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  • View in gallery

    Summary of the comparison of the different types of predictability over a 30-h forecast period illustrated in terms of vs t. The model predictability of the atmospheric state is characterized by computed between radar and radar-data-assimilating (radar DA) models and between radar and the nonradar DA C0. The predictability of the model state is characterized by computed between CN and radar DA models; was computed for instantaneous reflectivity fields with hourly resolution. The different color curves correspond to the different types of predictability as described in the legend. The thin lines correspond to the individual cases, while the thick lines represent averages over all cases.

  • View in gallery

    The predictability of precipitation for spring 2008. Each color line represents each of the 22 cases, while the thick black line corresponds to the average over all the cases. The different panels show the decorrelation scales as a function of forecast lead-time corresponding to (a) the entire ensemble, (c) Radar-CN, and (e) Radar-C0. (b) The normalized ensemble spread for the filtered fields, and the filtered corresponding to (d) Radar-CN and (f) Radar-C0.

  • View in gallery

    Scatterplots of the decorrelation scale vs and for low-pass filtered precipitation fields corresponding to (a) the ensemble, (b) CN-Radar, and (c) C0-Radar at different lead times (see legend for corresponding colors). The linear correlation coefficients between the different variables are shown in parentheses. The correlation coefficients are presented in bold lettering when statistically significant at the level.

  • View in gallery

    corresponding to CN-Radar vs normalized ensemble spread for low-pass filtered fields for different lead times as described in the legend. The linear correlation coefficients between the different variables are shown in parentheses. The correlation coefficients are presented in bold lettering when statistically significant at the level.

  • View in gallery

    The (a) convective-adjustment time scale and (b) CAPE averaged over all areas with rainfall higher than 1 mm h−1 as a function of forecast lead time in hours after 0000 UTC for each of the 22 cases as indicated in the legend.

  • View in gallery

    Time series of (see text for details) for different times: 0000 UTC (0-h forecast, black), 0600 UTC (6-h forecast, blue), 1200 UTC (12-h forecast, green), 1800 UTC (18-h forecast, orange), and 0000 UTC (24-h forecast, red) between 18 Apr and 25 Aug 2008. The more negative the value the stronger the large-scale forcing, while the more positive the value, the weaker the forcing.

  • View in gallery

    The fractional precipitation coverage corresponding to CN as a function of forecast lead time in hours after 0000 UTC for each of the 22 cases. The thick black line represents the average over all cases.

  • View in gallery

    Scatterplots of (a) vs , (b) fractional coverage vs , and (c) vs fractional coverage for different forecast times in hours after 0000 UTC, with the color legend at the right of each panel. The linear correlation coefficients between the different variables for the different times are indicated in parentheses. The correlation coefficients are presented in bold lettering when statistically significant at the level.

  • View in gallery

    Description of the widespread cases. Each row corresponds to a separate case. (left)–(right) The power spectrum of observed hourly rainfall accumulations with the colors representing different times as described at the top of the figure, 30-h accumulations derived from radar with the scale also located at the top of the figure, the fractional coverage as a function of forecast lead time for the CN reflectivity fields, and and CAPE as a function of forecast lead time. For the last three panels, the widespread cases are illustrated in thin red lines, the diurnally forced cases in thin blue lines, the average over all cases in black, and the particular case as a thick line.

  • View in gallery

    As in Fig. 9, but for the diurnally forced cases.

  • View in gallery

    Scatterplots of normalized spread against (a) , (b) , and (c) fractional coverage, for different lead times as indicated in the legend. The linear correlation coefficients between the different variables are indicated in parentheses. The correlation coefficients are presented in bold lettering when statistically significant at the level.

  • View in gallery

    Scatterplots of for (a)–(c) CN-Radar and (d)–(f) C0-Radar against (top) , (middle) , and (bottom) fractional coverage, for different lead times as indicated in the legend. The linear correlation coefficients between the different variables are indicated in parentheses. The correlation coefficients are presented in bold lettering when statistically significant at the level.

  • View in gallery

    Normalized ensemble spread for (a) filtered fields and (b) the decorrelation scale as a function of forecast time for the widespread (red) and diurnally forced (blue) cases. The thick lines represent averages over the two categories of cases, while the black thick line is the average over all cases.

  • View in gallery

    between CN-Radar (solid), C0-Radar (dotted), and CN-C0 (long dashes) as a function of bandpass scale at different forecast lead times (panels), averaged over the entire dataset (black), over the widespread cases (red), and over the diurnally forced cases (blue). Each point on the scale axis actually corresponds to the bandpass component between and , with the filtering being done using the Haar wavelet transform, sampling at scale of 4, 8, 16, 32, 64, 128, 256, and 512 km.

  • View in gallery

    between CN-Radar (solid) and MAPLE-Radar (dashed) as a function of bandpass scale at different forecast lead times (panels), averaged over the entire dataset (black), over the widespread cases (red), and over the diurnally forced cases (blue). Each point on the scale axis actually corresponds to the bandpass component between and , with the filtering being done using the Haar wavelet transform, sampling at scale of 4, 8, 16, 32, 64, 128, 256, and 512 km.

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The Case-to-Case Variability of the Predictability of Precipitation by a Storm-Scale Ensemble Forecasting System

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Abstract

This paper analyzes the case-to-case variability of the predictability of precipitation by a storm-scale ensemble forecasting (SSEF) system. Relationships are sought between ensemble spread and quantitative precipitation forecast (QPF) skill, and the characteristics of an event, such as the strength of the quasigeostrophic forcing for ascent, the presence of convective equilibrium, and the spatial extent of the precipitation system. It is found that most of the case-to-case variability of predictability is explained by the spatial coverage of the system. The relationship between convection and large-scale forcing seems to affect predictability mostly during the afternoon hours. While the relationships are weak for the entire dataset, two distinct types of cases are identified: widespread and diurnally forced cases. The loss of predictability at small scales, the effect of the radar data assimilation, and the comparison between forecasts from the SSEF and Lagrangian persistence forecasts are analyzed separately for these two types of cases. Despite overall predictability being better than average for the widespread cases, the loss of predictability with forecast time and spatial scale is just as rapid as for the other cases. For the diurnally forced cases, the radar data assimilation causes larger differences between the precipitation fields corresponding to the assimilating and nonassimilating members than for the widespread cases. However, the effect of radar data assimilation on QPF skill is similar for both types of cases. Also, for the diurnal cases, the models with radar data assimilation outperform very rapidly (after 2 h) the Lagrangian persistence forecasts.

Corresponding author address: Madalina Surcel, Dept. of Atmospheric and Oceanic Sciences, McGill University, 805 Sherbrooke St. W. 945, Montreal QC H3A2K6, Canada. E-mail: madalina.surcel@mail.mcgill.ca

Abstract

This paper analyzes the case-to-case variability of the predictability of precipitation by a storm-scale ensemble forecasting (SSEF) system. Relationships are sought between ensemble spread and quantitative precipitation forecast (QPF) skill, and the characteristics of an event, such as the strength of the quasigeostrophic forcing for ascent, the presence of convective equilibrium, and the spatial extent of the precipitation system. It is found that most of the case-to-case variability of predictability is explained by the spatial coverage of the system. The relationship between convection and large-scale forcing seems to affect predictability mostly during the afternoon hours. While the relationships are weak for the entire dataset, two distinct types of cases are identified: widespread and diurnally forced cases. The loss of predictability at small scales, the effect of the radar data assimilation, and the comparison between forecasts from the SSEF and Lagrangian persistence forecasts are analyzed separately for these two types of cases. Despite overall predictability being better than average for the widespread cases, the loss of predictability with forecast time and spatial scale is just as rapid as for the other cases. For the diurnally forced cases, the radar data assimilation causes larger differences between the precipitation fields corresponding to the assimilating and nonassimilating members than for the widespread cases. However, the effect of radar data assimilation on QPF skill is similar for both types of cases. Also, for the diurnal cases, the models with radar data assimilation outperform very rapidly (after 2 h) the Lagrangian persistence forecasts.

Corresponding author address: Madalina Surcel, Dept. of Atmospheric and Oceanic Sciences, McGill University, 805 Sherbrooke St. W. 945, Montreal QC H3A2K6, Canada. E-mail: madalina.surcel@mail.mcgill.ca

1. Introduction

Recent studies have shown that convection-allowing numerical weather prediction (NWP) models are more skillful than convection-parameterized models at representing convective mode and the diurnal cycle of precipitation (Clark et al. 2007). On the other hand, the gain in quantitative precipitation forecast (QPF) skill achieved by increasing model resolution to allow the explicit representation of convection is more modest than initially expected (Weisman et al. 2008). One of the main reasons why improving QPF skill is a very difficult task relates to the large variability of skill with spatial scale and from case to case (Clark et al. 2011; Done et al. 2012; Germann et al. 2006; Surcel et al. 2010). In a recent paper (Surcel et al. 2015, hereafter SZY15), we investigated how the predictability of precipitation by the storm-scale ensemble forecasting (SSEF) system run by the Center for the Analysis and Prediction of Storms (CAPS) during NOAA’s 2008 Hazardous Weather Testbed (HWT) Spring Experiment (Xue et al. 2008; Kong et al. 2008) is lost as a function of spatial scale and forecast lead time. The 2008 CAPS SSEF ensemble consists of 10 members: 2 control members and 8 members with initial and lateral boundary condition (IC/LBC) perturbations and varied model physics. All the perturbed members and one control member (CN) have mesoscale data assimilation, including radar, while one control member (C0) has no mesoscale data assimilation. The loss of predictability as a function of scale was quantified in terms of the decorrelation scale . For any two or more precipitation fields, is defined as a threshold scale such that for all scales the forecasts are fully decorrelated, indicating total loss of predictability, while for scales there is some correlation between the fields (see SZY15 for more details). In SZY15, was computed for precipitation forecasts from different ensemble members to characterize the predictability of the model state, and between forecasts and observations, characterizing the model predictability of the atmospheric state. The main results of SZY15 are reillustrated in Fig. 1.1

Fig. 1.
Fig. 1.

Summary of the comparison of the different types of predictability over a 30-h forecast period illustrated in terms of vs t. The model predictability of the atmospheric state is characterized by computed between radar and radar-data-assimilating (radar DA) models and between radar and the nonradar DA C0. The predictability of the model state is characterized by computed between CN and radar DA models; was computed for instantaneous reflectivity fields with hourly resolution. The different color curves correspond to the different types of predictability as described in the legend. The thin lines correspond to the individual cases, while the thick lines represent averages over all cases.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

Figure 1 shows computed for forecast–observation pairs (black and purple lines) and for forecasts from different ensemble members (green lines) for 22 cases during spring 2008 (thin lines correspond to the individual cases, while the thick lines represent the average over all cases) (for further details on this figure and how it was obtained the reader is referred to SZY15). On average, CN and the perturbed members of the CAPS SSEF become increasingly decorrelated with forecast lead time. After 2 h there is no similarity among them at meso-γ scales, while after 10 h the correlation at meso-β scales is lost (thick green line). When comparing the forecasts from the radar data assimilation members to observations, increases rapidly with time, reaching about 300 km after the first 3 h, while for the nonradar data assimilation member C0 is initially high and decreases to 300 km after the spinup time (thick black and purple lines, respectively).

While SZY15 focused on the average behavior of for the entire dataset, Fig. 1 indicates significant case-to-case variability of the decorrelation scale. Understanding the reasons for this variability is important for advancing the design of appropriate forecasting techniques, and it is one of the main objectives of this paper.

a. Background

Even for very simple dynamical systems governed by nonlinear differential equations, predictability (i.e., the rate of divergence of two states that initially are very close together) is dependent on the location of the initial states in the phase space [Palmer (1993) shows a nice illustration of the dependence of predictability on initial conditions for the Lorenz‘63 system]. If the atmosphere is treated as a nonlinear deterministic system, we expect a finite predictability limit and for this limit to be dependent on the initial conditions. However, the problem becomes more complicated for precipitation predictability at the mesoscale. Unlike the Lorenz (1963) model, a mesoscale NWP model has a phase space whose exploration “necessary to understand its behavior, is certainly beyond the scope of the individual, and, arguably, beyond the range of a single institute” (Palmer 1993). Hence, any study on the predictability of precipitation usually consists of characterizing the dependence of some predictability measure on the initial state for a limited number of cases. This was done in many previous studies, with some results summarized below.

For instance, Zawadzki et al. (1994) investigated how the prediction error by Lagrangian persistence (LP) for convective rainfall depends on some characteristics of the environment. Using results from numerical simulations of convective storms, they chose the vertical distribution of temperature advection, wind shear energy, helicity, geostrophic vorticity, the amount of convective available potential energy (CAPE) and the bulk Richardson number as main parameters characterizing the environment. Searching for statistically significant relationships between very-short-term (less than 6 h) forecast error and these parameters, only CAPE and helicity were found to partly explain the variability of skill from case to case.

Jankov and Gallus (2004) investigated the factors affecting the case-to-case variability of the accuracy of forecasts from convection-parameterized models. For a dataset of 20 mesoscale convective systems (MCSs), they found a strong relation between the strength of the large-scale forcing (inferred from the values of differential vorticity advection, frontogenesis, 700-hPa omega, 850-hPa temperature advection, and 200-hPa divergence) and the accuracy of the forecast.

The case-to-case variability of the predictability of precipitation was also reported in predictability studies using convection-allowing models. It was shown that the intensity of moist convection is a determining factor in affecting predictability (Walser et al. 2004; Zhang et al. 2003). However, Hohenegger et al. (2006) discussed “predictability mysteries,” defined as cases with similar intensity of moist convection but with very different predictabilities. For a small dataset (three cases) they found that the differences in the predictability could be best explained by how fast local perturbations can propagate across unstable regions.

Investigating the effect of different perturbing methodologies for a high-resolution NWP ensemble (with grid spacing of 2.5 km), Vié et al. (2011) noted that the relative importance of small-scale IC perturbations and large-scale LBC perturbations depends on the large-scale meteorological situation. When the synoptic-scale forcing was strong (evaluated subjectively in their study from upper-level maps), large-scale LBC perturbations played the major role, while the opposite stands for weakly forced cases at the synoptic scale. Similar case dependence of the sensitivity to IC/LBC perturbations was also reported by Johnson et al. (2014). Specifically, they found that when the weather feature of interest was occurring at the synoptic scale, large-scale IC perturbations sampled the main source of uncertainty. However, when the significant feature occurred at meso-γ or meso-β scales (i.e., individual convective systems), the type of perturbation methodology was no longer important, with all methodologies producing sufficient spread.

Recently, Duda and Gallus (2013) investigated whether there is a relationship between the accuracy of convection-allowing forecasts of convection initiation and evolution and the strength of the large-scale forcing. Their 36 cases were classified according to several measures of quasigeostrophic forcing for ascent and 700-hPa omega and 200-hPa divergence. No relationship was found between the skill of the model in initiating convection and the strength of the forcing. They comment that this may be an expected result given that convection initiation is often affected by local small-scale processes (such as storm-scale outflow, horizontal convective rolls in the boundary layer, and orographic circulations). On the other hand, a weak relationship was found between the strength of the forcing and model QPF skill.

Furthermore, several studies (Done et al. 2012; Keil et al. 2014; Zimmer et al. 2011; Done et al. 2006; Keil and Craig 2011) reported the sensitivity of the predictability of convective precipitation to whether or not convection is in equilibrium with the large-scale forcing (i.e., the generation of CAPE by large-scale processes is balanced by the consumption of CAPE by convection). According to Done et al. (2006), convective equilibrium cases are characterized by strong large-scale forcing for upward motion and by regions of CAPE that are collocated with regions of almost no convective inhibition (CIN). In these cases, as CAPE is continuously generated by the large scale, it is also spent through convection, which is free to act in the absence of CIN. Therefore, the mean properties of convection are set by the large-scale environment. In contrast, in cases without convective equilibrium, regions of CAPE are collocated with regions of CIN, and convection only occurs when surface mechanisms act to overcome the convective inhibition. As such, the properties of convection depend on the local characteristics of the environment. Note that in the aforementioned studies, the existence of convective equilibrium for a given case is characterized in terms of the convective-adjustment time scale (); is defined as the rate at which instability (CAPE) is removed by convective heating. Values of representative of convective equilibrium are on the order of 1 h, while values of suggestive of nonequilibrium are on the order of 1 day.

Analyzing the partitioning between parameterized and explicit precipitation in the simulation of two events, Done et al. (2006) found that both the partitioning and the behavior of convection depend on whether the convection is in equilibrium with the large-scale forcing. In a subsequent study, Done et al. (2012) also investigated the spread of an ensemble with small-scale perturbations to the boundary layer equivalent potential temperature for two cases with and without convective equilibrium. When convection is in equilibrium with the large-scale flow, there is a strong constraint on the total precipitation amount, which does not vary much between the ensemble members; the behavior of individual storms, however, is not constrained. On the other hand, in the nonequilibrium case, convection was triggered by orographic forcing. For this case, the location of convective rain was similar between the members, but the amount varied greatly. Based on the results of such studies, Keil and Craig (2011), Keil et al. (2014), and Zimmer et al. (2011) searched for a statistical relationship between and the predictability by an ensemble of precipitation forecasts using a large dataset. While the relationships obtained were usually weak, Keil et al. (2014) affirmed that “the convective adjustment time-scale τ represents an indicator of the practical predictability level of convective precipitation, in contrast to other instability indices that exhibit poor predictive skill.” Furthermore, by analyzing the effect of radar data assimilation in three precipitation events, Craig et al. (2012) found that correlated well with the duration of the effect of radar data assimilation.

b. Approach

In SZY15, we described how predictability is lost on average with forecast lead time and spatial scale. We noted that while there was some case-to-case variability between the cases, the shape of was consistent throughout the dataset, and we have shown snapshots of each of the 22 cases under consideration in terms of the total precipitation amount and the temporal evolution of the power spectrum of precipitation. Here, we further investigate the case-to-case variability of predictability and address the following questions:

  • Are cases that are highly predictable at large scales also characterized by enhanced predictability at small scales?
  • Are the overall characteristics of a case, be them dynamical (i.e., in terms of the large-scale forcing for ascent) or hydrometeorological (i.e., in terms of the statistical properties of the model rainfall fields), related to the predictability of precipitation?
If the overall characteristics of a forecast event (which could be described a priori) are indicators of predictability, then it may be known beforehand how much confidence can be placed in a forecast. Although ensemble dispersion is usually used as a measure of the uncertainty of a forecast, it was shown in previous studies (Johnson et al. 2014; Clark et al. 2011; SZY15) that it is common for storm-scale ensembles to be underdispersive. Also, the poor correlation between ensemble variance and mean-square error was previously reported (Jones et al. 2007), suggesting that ensemble variance might not be a good predictor of skill. Furthermore, investigating the case-to-case variability of predictability could identify the most problematic cases.

Here we analyze 22 cases, thus a thorough analysis and a subjective classification is doable for each case. Our purpose is to find an objective way of characterizing precipitation cases, so that relationships between predictability and case characteristics may be investigated over larger datasets, similarly to Keil et al. (2014). The results obtained here will then be extended to a larger dataset comprising of storm-scale ensemble forecasts from NOAA’s HWT Spring Experiments of 2009–13. The paper is organized as follows. The data are presented in section 2. Section 3 discusses the predictability measures for the 22 cases under study and discusses the relation between the various measures. Section 4 presents the classification of the cases. Section 5 investigates the relationships between the characteristics of the cases and their corresponding predictability. Conclusions are finally presented in section 6.

2. Data description

The model and verification data used here are the same as in SZY15. For a detailed description, the reader is referred to that paper.

Both hourly rainfall accumulations fields and instantaneous reflectivity fields were available for both forecasts and observations. For the forecasts, the precipitation fields are surface fields output by the model. In the case of observations, the precipitation fields are estimates derived from radar reflectivity mosaics at 2.5-km altitude obtained from the National Severe Storms Laboratory (NSSL; Zhang et al. 2005). These estimates were chosen over other existing quantitative precipitation estimation (QPE) products for their better quality as explained by Surcel et al. (2010), particularly in terms of the effects of nonmeteorological targets evident in other products. To obtain the radar derived hourly accumulation fields, the 2.5-km reflectivity mosaics with a temporal resolution of 5 min and a spatial resolution of 1 km were converted to rain rate using a standard ZR relationship () and then accumulated. A threshold of 15 dBZ (0.2 mm h−1) was used to differentiate between the rain/no rain areas for the reflectivity (hourly accumulations) maps. The entire analysis was performed on both the hourly accumulation and the instantaneous reflectivity fields at the top of each hour, with consistent results. Thus, only the results corresponding to instantaneous reflectivity fields will be presented here, and hereafter, the term precipitation fields will refer to the instantaneous reflectivity fields. 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 the eastern United States (extending from 102° to 78°W in longitude and from 32° to 45°N in latitude). The dataset consists of 22 precipitation cases from 18 April to 6 June.

The CAPS SSEF system uses the Advanced Research version of the Weather Research and Forecasting (WRF) Model (ARW; Skamarock et al. 2008), version 2.2, and consists of 10 members with different physical schemes, mesoscale data assimilation (including radar), and perturbed IC/LBCs. The background ICs are interpolated from the North American Mesoscale Forecast System (NAM; Janjic 2003) 12-km analysis, and the IC/LBC perturbations are obtained directly from the short-range ensemble forecasting (SREF) system that runs operationally at NCEP (Du et al. 2009). In addition to IC/LBC perturbations, the ensemble members have different microphysical schemes, planetary boundary layer (PBL) schemes, and shortwave radiation schemes. Thirty-hour forecasts on a 4-km grid were performed almost daily from April to June 2008. Two of the members (control members C0 and CN) do not have SREF-based IC/LBC perturbations and have identical model configurations, but convective-scale observations from radar are assimilated only within CN.

3. Characterizing the predictability of precipitation by the CAPS SSEF during spring 2008

In SZY15, the decorrelation scale was used to quantify the loss of the practical predictability by NWP with spatial scale and forecast lead time. Two types of predictability were investigated: the predictability of the model state, corresponding to the resemblance between forecasts obtained with slightly different model configurations, and the model predictability of the atmospheric state, corresponding to the resemblance between forecasts and observations. Two other predictability measures are introduced in this section, namely the normalized ensemble spread and the normalized root-mean-square error (). These metrics quantify the domainwide dispersion and skill of the ensemble forecasts.

For an ensemble of N precipitation forecasts, the normalized ensemble spread is defined as follows:
e1
where
e2
Here , is the forecast corresponding to an ensemble member, N is the number of members, and are the spatial dimensions of the forecast. The normalized ensemble spread is computed for the CAPS SSEF ensemble forecasts of simulated radar reflectivity with an hourly resolution. This definition of ensemble spread is similar to that of Hohenegger et al. (2006). Normalized measures are preferred in order to eliminate as much as possible the effect of the diurnal cycle of precipitation and of the case-to-case variability of rainfall amounts.
The (Surcel et al. 2014) between two precipitation fields is defined as
e3
where X and Y are two different precipitation fields of spatial dimensions I and J.

As shown in SZY15, the predictability at meso-γ and meso-β scales (i.e., less than 200 km) is lost very rapidly. Thus, we only present and for scales that exhibit some predictability, computing these measures for filtered precipitation fields from which the information at scales smaller than 256 km has been removed using a Haar low-pass filter. Figure 2 shows all the predictability measures as a function of forecast lead time for each of the 22 cases (colored lines) and averaged for all cases (thick black line). As shown in Fig. 2b, the values of increase steadily with time, with a slight indication of the effect of the diurnal cycle being visible at 2000 UTC (20-h lead time).2 On the other hand, changes very rapidly in the first three hours, corresponding to the model spinup (Figs. 2d,f). This is the case for both the radar data assimilation member CN and for the nonradar data assimilation member C0. After the initial variability, appears to be constant.

Fig. 2.
Fig. 2.

The predictability of precipitation for spring 2008. Each color line represents each of the 22 cases, while the thick black line corresponds to the average over all the cases. The different panels show the decorrelation scales as a function of forecast lead-time corresponding to (a) the entire ensemble, (c) Radar-CN, and (e) Radar-C0. (b) The normalized ensemble spread for the filtered fields, and the filtered corresponding to (d) Radar-CN and (f) Radar-C0.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

The case-to-case variability is evident in Fig. 2 for , , and . To determine whether this variability is consistent among the measures, Figs. 3 and 4 show scatterplots between them. Figure 3 illustrates how relates to the global measures of spread and skill. The plots are shown for different forecast lead times (different colors). The linear correlation coefficients between and or are shown in the parentheses in the legend. Statistically significant correlation values at indicated in bold lettering for an level. According to Fig. 3a, there is no relation between the predictability of precipitation at scales larger than 256 km and the loss of predictability with time and scale. A weak but significant relationship seems to appear at late lead times ( for 30 h). However, the inspection of each individual case revealed that cases with very high or very low values do not have consistent values, and hence this relationship is considered nonsignificant. The overall predictability by the ensemble characterized in terms of is not related to how predictability is lost at meso-γ and meso-β scales.

Fig. 3.
Fig. 3.

Scatterplots of the decorrelation scale vs and for low-pass filtered precipitation fields corresponding to (a) the ensemble, (b) CN-Radar, and (c) C0-Radar at different lead times (see legend for corresponding colors). The linear correlation coefficients between the different variables are shown in parentheses. The correlation coefficients are presented in bold lettering when statistically significant at the level.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

Fig. 4.
Fig. 4.

corresponding to CN-Radar vs normalized ensemble spread for low-pass filtered fields for different lead times as described in the legend. The linear correlation coefficients between the different variables are shown in parentheses. The correlation coefficients are presented in bold lettering when statistically significant at the level.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

On the other hand, Figs. 3b and 3c show that in the case of model–radar comparison, there is a correlation between and . This might be related to the values being higher than 256 km for the model–radar comparison, which means that the filtered values may be impacted by the unpredictable scales.

Figure 4 illustrates the relationship between spread and skill for this dataset. As confirmed by the correlation coefficients, correlates well with the corresponding to CN-Radar for most lead times. The correlation coefficient is lower during the afternoon hours (1500–2000 UTC, shown only at 1800 UTC). This might be an effect of the diurnal cycle of precipitation, which is characterized by an afternoon maximum of precipitation, which shows case-to-case variability, and which has an impact on forecasting skill as shown by Berenguer et al. (2012).

4. Description of cases

We seek objective methods of classification to investigate whether the predictability of a precipitation case might be related to some average properties of that case. As discussed in the introduction, some studies have reported that the predictability of precipitation is related to the influence of the large-scale forcing on the evolution of the precipitation system. Thus, a classification based on this criterion is attempted. Although the term “large-scale forcing” is commonly used, its meaning remains unclear. Generally, it refers to nonlocal reasons for upward motion, such as quasigeostrophic forcing for ascent or meso-α meteorological features such as fronts. Proposing an objective definition of large-scale forcing is outside of the scope of this study. Instead, the cases are classified according to the convective-adjustment time scale , as described by Keil et al. (2014), and to the strength of the quasigeostrophic ascent over precipitation areas. Finally, a classification based solely on the statistical properties of the forecast precipitation pattern is also presented. The different case classifications are described below.

a. Classification based on the convective-adjustment time scale

As already mentioned, several studies indicated that both the evolution and the predictability of convective precipitation appears to be related to whether convection is in equilibrium or not with the large-scale forcing. To differentiate objectively between equilibrium and nonequilibrium cases, Done et al. (2006) proposed the convective-adjustment time scale , defined there as the ratio between CAPE and the rate of change of CAPE:
e4
in which is defined as
e5
Here, g is the acceleration of gravity, T is the environmental temperature, is the temperature of a pseudoadiabatically lifted boundary layer parcel, and is a constant reference temperature (a value of 280 K is used here).
Here, is calculated as a function of forecast lead time using data from the NAM model, which is available every 3 h. The NAM forecasts and analysis, which provide the background ICs and the LBCs for the CAPS SSEF, are freely available from the NOAA/National Climatic Data Center (NCDC) National Operational Model Archive and Distribution System (NOMADS). The NAM model is run with a horizontal grid spacing of 12 km, and thus convection is parameterized using the Betts–Miller–Janjić (BMJ; Janjić 1994). The poor representation of precipitation by convection-parameterized models is well documented in the literature (e.g., Berenguer et al. 2012; Clark et al. 2007; Davis et al. 2003). Thus, whereas in previous work by Done et al. (2006) and Keil et al. (2014), the rate of change of change of was calculated from the precipitation rate, here we compute the rate of removal of CAPE explicitly from the 3-hourly data:
e6
To obtain daily values, we average the resulting fields obtained using Eq. (6) over all data points with nonzero values of 3-h accumulations of convective rainfall and with a decrease of CAPE.

Note that the formula used for the CAPE calculation could impact the resulting values. Here we use the CAPE values available in the NAM files, which are calculated based on the properties of a surface parcel. Since the interest is not in the nominal value of , but on comparing the values between the different cases, keeping the CAPE formula consistent in our calculations is sufficient to accomplish our objective.

Figure 5 shows the evolution of with forecast time for all cases. Values of vary between 0 and 20 h, but significant diurnal variability is evident. Moreover, this variability appears more pronounced for cases occurring in late spring. Figure 5b shows the average CAPE values over precipitation area as a function of forecast lead time. Indeed, there seems to be a clear difference between the earliest cases (blue lines) and the later cases (red lines). Over the continental United States during summer, strong solar heating at the surface makes the atmosphere conditionally unstable. Because of the terrain characteristics, the conditionally unstable region expands over most of the continental United States east of the Rockies (Dai et al. 1999). While early-spring CAPE is associated mostly with synoptic cyclones, late-spring CAPE evolution is controlled by the diurnal cycle of solar heating (not shown). Therefore, for the late-spring cases, values can remain high even when rainfall is associated with midlatitude cyclones (e.g., on 6 June 2008).

Fig. 5.
Fig. 5.

The (a) convective-adjustment time scale and (b) CAPE averaged over all areas with rainfall higher than 1 mm h−1 as a function of forecast lead time in hours after 0000 UTC for each of the 22 cases as indicated in the legend.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

b. Classification based on the large-scale forcing for ascent

Similarly to Duda and Gallus (2013), we consider large-scale forcing to be limited to quasigeostrophic (QG) forcing for ascent [as proposed by Doswell (1987)]. According to the QG omega equation, the vertical motion is directly affected by the differential vorticity advection and by the temperature advection by the geostrophic wind (Bluestein 1992). To facilitate the use of the ω equation, Hoskins et al. (1978) proposed the Q-vector form, which reads
e7
where
e8
Here is the pressure vertical velocity, is the Coriolis parameter, σ is the static stability.

This equation indicates that QG forcing is related to the horizontal gradient of the geostrophic wind, which must have a component parallel to the horizontal temperature gradient. Then, by Eq. (7), Q-vector convergence indicates forcing for ascent, while Q-vector divergence implies forcing for descent. Note that the Q-vector divergence only indicates the forcing for vertical motion and does not give an estimate of its magnitude (Doswell 1987). For the purpose of the analysis presented herein, we are only interested in the synoptic forcing for ascent and not in the magnitude of the vertical motion induced by that forcing.

The Q-vector divergence is calculated using the NCEP reanalysis data. These data were preferred to other higher-resolution analyses as we are only interested in quasigeostrophic dynamics. Given the coarse resolution of the NCEP data (210 km), mesoscale features that could contaminate the QG analysis are avoided.

To explore the relation between large-scale forcing and predictability, we are interested in having a single forcing indicator for a given case, the large-scale forcing indicator (). Therefore, to obtain , we first take the spatial average of the at t − 6 h over the precipitation regions at time t at different vertical levels:
e9
where z represents height. Then, is defined as the vertical maximum of .

Other definitions for have been considered, but using the vertical maximum of was found to agree best with the visual inspection of surface analyses and upper-level maps. The is computed every 6 h (the temporal resolution of the NCEP data) for most of the spring and the summer of 2008. This is done to investigate the ability of this index to differentiate between the strength of the forcing during the two seasons. Surcel et al. (2010) investigated the seasonal variability of the diurnal cycle of precipitation in 2008, mentioning that the difference between the two seasons could be due to the relative importance of large-scale and thermal forcing. Indeed, Fig. 6 confirms that the strength of the large-scale forcing as quantified by the is very different between spring and summer, in terms of both average value and variance.

Fig. 6.
Fig. 6.

Time series of (see text for details) for different times: 0000 UTC (0-h forecast, black), 0600 UTC (6-h forecast, blue), 1200 UTC (12-h forecast, green), 1800 UTC (18-h forecast, orange), and 0000 UTC (24-h forecast, red) between 18 Apr and 25 Aug 2008. The more negative the value the stronger the large-scale forcing, while the more positive the value, the weaker the forcing.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

c. Classification based on the statistical properties of precipitation

SZY15 showed how the 22 cases differ in terms of the total precipitation amount and in terms of the temporal evolution of the power spectrum of the precipitation field. The cases with larger-scale precipitation systems characterized by widespread rainfall usually showed little variability in the power spectrum of precipitation with forecast time. In contrast, cases that were mostly driven by the diurnal cycle of solar heating showed higher variability in the power spectrum. Here, to quantify the evolution of the precipitation field, the fractional precipitation coverage is computed as a function of forecast lead time for simulated reflectivity fields from the CN member (although any other member would show a very similar evolution).3 The fractional precipitation coverage is defined simply as the proportion of the model domain with rainfall rates higher than 15 dBZ.

Figure 7 shows the fractional precipitation coverage as a function of forecast lead time for the different cases. There is a large case-to-case variability between the cases, both in the average value and in the temporal evolution. As for , some cases (especially the later ones) are marked by a clear diurnal cycle.

Fig. 7.
Fig. 7.

The fractional precipitation coverage corresponding to CN as a function of forecast lead time in hours after 0000 UTC for each of the 22 cases. The thick black line represents the average over all cases.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

d. Comparison of the different ways of classifying the cases

We have three different ways of characterizing the cases. Next, we verify whether a relationship exists between , , and precipitation coverage. Relationships between and the properties of the rainfall system have been reported previously by Molini et al. (2011). For extreme rainfall events in Italy, it was found that 90% of long-lasting events were associated with convective equilibrium cases (i.e., h), while 70% of the short-lasting events were associated with nonequilibrium cases (i.e., h).

The simplest way to verify whether a relationship exists is to compute the correlation coefficients between the different variables. Figure 8 shows scatterplots between and , coverage and , and and coverage. As indicated by the correlation coefficients at the right of each panel, no statistically significant relationships exist between the variables (the correlation coefficients are presented in bold lettering when statistically significant at the level).

Fig. 8.
Fig. 8.

Scatterplots of (a) vs , (b) fractional coverage vs , and (c) vs fractional coverage for different forecast times in hours after 0000 UTC, with the color legend at the right of each panel. The linear correlation coefficients between the different variables for the different times are indicated in parentheses. The correlation coefficients are presented in bold lettering when statistically significant at the level.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

Despite the lack of stronger relationships, two different types of cases with some consistency of these variables can be identified from the dataset. First, there are widespread precipitation events associated with midlatitude cyclones, characterized by strong large-scale forcing, large precipitation coverage, and convective equilibrium. Such cases occur on 24 and 25 April and 2, 7, and 8 May. For these, the value of the precipitation coverage is higher than 0.15, the convective-adjustment time scale is less than 6 h, and areas of -vector convergence are associated with the precipitation region. On the other hand, there are cases marked by a clear diurnal cycle of CAPE and , such as 30 May and 2, 4, 5, and 6 June. These cases are also marked by a diurnal cycle of rainfall coverage, which maximizes during the afternoon. However, for these cases, the values of are not consistent with weak forcing. There are 10 remaining cases for which nothing clear can be said about the evolution of the three variables.

The widespread and the diurnally forced cases are described in Figs. 9 and 10, respectively. As in SZY15, the power spectrum of observed hourly accumulations is shown to illustrate the daily variability of rainfall. Comparing the two figures shows that the diurnally forced cases show indeed more variability of the power spectrum of precipitation with time of day than the widespread cases. Also, a large difference between Figs. 9 and 10 can also be seen in the spatial coverage of the 30-h rainfall accumulations. Finally, the last three panels of each row show the fractional coverage, , and CAPE for the two types of cases (widespread: red, and diurnally forced: blue). For the widespread cases, the coverage is by definition higher than the average coverage, but also, it varies less throughout the day. On the other hand, the diurnally forced cases show a lower average coverage but with a pronounced diurnal variability. These cases are also characterized by higher-than-average and CAPE values. Conversely, is not statistically different between the two types of cases (not shown). Moreover, the diurnally forced cases show large values of -vector convergence over the precipitation areas, but it appears that the evolution of the precipitation system is modulated by the diurnal cycle of solar heating (Surcel et al. 2010).

Fig. 9.
Fig. 9.

Description of the widespread cases. Each row corresponds to a separate case. (left)–(right) The power spectrum of observed hourly rainfall accumulations with the colors representing different times as described at the top of the figure, 30-h accumulations derived from radar with the scale also located at the top of the figure, the fractional coverage as a function of forecast lead time for the CN reflectivity fields, and and CAPE as a function of forecast lead time. For the last three panels, the widespread cases are illustrated in thin red lines, the diurnally forced cases in thin blue lines, the average over all cases in black, and the particular case as a thick line.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

Fig. 10.
Fig. 10.

As in Fig. 9, but for the diurnally forced cases.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

Given the lack of substantial correlation between each of the three parameters, the relationships between each of them and precipitation predictability are explored next.

5. The case-to-case variability of predictability

a. Overall relationships

This section presents how the predictability of precipitation—quantified in terms of spread and skill—is related to the type of case. As in the previous subsection, we first show the scatterplots of predictability indicators against , , and fractional coverage. Figure 11 shows the normalized ensemble spread versus (Fig. 11a), (Fig. 11b), and fractional coverage (Fig. 11c) as a function of forecast lead time. The only clear relationship appears to be between and precipitation coverage. The normalized ensemble spread at scales larger than 256 km is inversely proportional to the precipitation coverage, such that the higher the coverage the smaller the spread, and the smaller the spatial extent of the system, the higher the average spread. During the afternoon hours, the evolution of the precipitation systems is mostly affected by the diurnal cycle of solar heating [Berenguer et al. (2012), who showed also that the SSEF members had difficulties reproducing the observed diurnal evolution of precipitation]. Nevertheless, the correlation coefficients indicate a weak but significant relationship between ensemble spread and for later forecast hours (). Cases characterized by strong synoptic forcing for ascent usually show smaller spread. However, there is no statistically significant relationship at later forecast times between and . This is somewhat in disagreement with Keil et al. (2014), who report a weak relationship between ensemble spread and the convective-adjustment time scale, even though their determination coefficient is only and they do not stratify by forecast lead time.

Fig. 11.
Fig. 11.

Scatterplots of normalized spread against (a) , (b) , and (c) fractional coverage, for different lead times as indicated in the legend. The linear correlation coefficients between the different variables are indicated in parentheses. The correlation coefficients are presented in bold lettering when statistically significant at the level.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

The relationship between model performance and the different classifications is also investigated. Figure 12 shows the plots of the computed for low-pass filtered ( km) reflectivity fields corresponding to CN-Radar and C0-Radar versus , , and fractional coverage. The effect of large-scale forcing is less evident for QPF skill than for ensemble spread. It seems that for our dataset, solely the presence of quasigeostrophic forcing for ascent does not necessarily result in better QPF skill. This is in disagreement with Jankov and Gallus (2004), who showed that the QPF skill of a convection-parameterized model was significantly better for the strongly forced than for the weakly forced cases. In contrast, Duda and Gallus (2013) only reported a very weak correlation between the QPF skill of a convection-allowing model (in forecasting the upscale evolution of convection into an MCS) and the strength of the large-scale forcing. Also in disagreement with the results of Keil et al. (2014), no relationship can be reported between and QPF skill. Finally, it still appears that the fractional coverage of the precipitation pattern accounts for most of the case-to-case variability in QPF skill (Figs. 12c,f).

Fig. 12.
Fig. 12.

Scatterplots of for (a)–(c) CN-Radar and (d)–(f) C0-Radar against (top) , (middle) , and (bottom) fractional coverage, for different lead times as indicated in the legend. The linear correlation coefficients between the different variables are indicated in parentheses. The correlation coefficients are presented in bold lettering when statistically significant at the level.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

b. Intercomparison of various forecast sensitivities between the widespread and the diurnally forced cases

On average, any relationships between predictability and forcing are very weak. But as mentioned previously, there are two different types of cases that stand out from the dataset. First, there are the strongly forced, widespread cases, comprising 24 and 25 April and 2, 7, and 8 May, and second, there are the diurnally forced cases: 30 May and 2, 4, 5, and 6 June. We now examine whether some consistency between the strength of the forcing and predictability exists for these cases. The following questions are addressed:

  • Is the loss of predictability with spatial scale and forecast lead time related to the overall predictability of the case? Differently said, are cases characterized by good overall predictability (as quantified by the normalized ensemble spread ) also predictable for longer times at small scales (as quantified by the decorrelation scale )?
  • The success of radar data assimilation also depends on predictability considerations. Is the effect of the radar data assimilation different between the widespread and the diurnally forced cases, as it would be expected from the results of Craig et al. (2012)?
  • Another less computationally expensive option for very-short-term precipitation forecasting consists of radar-based Lagrangian persistence forecasts. The comparison of QPF skill between LP and NWP forecasts was addressed previously for this dataset by SZY15. Here, we will further investigate whether the difference in performance between LP and forecasts from CN differs between the widespread and the diurnally forced cases.

1) The case-to-case variability of and

Figure 13 shows and (Fig. 13b) for the widespread (blue) and the diurnally forced (red) cases. A clear difference in exists between the two types of cases, with the widespread cases showing less dispersion. The diurnally forced cases show a less-than-average spread during the first few forecast hours, while the spread increases rapidly after 0500 UTC, associated with the decay of precipitation (shown by the decrease in fractional precipitation coverage: Fig. 10). At 1500 UTC, a slight plateau is noticed in , but as the new systems start developing, the rate of increase of spread changes again. The modulation of the ensemble dispersion by the diurnal cycle of precipitation has been previously reported by Johnson et al. (2014). This is in agreement with the results of previous studies (Zhang et al. 2003; Hohenegger et al. 2006; Walser et al. 2004), which showed that the moist convection is the main mechanism of error growth at the mesoscale.

Fig. 13.
Fig. 13.

Normalized ensemble spread for (a) filtered fields and (b) the decorrelation scale as a function of forecast time for the widespread (red) and diurnally forced (blue) cases. The thick lines represent averages over the two categories of cases, while the black thick line is the average over all cases.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

Despite the normalized spread being different between the two types of cases, there is no significant difference in the evolution of (Fig. 13b). Although the widespread cases are more predictable than average at scales larger than 256 km (quantified in terms of ), the loss of predictability at small scales with time is as rapid as on average (the red, blue, and black line overlap in Fig. 13b).

2) The case-to-case variability of the effect of radar data assimilation

The case dependence of the effect of radar data assimilation was reported by Craig et al. (2012). They showed that the duration of the effect of radar data assimilation correlated well with for three convective events in Germany. The regime dependence of the effect of radar data assimilation for the SSEF forecasts is analyzed next. Figure 14 shows the as a function of bandpass scale for CN-Radar, C0-Radar, and CN-C0. The different panels correspond to forecast lead time, and the different lines correspond to averages over the types of cases: red for the widespread cases, blue for the diurnally forced cases, and black for the entire dataset. The evaluation is done in terms of for different bandpass scales, since predictability was shown to be scale dependent (Surcel et al. 2014; Stratman et al. 2013; etc.). The filtering is performed using the Haar wavelet transform (Turner et al. 2004) with scale samplings of 4, 8, 16, 32, 64, 128, 256, and 512 km. Figure 14 shows that from the first forecast hour, CN outperforms C0 at all scales, and the error in CN (as quantified by the between CN and Radar) is smaller than the difference between the two members. After 3 h, CN and C0 become more similar to each other at meso-γ and meso-β scales rather than to the radar observations, while CN still has better skill. No regime dependence is evident up to 3 h. After 6 h some regime dependence becomes evident, but only in terms of the difference in model performance between widespread and diurnally forced cases, and not in terms of the difference in performance between CN and C0 (which seems the same between the two types of cases). By comparing the red and blue dashed lines in Fig. 14, it can be said that the effect of the radar data assimilation (if considered as an IC perturbation) is larger for the widespread cases between 0600 and 1500 UTC. This may be due to the lower precipitation amounts during the late night and early morning for the diurnally forced cases. In agreement with Stratman et al. (2013), an improvement in the at scales of the order of 200 km is noticeable up to 15 h, especially for the widespread cases. Also, up to 18 h, CN forecast error is comparable to the difference between CN and C0 for the widespread cases, but not for the diurnal cases. After 21 h, there is no longer any regime dependence of the corresponding to CN-C0. Also, at 24 h, the performances of CN and C0 are identical for all types of cases.

Fig. 14.
Fig. 14.

between CN-Radar (solid), C0-Radar (dotted), and CN-C0 (long dashes) as a function of bandpass scale at different forecast lead times (panels), averaged over the entire dataset (black), over the widespread cases (red), and over the diurnally forced cases (blue). Each point on the scale axis actually corresponds to the bandpass component between and , with the filtering being done using the Haar wavelet transform, sampling at scale of 4, 8, 16, 32, 64, 128, 256, and 512 km.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

We note that for this dataset, the forecasts were initialized at 0000 UTC, a time when the diurnally forced convection was already initiated and organized. The impact of radar data assimilation could be different if the assimilation was performed at other times in the life cycle of the precipitation systems.

3) The case-to-case variability of the comparison in performance between LP and CN forecasts

SZY15 also discussed the predictability by the McGill Algorithm for Precipitation nowcasting by Lagrangian Extrapolation (MAPLE) (Turner et al. 2004). MAPLE is a statistical forecasting method, and it was shown to outperform NWP models for very-short-term forecasting for about 3 h for the radar-data-assimilating models and about 5 h for the nonradar-data-assimilating members (Berenguer et al. 2012; SZY15). The case dependence of the difference in performance between MAPLE and CN is also addressed, employing the same evaluation metric. Figure 15 shows the as a function of bandpass scale for lead times of 0–5 h for CN-Radar (solid) and MAPLE-Radar (dashed). On average, CN becomes as skillful as MAPLE at all scales at and at h for diurnally forced cases. Note that whereas there is no regime dependence in the CN skill (red and blue solid lines superimpose), there is a difference in the MAPLE skill between the widespread and the diurnally forced cases. MAPLE uses the concept of Lagrangian extrapolation, which assumes that the evolution of precipitation patterns is dominated by advection or by systematic growth and decay (Germann et al. 2006). Therefore, when a precipitation case is dominated by growth and decay processes (which is likely the case for the diurnally forced cases, and less so for the widespread cases), MAPLE is not able to capture the evolution of the precipitation system, and the resulting QPF skill is poor. Thus, for the diurnally forced cases, using a radar-data-assimilating convective-allowing model results in forecasts more skillful than Lagrangian persistence forecasts after only 2 h.

Fig. 15.
Fig. 15.

between CN-Radar (solid) and MAPLE-Radar (dashed) as a function of bandpass scale at different forecast lead times (panels), averaged over the entire dataset (black), over the widespread cases (red), and over the diurnally forced cases (blue). Each point on the scale axis actually corresponds to the bandpass component between and , with the filtering being done using the Haar wavelet transform, sampling at scale of 4, 8, 16, 32, 64, 128, 256, and 512 km.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0232.1

6. Conclusions

This paper further investigates the mesoscale predictability of precipitation by the CAPS SSEF system during spring 2008. Using a dataset of 22 precipitation events, we have attempted to explain the case-to-case variability of predictability. Our approach consists of characterizing first the predictability of precipitation by the CAPS SSEF and then relating it to some characteristics of the events (which can be described a priori). By analogy to the Lorenz (1963) chaotic system, this would result in mapping the predictability estimates associated with different regions of the attractor. But as mentioned before, it is impossible to map the entire phase space of an atmospheric model. For this reason, we attempt to find certain parameters that could represent different states of the atmospheric model and estimate the predictability for different values of these parameters. The predictability was characterized in terms of the normalized ensemble spread , the QPF skill of the control member CN (quantified by ), and the decorrelation scale . The parameters chosen for the case classification were the , related to the strength of the quasigeostrophic forcing for ascent; the convective-adjustment time-scale , related to whether convection is or not in equilibrium with the large-scale flow; and the fractional coverage of precipitation.

The results show that, on average, large-scale forcing (either or ) is only weakly related to ensemble spread and unrelated to QPF skill. When a relationship does exist, it shows diurnal variability, being most important in the afternoon hours. Additionally, the spatial extent of the precipitation system, as quantified by the fractional precipitation coverage, was shown to explain most of the case-to-case variability of precipitation predictability. Many factors may account for the lack of stronger relationships. The evolution of convective precipitation systems is mostly controlled by mesoscale features (Duda and Gallus 2013), and hence a clear relationship between the strength of the forcing and the predictability of precipitation is not necessary. With respect to , equilibrium and nonequilibrium regimes represent extremes of a continuous distribution of values (Zimmer et al. 2011). Therefore, while a relationship might not be strong on average, the two extremes should show significantly different predictability estimates.

From the dataset of 22 cases, two types of cases show a distinct behavior in terms of both their characteristics and their predictability. First, there are the widespread precipitation systems, which are characterized by their large spatial extent and lower values. Second, there are cases that are modulated by the diurnal cycle of solar heating. These cases have large values of and a clear diurnal cycle of fractional precipitation coverage.

The predictability and certain model sensitivities were analyzed separately for the two types of cases. The widespread cases were found to have a much better predictability both in terms of ensemble spread and QPF skill. However, despite these cases being more predictable than average, the predictability at meso-γ and meso-β scales is lost just as rapidly as for the rest. It seems that the decorrelation of the ensemble members with forecast lead time and spatial scale depends only on the type of IC/LBC perturbations and not on the actual initial conditions.

On the other hand, the second type shows a larger ensemble spread with a marked diurnal cycle, emphasizing again the importance of properly representing the diurnal cycle of precipitation in NWP models (Berenguer et al. 2012; Johnson et al. 2014; Clark et al. 2007).

The dependence of the effect of the radar data assimilation on the type of case was also investigated. In contrast to Craig et al. (2012), for this dataset, the effect of radar data assimilation on QPF skill does not depend on the type of event. However, the forecasts analyzed herein were initialized at 0000 UTC, when precipitation systems, even the diurnally forced, are already well organized. A different dependence of the effect of radar data assimilation on the forecast regime is expected for forecasts initialized at a time when storms are initiated or decaying.

The comparison between Lagrangian persistence nowcasts and forecasts from CN was revisited with a focus on the regime dependence. It was found that while CN skill does not show a case dependence during the first 6 h, MAPLE does, resulting in CN outperforming MAPLE after only 2 h for diurnally forced cases.

Some limitations of this analysis are noted. Here, is computed using NAM output, while it may have been computed using the CN output. While this choice was due to the availability of the data, using the NAM output is justified given that the control member of the SSEF system is driven by NAM. It is less credible to use the NCEP data to compute the . However, we tried using NAM analysis and forecasts to calculate but we found that the high resolution of the NAM output (12-km grid spacing) results in very noisy -vector fields (even after filtering out the mesoscale detail). Despite these limitations, we believe that the lack of stronger relationships presented in this paper is mainly due to the complexity of the physics of the precipitation systems themselves.

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. This work was funded by the Natural Science and Engineering Research Council of Canada (NSERC) and Hydro-Quebec through the IRC program. Dr. Konstantinos Menelaou is acknowledged for the careful proofreading of the manuscript.

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1

SZY15 presents results for forecasts of hourly accumulations. For Fig. 1, the decorrelation scale was computed for forecasts of simulated instantaneous reflectivity with a temporal resolution of 1 h.

2

The forecasts are initialized daily at 0000 UTC. Hence, 2000 UTC is equivalent to a 20-h lead time. Unless otherwise mentioned, whenever the UTC time is used it corresponds to the equivalent forecast hour.

3

The distribution of coverage might differ for observed precipitation, but we are interested in having an indicator of the type of case before the verification becomes available.

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