for the CAPS SSEF
As mentioned in the introduction, by predictability of the model state, we mean the extent to which forecasts from models with slight differences in the model formulation and in the IC–LBCs resemble each other. Usually, the predictability of the model state is characterized in terms of the ensemble spread. In this sense, computing the decorrelation scale for the entire ensemble is equivalent to determining the range of scales over which the ensemble has as much spread as that of an ensemble of random precipitation fields.
(a) Power ratios for the nine-member ensemble averaged over the 22 cases. The colored lines represent the different lead times from the beginning (T = 0 h, blue) to the end (T = 29 h, red) of the forecast. (b) The decorrelation scale as a function of forecast lead time. The black line resulted from averaging the daily 
Citation: Journal of the Atmospheric Sciences 72, 1; 10.1175/JAS-D-14-0071.1
Example of hourly precipitation forecasts and observations (left) on a large subdomain (1300 km × 1300 km) and (right) on a small subdomain (300 km × 300 km) to illustrate the difference between the forecasts as a function of spatial scale.
Citation: Journal of the Atmospheric Sciences 72, 1; 10.1175/JAS-D-14-0071.1
(a) The decorrelation scale as a function of forecast lead time for the entire ensemble (black) and for two pairs of models: CN–N2 (blue, IC–LBC perturbations) and CN–C0 (orange, radar DA) averaged over all 22 cases. The best power-law fits for each of the three thick lines are on the bottom-right side of the graph. The shading represents the variability around the mean values of 
Citation: Journal of the Atmospheric Sciences 72, 1; 10.1175/JAS-D-14-0071.1
On the other hand, the decorrelation scale between CN and C0 is constant throughout the forecast period. This is particularly clear in Fig. 7c, which shows the scatterplot between 
This subsection only deals with the effect of certain sources of error on the forecasts themselves, not on forecasting skill. The next section discusses the SSEF’s predictability of the atmospheric state as evaluated against radar rainfall estimates.
b. CAPS SSEF’s model predictability of the atmospheric state
(a) The decorrelation scale as function of forecast lead time for the radar and each of the ensemble members (gray lines) averaged over all cases. The black line represents the mean over all ensemble members. The gray shading shows the variability around the mean (
Citation: Journal of the Atmospheric Sciences 72, 1; 10.1175/JAS-D-14-0071.1
To better characterize the effect of assimilating radar observations, Fig. 8b shows 
Our results are comparable to Roberts (2008), who found that even at the beginning of the forecast, useful skill is exhibited only at scales larger than 100 km for forecasts of localized rainfall. This result appears grim in terms of the ability of convection-allowing models to predict precipitation. However, this metric does not measure the actual predictive skill that these models have at scales where they exhibit some predictability. In fact, our results suggest that to properly evaluate the importance of radar data assimilation for model skill, all features in the precipitation field occurring at scales lower than 
c. Predictability by EP and LP
(a) Power ratios for EP–radar for hourly rainfall accumulations for all different lead times (0–7 h, colors), averaged over all 22 cases for forecasts initialized at 0000 UTC. (b) The decorrelation scale averaged over all cases (black line), power-law fit to the average 
Citation: Journal of the Atmospheric Sciences 72, 1; 10.1175/JAS-D-14-0071.1
A better model for precipitation nowcasting is LP. Radar-based extrapolation algorithms using this principle are commonly used for very-short-term forecasting (0–6 h). Figure 10 shows the power ratios as a function of scale (Fig. 10a) and the decorrelation scale as a function of lead time (Fig. 10b) averaged over all cases for hourly accumulation forecasts initialized at 0000 UTC and produced by MAPLE. The decorrelation scale is increasing with lead time for LP forecasts as well, following a power law 
As in Fig. 9, but for LP (MAPLE) forecasts.
Citation: Journal of the Atmospheric Sciences 72, 1; 10.1175/JAS-D-14-0071.1
Figure 10c shows 
The predictability of precipitation by LP has been thoroughly studied by Germann and Zawadzki (2002, 2004) and Germann et al. (2006). Despite using a different methodology, Germann et al. (2006) nonetheless obtained results similar to ours, with predictability estimates of about 3 h for scales of O(100) km, as quantified in terms of the lifetime of bandpass scales.
d. The effect of observational uncertainty on predictability estimates
The entire analysis presented in this paper is based on the comparison of forecasts to a particular set of radar-derived quantitative precipitation estimates (QPEs) described in section 2b. While this verification dataset has been chosen for its quality as mentioned by Surcel et al. (2010), it is still reasonable to question how the uncertainty of these products affects the estimates of the decorrelation scale. Radar QPE is affected by many sources of error, and a proper error characterization is complicated, as shown by Berenguer and Zawadzki (2008, and references therein). Therefore, rather than attempting to characterize the error of the radar QPEs used here, we investigate the effect of observational uncertainty simply by comparing our verification dataset to another set of rainfall estimates. The additional verification dataset is NCEP’s Stage IV multisensor precipitation product (Baldwin and Mitchell 1997), available as hourly rainfall accumulations on a 4-km polar stereographic grid. The Stage IV data were obtained from the NCEP website (www.emc.ncep.noaa.gov/mmb/ylin/pcpanl/) and were remapped on the analysis grid (section 2) using nearest-neighbor interpolation. The two precipitation datasets are compared in terms of the decorrelation scale.
Figure 11a shows the power ratios 
(a) Power ratios for radar–Stage IV averaged over the 22 cases. The colored lines represent the different times every hour starting at 0000 UTC. (b) The decorrelation scale as a function of forecast lead time. The black line resulted from averaging the daily 
Citation: Journal of the Atmospheric Sciences 72, 1; 10.1175/JAS-D-14-0071.1
5. Discussion on the comparison between various forecast methods
Figure 12a summarizes the results previously presented about the predictability of precipitation for lead times of 0–30 h. This figure shows 
Summary of the comparison of the different types of predictability (a) over a 30-h forecast period and (b) only for very short lead times, illustrated in terms of 
Citation: Journal of the Atmospheric Sciences 72, 1; 10.1175/JAS-D-14-0071.1
The region in Fig. 12a between the two lines represents scales where there is some predictability of the modeled state, but no predictability of the atmospheric state by NWP. Ideally, the two lines should lie together, as the truth, approximated here by observations, should be a sample of the ensemble probability density function. We have explained in the previous section that the evolution of 
Figure 12b focuses on the time range (0–6 h) of very-short-term precipitation forecasts and adds the decorrelation scale curves for LP and EP. This figure shows that after the first two hours, no forecasting method exhibits any predictability as characterized with respect to radar observations at meso-γ and meso-β scales. Furthermore, while at the beginning of the forecast period, the decorrelation scale corresponding to LP is lower than that of the model, with the crossover time being 3 h, both forecasting methods (NWP and LP) are very similar during this time. Also, both the CAPS SSEF and the MAPLE LP algorithm exhibit better predictability than simple EP in terms of the scales that they can predict. It is therefore clear that EP is no longer necessary as a baseline for evaluating precipitation forecasts from LP algorithms or from radar data–assimilating models, as these other methods consistently outperform EP.
The dashed lines in the two figures illustrate the effect of the uncertainty in our verification data on the results, which is exhibited at scales smaller than 12 km. As this scale is very low compared to the model predictability limits, there is a large range of scales over which improvement is necessary before reaching this “observational” limit.
6. Conclusions
This paper builds on the results of Surcel et al. (2014) to propose and use a methodology for analyzing the scale dependence of the predictability of precipitation fields over the continental United States during spring 2008 by various forecasting methods. There have been many efforts to understand mesoscale predictability in the past few decades, and our study contributes to this by
offering a quantitative measure of the evolution of the decorrelation scale, and hence of the range of scales at which a given method exhibits a lack of predictability, with forecast lead time;
computing this measure for precipitation forecasts and observations for a dataset of a reasonable size (22 cases during spring 2008), rather than for only a few cases, thus verifying and complementing the results obtained by previous predictability studies that used a case study approach (Walser et al. 2004; Zhang et al. 2002, 2003, 2006, 2007; Bei and Zhang 2007);
using the decorrelation scale to intercompare the predictability of the model state to the model predictability of the atmospheric state, hence providing a measure of ensemble consistency as a function of scale for a storm-scale ensemble; and
intercomparing the predictive ability of statistical and dynamical methods for short-term precipitation forecasting as a function of scale.
Our results show that for all forecasting methods there is a range of scales over which the method displays a complete lack of predictability of the atmospheric state, the upper limit of which is the decorrelation scale 
The comparison among EP, LP forecasts, and radar DA models, meant to complement the study of Berenguer et al. (2012), confirms that, given the better performance shown by radar DA models, a better baseline for model evaluation would be LP rather than EP.
On the other hand, we found that the uncertainties currently accounted for in the CAPS SSEF appear to not generate sufficient spread at forecast hours less than 18 h at meso-β scales, demonstrated by the difference in 
The predictabilities discussed here correspond to what Lorenz (1996) and later studies (e.g., Zhang et al. 2006; Bei and Zhang 2007; Melhauser and Zhang 2012) refer to as practical predictability. For example, the value of the decorrelation scale between forecasts and observations is more likely due to large IC and model errors than to the amplification of small IC errors through nonlinear dynamics. In the case of the decorrelation scale between the ensemble members as well, the initial perturbations were derived from a regional-scale ensemble, and it can therefore be expected that the predictability limit for this ensemble might be different than that of an ensemble that samples only very small IC errors. The intrinsic predictability of the atmosphere would have an effect on the estimates of practical predictability if the model captures the appropriate nonlinear dynamics. For example, cases with intense moist convection (highly nonlinear processes) are usually more unpredictable from a practical point of view as well. On the other hand, even for cases that exhibit strong intrinsic predictability, it is possible that model deficiencies and analysis errors might lead to poor forecast results. Investigating the intrinsic predictability for our set of cases would have demanded setting up additional experiments, thus needing significant computational resources, and is therefore better suited for future work. However, the results that we have obtained qualitatively agree with those of Bei and Zhang (2007) and Melhauser and Zhang (2012), who looked at the relationship between practical and intrinsic predictability for two case studies. By reducing the IC errors considered in their ensemble simulations, they noted linear gains in predictability, but they also found that the effect of moist convective processes on error growth sets an inherent predictability limit at the mesoscale.
Our study provides a quantitative estimate of the range of spatial scale over which the very detailed information that a forecasting method can provide is in fact unpredictable given the errors both in the modeling approach and in the initial conditions. However, this decorrelation scale focuses on the agreement between entire two-dimensional precipitation maps, and therefore it is sensitive to displacement errors. In an operational setting, forecasters might still find useful the information provided by a model in terms of storm characteristics at scales lower than 200 km, and our methodology does not account for these cases. However, our methodology is useful for the many applications that use all of the information in a two-dimensional QPF, such as blending applications, ensemble averaging, or hydrological modeling. For these applications, it is useful to know that scales lower than 
Finally, we remind the reader that the results presented in this study are dependent on the forecasting systems under study. We are in the process of extending the methodology herein to the Spring Experiment data of 2009–13. Our preliminary results indicate that the sensitivity to the IC–LBC perturbations analyzed here is consistent from year to year and that the decorrelation scale shows great sensitivity to the type of perturbations; that is, different errors propagate differently upscale. A paper describing these new findings is in preparation.
Acknowledgments
We are greatly indebted to Ming Xue and Fanyou Kong from CAPS for providing us the ensemble precipitation forecasts. The CAPS SSEF forecasts were produced mainly under the support of a grant from the NOAA CSTAR program, and the 2008 ensemble forecasts were produced at the Pittsburgh Supercomputer Center. Kevin Thomas, Jidong Gao, Keith Brewster, and Yunheng Wang of CAPS made significant contributions to the forecasting efforts. M. Surcel acknowledges the support received from the Fonds de Recherche du Québec–Nature et Technologies (FRQNT) in the form of a graduate scholarship. This work was also funded by the Natural Science and Engineering Research Council of Canada (NSERC) and Hydro-Quebec through the IRC program. We acknowledge the comments and suggestions of three anonymous reviewers that helped improve the paper.
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In fact, the NOAA HWT Spring Experiments have been taking place since 2000, but ensemble forecasts have been produced as part of the experiments only since 2007.
