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  • View in gallery

    Surface elevation (m) above sea level in the inner-model domain. The Front Range of the Rocky Mountains is not labeled but encompasses the western third of the domain. Arrows denote typical low-level flow direction with notable terrain-induced circulations.

  • View in gallery

    Outer domain with inner domain (d02) shown in white box.

  • View in gallery

    An example of CI algorithm output from 2300 UTC 1 Aug 2010 showing (a) interpolated observed reflectivity and (b) identified CA objects.

  • View in gallery

    Performance (Roebber) diagram for each CI case showing SR on the x axis and POD on the y axis. Bias is shown through dashed lines and CSI through curved, solid lines. Circles are filled based upon the case-averaged object C error. Green denotes C errors less than 55 km, yellow 55–70 km, and red greater than 70 km.

  • View in gallery

    Linear fit between domain-averaged vertical velocity (Pa s−1; vertical axis) and (top) average case-object C error and (bottom) CSI.

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    (top) Temporal distribution of all CI objects in 5-min bin intervals, (middle) its fitted polynomial distribution, and (bottom) the fitted polynomial of the ratio of modeled to observed object count.

  • View in gallery

    Weighted spatial density difference in CI occurrence between model and observed, where positive (negative) values denote more modeled (observed) CI objects.

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Assessing the Predictability of Convection Initiation in the High Plains Using an Object-Based Approach

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  • 1 Atmospheric Science Group, Department of Mathematical Sciences, University of Wisconsin–Milwaukee, Milwaukee, Wisconsin
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Abstract

This study investigates the short-range (0–12 h) predictability of convection initiation (CI) using the Advanced Research Weather Research and Forecasting (WRF) Model (ARW) with a horizontal grid spacing of 429 m. A unique object-based method is used to evaluate model performance for 25 cases of CI across the west-central high plains of the United States from the 2010 convective season. In the aggregate, there exists a high probability of detection but, due to the significant overproduction of CI events by the model, high false alarm and bias ratios that lead to modestly skillful forecasts. Model CI objects that are matched with observed CI objects show, on average, an early bias of about 3 min and distance errors of around 38 km. The operational utility and inherent biases of such high-resolution simulations are discussed.

Current affiliation: Atmospheric Science Group, Department of Geosciences, Texas Tech University, Lubbock, Texas.

Corresponding author address: Dr. Clark Evans, Dept. of Mathematical Sciences, University of Wisconsin–Milwaukee, P.O. Box 413, Milwaukee, WI 53201-0413. E-mail: evans36@uwm.edu

Abstract

This study investigates the short-range (0–12 h) predictability of convection initiation (CI) using the Advanced Research Weather Research and Forecasting (WRF) Model (ARW) with a horizontal grid spacing of 429 m. A unique object-based method is used to evaluate model performance for 25 cases of CI across the west-central high plains of the United States from the 2010 convective season. In the aggregate, there exists a high probability of detection but, due to the significant overproduction of CI events by the model, high false alarm and bias ratios that lead to modestly skillful forecasts. Model CI objects that are matched with observed CI objects show, on average, an early bias of about 3 min and distance errors of around 38 km. The operational utility and inherent biases of such high-resolution simulations are discussed.

Current affiliation: Atmospheric Science Group, Department of Geosciences, Texas Tech University, Lubbock, Texas.

Corresponding author address: Dr. Clark Evans, Dept. of Mathematical Sciences, University of Wisconsin–Milwaukee, P.O. Box 413, Milwaukee, WI 53201-0413. E-mail: evans36@uwm.edu

1. Introduction

The development of deep, moist convection, known as convection initiation (CI), refers to the process in which, through a series of cumulus congestus plumes, conditionally unstable air parcels penetrate and overcome a layer of relative negative buoyancy to buoyantly accelerate upward and develop a deep, mature, and precipitating updraft. Conversely, failed CI is the process in which CI is not fully achieved (shallow or nonsustained deep updraft). Failures in CI forecasts (false positive) and nonforecasts (false negative) are two of the most vexing occurrences in operational meteorology and their impacts on later forecasts can be significant. This is attributable to errors from incorrect redistributions of heat and moisture that quickly grow upscale, manifesting through nonlinear moist dynamics (e.g., Zhang et al. 2003). Once deep, moist convection has initiated in a favorable atmospheric environment, it may become severe, producing flash flooding, damaging straight-line winds, large hail, and tornadoes. The societal and economic impacts of severe convection are enormous, resulting in more than $4 billion in damage annually across the United States (Weiss et al. 2011).

Deep, moist convection can be initiated by a large array of different atmospheric circulations, oftentimes referred to as triggering mechanisms. This is one of three basic elements in developing convection, in addition to the presence of convective available potential energy (CAPE) and a sufficient moist layer at low to midlevels of the troposphere (Doswell 1987; Johns and Doswell 1992). The most common triggering mechanisms, as summarized by Jorgensen and Weckwerth (2003) and Weckwerth and Parsons (2006), include frontal boundaries, drylines, elevated convergence zones, gust fronts, sea breezes, orographic circulations, undular bores, and horizontal convective rolls. Further, inhomogeneities in land surface characteristics can produce sufficient vertical motion to force CI, particularly under strong surface heating (Kang and Bryan 2011).

The occurrence of CI, however, requires more than just the presence of a triggering mechanism. Rather, a favorable interaction between phenomena on multiple atmospheric scales is required to establish an environment favorable for deep, moist convection and then to force CI. The synoptic to mesoalpha scales are responsible for the creation of an environment favorable for convection (e.g., Doswell 1987; Schumann and Roebber 2010). This is accomplished through the advection of higher potential energy air at low levels and ascent that acts to increase CAPE and reduce convective inhibition (CIN). Mesobeta- to microscale phenomena [spatial order 10 km–100 m; Fujita (1981)] are responsible for the local deepening and moistening of the convective boundary layer (CBL) through ascent that can ultimately eliminate CIN (e.g., Wilson et al. 1992; Doswell 1987; Markowski et al. 2006; Weckwerth et al. 2008). This is primarily accomplished through local moisture convergence that results in localized preferred areas of CI where parcels are most likely to reach their level of free convection (LFC). In general, CI is most likely when deep CBL plumes occur in regions of mesoscale ascent (e.g., Ziegler and Rasmussen 1998; Markowski et al. 2006).

The occurrence of CI is also remarkably sensitive to its thermodynamic environment (e.g., Lee et al. 1991; Mueller et al. 1993; Crook 1996; Weckwerth 2000; Houston and Niyogi 2007). This is true regarding the low-level thermodynamic state, as demonstrated by Crook (1996), who showed that a temperature change of 1°C in a mixed surface layer determined whether CI did or did not occur. Furthermore, if CI did occur, a 1 g kg−1 change in low-level moisture content resulted in considerable variability in storm intensity. This is one of the leading factors that limit the practical predictability of CI because such variability falls into the range of observational uncertainty and local variability (e.g., Zhang et al. 2002, 2003; Weckwerth and Parsons 2006). Furthermore, although our current observational network can resolve synoptic- and, to a degree, mesoscale features, it is unable to properly resolve (if at all) smaller-scale details that are known to be crucial to CI (Weckwerth et al. 1999; Weckwerth and Parsons 2006).

Previous studies (e.g., Fowle and Roebber 2003; Weisman et al. 2008) have demonstrated that credible numerical forecasts of warm-season CI events are possible at lead times of 24–48 h, even if the initial conditions for the numerical forecasts contain only synoptic-scale information. In addition, two recent studies have provided insight into the ability of state-of-the-art numerical models to predict CI. In an examination of the predictive capabilities of an ensemble of convection-permitting numerical simulations of CI events, Kain et al. (2013) found that models with horizontal grid spacing Δx ≈ 4 km represent many aspects of CI explicitly with a high probability of detection and no systematic bias in the timing of the day’s first CI event. Duda and Gallus (2013) showed that 3-km horizontal grid-spacing Weather Research and Forecasting (WRF) simulations for convective events across the United States demonstrated good skill in matching observed CI objects prior to the upscale growth of convection. They found that on average model simulations showed no discernible timing error in forecasted CI with a mean spatial displacement error of 105 km.

It is well documented that convection-permitting model forecasts typically demonstrate better short-range (0–24 h) predictions of deep, moist convection compared to their convection-parameterizing counterparts regarding rainfall amounts, storm mode, and mesoscale circulations (e.g., Weisman et al. 1997; Mass et al. 2002; Fowle and Roebber 2003; Done et al. 2004; Roebber et al. 2004; Wilson and Roberts 2006; Kain et al. 2008; Weisman et al. 2008; Clark et al. 2010). However, beyond 24 h, error growth and spread tend to increase faster in convection-permitting models than in coarser convection-parameterizing models. This issue was well described by the works of Zhang et al. (2002, 2003), which demonstrated how small-scale errors, in the form of misplaced and misrepresented deep, moist convection, can quickly spread and amplify through the convective to mesoscales before growing upscale to the synoptic scale. This places a stricter intrinsic time limit of predictability in convection-permitting models than in coarser convection-parameterizing models, where the greater range of resolved scales in convection-permitting models can allow for the greater net accumulation and subsequent upscale growth of errors.

Although horizontal grid spacings in operational convection-permitting models allow for the resolution of at least larger storm-scale features, they remain too coarse to truly resolve all scales of motion in deep, moist convection. Bryan et al. (2003) demonstrated that horizontal grid spacing on the order of 100 m is necessary to adequately resolve the structure and dynamics of squall lines. However, better resolving deep, moist convection may not be as simple as decreasing the grid spacing to order 100 m. This was demonstrated by Adlerman and Droegemeier (2002), who found that the dynamics of an idealized modeled supercell change drastically as the horizontal grid spacing is decreased from 2 to 0.5 km but that the results are very sensitive to only minor changes in physical parameter constants such that results become nearly interchangeable. Furthermore, Kain et al. (2008), Schwartz et al. (2009), and Clark et al. (2012a) suggest that forecast skill is not improved by moving to a horizontal grid spacing of 1–2 km from a horizontal grid spacing of 4 km. This may arise in part due to a need for more sophisticated physical parameterizations at high resolution (e.g., Coniglio et al. 2010, 2013) or an inability to accurately depict these scales within operational analyses of the atmospheric state (Roebber et al. 2004). Consequently, it is unclear as to whether forecasts of CI timing and location can be improved solely by reducing the horizontal grid spacing to a value that is capable of explicitly resolving many plausible physical triggering mechanisms.

Given the above, this study focuses upon examining and quantifying the short-range (0–12 h) forecast skill of a series of very high-resolution numerical model simulations in predicting the timing and location of CI events. The remainder of this manuscript is structured as follows. The methodology behind the study, including the numerical model configuration and case selection, is presented in section 2. Primary results from this work are presented in section 3. The key results are summarized, and implications from this work are presented in section 4.

2. Methodology

In this work, we investigate CI events that occurred during the 2010 convective season across the west-central high plains of the United States. The west-central high plains contain a diverse set of topographic features that, through synoptic flow interaction and sensible heating, develop mesoscale circulations that are capable of priming the environment for and, in some cases, initiating deep, moist convection (Fig. 1). The most notable topographic features in this region are the Black Hills in western South Dakota and the Front Range of the Rocky Mountains across central and eastern Colorado and Wyoming. Two additional smaller-scale topographic features adjacent to the Front Range include the Palmer Divide and Cheyenne Ridge, which are responsible for developing the Denver convergence vorticity zone (e.g., DCVZ; Denver cyclone) and Chugwater circulation, respectively (Abbs and Pielke 1986; Wilczak and Christian 1990; Schreiber-Abshire and Rodi 1991; Davis 1997). In addition, the 2010 convective season from mid-May through mid-August was anomalously active across the selected region (e.g., Blunden et al. 2011), with numerous cases of CI forced by a large variety of triggering mechanisms.

Fig. 1.
Fig. 1.

Surface elevation (m) above sea level in the inner-model domain. The Front Range of the Rocky Mountains is not labeled but encompasses the western third of the domain. Arrows denote typical low-level flow direction with notable terrain-induced circulations.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-13-00089.1

In this study, we focus on quantifying the ability of a high-resolution numerical model to accurately forecast CI when only synoptic- to mesoalpha-scale features are resolved within its initial and lateral boundary conditions. A radar-reflectivity-based method (e.g., Gremillion and Orville 1999) is used to objectively define both observed and simulated CI objects (additional details in section 2b). The ability to replicate CI occurrence within realistic time and distance constraints is the criterion through which model performance is evaluated.

a. Model description

We create a sample set of CI cases by first identifying all days between 15 May and 15 August 2010 on which CI occurred across the west-central high plains. From this sample set, only cases in which the first CI event occurred between the late morning and late afternoon hours (between 1600 and 0000 UTC) in a convectively undisturbed (e.g., pristine, “clean slate,” or nonoverturned) environment are kept. Subsequently, a set of 23 days is randomly selected for further examination. Also included in the sample set are two “null” cases that were most apparent during the selected time period in which CI was expected in operational forecasts 6–12 h prior but was not observed (based upon Storm Prediction Center day 1 convective outlook discussions). The reason for choosing these “null” CI cases is to investigate, at least to a small degree, the model’s ability to properly not produce CI. Although worthy of study in their own right, only two null cases are selected due to our desire to focus primarily on events where potentially hazardous deep, moist convection was observed. We note that this study considers all CI events that occurred on a given day, and not just the first (Duda and Gallus 2013) and/or severe (Kain et al. 2013) CI events on that day. The sample size of 25 cases is chosen to give a reasonable statistical sample while taking into consideration the computational requirements for such high-resolution simulations. Table 1 shows the selected dates of CI events with the approximate time of the first observed CI event and primary triggering mechanism(s). To assess deep, moist convection development and select CI cases, we use the Storm Prediction Center’s storm event archive (http://www.spc.noaa.gov/exper/archive/events) and the University Corporation for Atmospheric Research (UCAR) archive (http://locust.mmm.ucar.edu) of composite radar reflectivity, satellite, and surface observations.

Table 1.

Selected dates, start times, and apparent triggering mechanisms of CI events.

Table 1.

Version 3.3.1 of the Advanced Research Weather Research and Forecasting (WRF) Model (ARW; Skamarock et al. 2008) is used to simulate each of the 25 selected cases. To resolve CI triggering mechanisms down to spatial scales on the order of 1 km, simulations use 429-m (3/7 km) horizontal grid spacing in a nested inner grid across the western high plains. This inner domain is embedded within an outer domain with 3-km horizontal grid spacing that extends across the western two-thirds of North America (Fig. 2), with two-way feedback permitted between domains. This work only assesses the model performance of CI in the very high-resolution inner domain, as the outer domain’s sole purpose is to provide lateral boundary conditions with mesoscale-resolved details. Both domains use a hybrid sigma-pressure vertical coordinate with 35 vertical levels, including 8 within the lowest 1 km above ground level. Although we use fewer vertical levels than a similar sub-3-km (Δx = 1 km) CI modeling study by Xue and Martin (2006), which employed 50 vertical levels, we feel our selection is sufficient, at least broadly, to resolve vertical structures critical to the CI problem [i.e., dry air entrainment; Markowski et al. (2006)]. For all cases, simulations are initialized at 1200 UTC on the day of a CI event to specifically investigate the short-range (6–12 h) forecast performance of the high-resolution model configuration. Sufficient model spinup is allowed from the cold-start initialization given an average CI start time for all simulated objects of 1936 UTC. Model simulations are initialized with the 1200 UTC 12-km North American Mesoscale (NAM) model operational analysis. Subsequent NAM analyses at 6-h intervals provide lateral boundary conditions on the outer domain. This is not believed to influence the results presented herein given sufficient spatial displacement (approximately 1000–1500 km) between the inner and outer domain’s respective lateral boundaries.

Fig. 2.
Fig. 2.

Outer domain with inner domain (d02) shown in white box.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-13-00089.1

Because of the grid spacing chosen for both model domains, no cumulus parameterization is used. The Rapid Radiative Transfer Model (RRTM; Mlawer et al. 1997) and Dudhia (1989) schemes are used to parameterize long- and shortwave radiation, respectively. The Noah land surface model (Ek et al. 2003) is used with four soil levels. The Thompson et al. (2008) bulk microphysical parameterization handles microphysical species, which include the mixing ratio of six different microphysical species and number concentration for cloud water and ice. The local mixing Mellor–Yamada–Janjić (MYJ) scheme (Janjić 2001) is used to parameterize planetary boundary layer (PBL) turbulent energy. While a horizontal grid spacing of 429 m (and a vertical grid increment on the order of 100–150 m within the PBL) is able to crudely resolve the largest eddies within the PBL, a PBL parameterization is nevertheless necessary to reasonably represent the presence and impacts of smaller-scale PBL eddies. Although certain biases exist in model solutions with the use of the MYJ PBL scheme (see section 3b for details), we choose this rather than another parameterization due to its widespread use in quasi-operational and research simulations of deep, moist convection (e.g., Duda and Gallus 2013; Kain et al. 2013). Model output every 5 min in the inner domain complements the high spatial resolution of the model configuration to analyze the evolution of convective-scale features and verify the simulated reflectivity at the sampling frequency of the observed radar reflectivity.

b. CI object algorithm

In this study, radar reflectivity at the −10°C level is used to assess CI (e.g., Kain et al. 2008 and references therein). We chose this method to remove the effects of bright branding that may falsely amplify the occurrence and areal coverage of deep, moist convection (Gremillion and Orville 1999). For the numerical simulations, radar reflectivity at the −10°C level is obtained using version 4 of the Read/Interpolate/Plot (RIP) postprocessing software. It should be noted that the radar reflectivity calculation used in RIP does not utilize the second-moment number concentration provided by model output with versions above 3.4.1 of ARW. Although such differences in calculation method will impact the postprocessed representation of radar reflectivity to some degree (e.g., Clark et al. 2012a), the general characteristics and the ability to produce CI objects are expected to be meaningfully represented by the chosen simulated reflectivity calculation algorithm.

To calculate the observed reflectivity at the −10°C level, we first trim three-dimensional radar data, obtained from the National Severe Storms Laboratory’s National Mosaic and Multisensor Quantitative Precipitation Estimate (NMQ) system (Zhang et al. 2005; Langston et al. 2007), which is on a 0.01° latitude × 0.01° longitude grid, and fit it to the area of the inner-model domain. Second, we obtain hourly Rapid Update Cycle (RUC; Benjamin et al. 2004) model analysis data and interpolate it onto the NMQ grid. The third step uses linear interpolation to find the approximate height of the −10°C isotherm at each observed grid point in the RUC dataset, to which the observed reflectivity data are subsequently interpolated. Note that both observed and modeled −10°C radar reflectivity data are interpolated onto a common 1.29-km (9/7 km) horizontal spaced grid. The common grid spacing is chosen such that it is a integer multiple of the inner-model domain grid spacing, allowing for simple spatial interpolation of model data while requiring minimal up interpolation of data from the NMQ grid.

To examine the spatiotemporal numerical skill of CI, we use an object-based approach to verify CI. Following the work of Clark et al. (2012a,b, 2013) and Kain et al. (2013), we develop an algorithm to identify convectively active (CA) objects based upon radar reflectivity and we track them through time and space. This idea stems from initial work by Davis et al. (2006a,b, 2009), who developed a method to verify precipitation forecasts through the characteristics of observed and forecast rainfall objects. Our algorithm starts by applying a Gaussian smoother (σ = 1.25; the standard deviation of the number of grid cells to the half-width of the Gaussian function) to the observed and simulated reflectivity fields on the common 1.29-km grid. Grid cells that do not have both an interpolated reflectivity ≥35 dBZ and smoothed reflectivity ≥25 dBZ are filtered out, helping to remove very small, borderline CA objects (i.e., localized, transient ≥35-dBZ cells in stratiform precipitation). From here, the algorithm uses an expanding radial search function to cluster CA grid cells that are within two grid points (~2.58 km) from the last cell clustered into an object. This radial distance threshold of two grid points is found to group cells into an object that may contain a small discontinuity in 35-dBZ reflectivity but still properly separates close neighboring objects.

To ensure these groupings of CA grid cells (objects) are resolvable features, we require that they have an area equal to or greater than the length scale of resolvability [approximately eight model gridpoint lengths; e.g., Skamarock (2004)] squared, where an 8 × 8 model gridcell area is ~12 km2. Although lesser values can be sufficient to resolve features of interest, we choose eight gridpoint lengths to create a proxy length scale of resolvability between the model and the coarser observed reflectivity. The algorithm discards objects with areas less than the size threshold, as we consider these objects poorly resolved or altogether unresolved by the model and/or by radar observations and thus not indicative of true, sustained deep, moist convection features. This series of steps is performed on a time interval of 5 min for both observed and model-simulated data. Figure 3 shows the beginning and end results of the CI algorithm for a representative example from 1 August 2010.

Fig. 3.
Fig. 3.

An example of CI algorithm output from 2300 UTC 1 Aug 2010 showing (a) interpolated observed reflectivity and (b) identified CA objects.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-13-00089.1

Next, the CI algorithm attempts to match each CA object to CA objects present at previous times. The algorithm matches CA objects to previous objects when they are within 10 min of each other and their centroids are within 10 interpolated gridpoints (~13 km) distance. If there are multiple matches at some time to one object from the previous time, the algorithm matches the object that is spatially closest. The distance-matching requirement helps us to properly match objects through time by constraining how far the algorithm should search, but is within the realistic bounds of distance covered by very fast storm movement in 5–10 min (i.e., ≤78 km h−1). The algorithm declares CI once an object has been matched for 30 min, demonstrating a mature, sustained, precipitating deep, moist convective updraft. We set the timing and centroid location of CI from when the object was first identified to be CA. One shortcoming of this methodology is the potential for a sequence of pluming cells that do not reach maturity to be incorrectly flagged as CI if their proximity in time and space is constrained within threshold values. However, we anticipate any resultant issue to be minimal as the same algorithm methodology is applied to both observations and model simulations.

c. Verifying CI objects

Once a list of observed and modeled CI objects is created for a case, we compare each set of objects to each other for model verification purposes. We develop a verification algorithm to go through each observed CI object in a case, creating a sorted list of model CI objects that are closest in time and space. A flow-dependent error metric is then used to order the closeness of the modeled and observed CI objects. This error metric is given as
e1

For simplicity, we represent this spatiotemporal error as C in units of kilometers. The spatial difference in forecast and observed CI objects Errord and temporal difference Errort in (1) have units of kilometers and hours, respectively. The characteristic velocity Velocityc (km h−1) approximates to first order the speed of the storm object. We approximate Velocityc by calculating the layer-averaged horizontal wind between the 4th and 16th vertical levels in model output at hourly intervals, approximating the magnitude of the deep-layer advecting winds in the 850–450-hPa layer. We apply maximum distance and time error thresholds to matched objects such that if a modeled CI object exceeds either a timing error of 60 min or a distance error of 100 km, it is unmatched and classified as a false negative. These values are chosen to represent a balance in time and space errors on the mesoscale, accounting for maximum CI object motions of up to 100 km h−1 (~54 kt; 1 kt = 0.51 m s−1). Although the time and space scales of CI are smaller than these thresholds, they are necessary to quantify the error statistics of CI forecasts that can often be larger than the scales of the CI themselves. The selection of these thresholds in relation to statistical results is discussed more in the following section. In such a case, this means that the closest model CI object cannot be matched to an observed CI object and demonstrates the model’s inability to reasonably produce a CI object in spatiotemporal proximity to an observed one. These error thresholds correspond to a C error of 104 km for an average characteristic velocity from all CI cases of around 8 m s−1 (~29 km h−1).

Because we are specifically focusing on the spatiotemporal character of CI objects, rather than just its occurrence alone, CI objects that develop in the outer simulation domain may incorrectly be flagged as CI if and when they move into the inner simulation domain. To mitigate this problem, we apply a “buffer zone” near all lateral boundaries of the common grid domain in which all CI objects occurring within the buffer zone are discarded. To account for potentially fast storm motions (up to 43 m s−1), we use a buffer zone of 10 interpolated gridpoints (~13 km) inward from the lateral boundaries for all 25 CI cases.

3. Results

a. Model verification

Although not all aspects of CI can be realistically verified by standard skill scores, as has been done with coarser grid-spacing convection-permitting model studies (e.g., Fowle and Roebber 2003; Weisman et al. 2008; Stratman et al. 2013), we are able to verify the occurrence of CI through the use of a contingency table (Wilks 1995; Fowle and Roebber 2003; Kain et al. 2013). Through all 25 cases, model simulations produce a total of 651 CI objects compared to only 249 observed CI objects. A total of 126 model CI objects are successfully matched with observed CI objects within the spatial and temporal error thresholds mentioned above and are considered true positives or hits (a). Another 411 modeled CI objects are considered false positives (b) that the model produces in excess of actual CI objects observed. This overproduction of CI is further discussed later in this paper. An additional 123 model CI objects fail the temporal and spatial error thresholds described in section 2 and, thus, are not successful matches (c). A fourth classification in the contingency table is the true negative or nonevents (d), which on a grid-to-grid verification is when forecast occurrence is correctly negative. Note that letters (a)–(d) [where (a) = true positives, (b) = false positives, (c) = false negatives, and (d) = true negatives] correspond to their appearance in Eqs. (2)(5) and follow from the contingency table cited at the outset of this section. In this study, however, this classification is inapplicable as there is no way of forecasting a nonexistent object.

The collection of classified objects on a contingency table can then be used to quantify the overall model performance in forecasting the occurrence of CI (e.g., Murphy 1993; Roebber 2009). One metric used is the probability of detection [POD; Eq. (2)], a ratio that ranges from 0 to 1 and represents the ratio of correctly forecasted objects to the total number of observed CI objects. Another metric is the false alarm ratio [FAR; Eq. (3)], the ratio of overproduced forecast CI objects to the total number of CI objects forecasted, which also ranges from 0 to 1. A third metric is the bias score, which is the ratio of the positive CI forecasts (both true and false) to the number of observed CI objects [Eq. (4)]. The final metric used is the critical success index [CSI; Eq. (5)], which is the ratio of correctly forecasted objects to the total number of observed and forecast CI objects:
e2
e3
e4
e5

The POD for all CI events occurring within the 25 cases examined is 0.506. This, however, comes at the expense of an FAR of 0.765, a result of the significant overproduction of modeled CI, as inferred from a bias score of 2.16. Further, the model produces a CSI of 0.191. Although these performance measures generally fall short of the performance of both operational and nonoperational models alike (e.g., Fowle and Roebber 2003; Roebber 2009; Duda and Gallus 2013; Kain et al. 2013), we emphasize that they should be viewed in light of the rather strict spatiotemporal thresholds, relative to past works, used to match forecast to observed CI objects [e.g., Fowle and Roebber (2003), CI within domain; Wilson and Roberts (2006), CI within 250 km and 3 h; Duda and Gallus (2013), CI within domain; Kain et al. (2013), CI within 5 h].

Results from the 23 cases in which CI was observed are plotted (Fig. 4) on a Roebber performance diagram (Roebber 2009). This diagram visualizes multiple measures of forecast quality, including POD, success ratio (SR; 1-FAR), bias, and CSI. Additionally, each point for each case is color coded based upon the case-average C error. In general, better-performing forecasts will be positioned toward the top-right portion of the diagram near the y = x line, indicating high POD and SR and near unity bias. Most points on this diagram are skewed to the top to middle left, indicating higher values of POD and bias, coinciding with the model overproduction of CI objects. Several points are seen along the right y axis, indicating an FAR of 0, but relatively lower values in POD. These points represent four cases in which model simulations actually underforecast the number of observed CI objects.

Fig. 4.
Fig. 4.

Performance (Roebber) diagram for each CI case showing SR on the x axis and POD on the y axis. Bias is shown through dashed lines and CSI through curved, solid lines. Circles are filled based upon the case-averaged object C error. Green denotes C errors less than 55 km, yellow 55–70 km, and red greater than 70 km.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-13-00089.1

As mentioned earlier, we use a flow-dependent error metric to quantify the spatial and temporal errors of a modeled CI object. This flow dependence serves two purposes: to allow for consistency in dimensional units between the spatial and temporal errors and to correctly characterize timing errors associated with differences in storm motion. For the 126 model CI objects matched, an average error of −2.78 min (negative denotes early) is seen, with the distribution of timing errors near-normally distributed about zero time error. The average distance error for modeled CI objects is 38.4 km. The average C error is approximately 48 km per object. Average object C errors for all cases range from about 14 to 91 km. CI occurrence and spatial–temporal error results are shown in Table 2. The average CI object distance error we find and, to a lesser extent, the average timing error are similar to the results of Duda and Gallus (2013), who found average model object errors of −0.56 min and 105 km for initial CI events occurring prior to the subsequent upscale growth of deep, moist convection.

Table 2.

Case-averaged model performance, as ordered by CSI score. Neither of the two null cases, each with a CSI of 0, is not shown below.

Table 2.

Relaxing the spatiotemporal-matching thresholds utilized herein helps to slightly improve CI occurrence skill scores. However, a delicate balance exists between the statistical occurrence scores (e.g., POD, CSI) and spatiotemporal object scores (e.g., average C errors) based upon the choice of spatial- and temporal-matching thresholds. Less strict thresholds yield more matches between observed and modeled CI objects, helping to bolster the statistical occurrence scores; however, these matches are more distant in time and space, thus reducing average object C errors for CI cases. The opposite is true for stricter matching thresholds. If we use stricter space (50 km)- and time (30 min)-matching thresholds, POD and CSI decrease (from 0.506 to 0.233 and from 0.191 to 0.088, respectively) while FAR increases (from 0.765 to 0.876). The decreased number of matches (from 126 to 58) though contains lower space and time errors (from 38.4 to 26.3 km and from −2.78 to −0.26 min, respectively) and lower average C errors (from 48.0 to 30.2 km). Using even stricter space (25 km)- and time (15 min)-matching thresholds, POD and CSI decrease further (to 0.080 and 0.030, respectively) with FAR again increasing (to 0.954). The further decreased number of matches (20) again contains even smaller space and time errors (from 38.4 to 15.8 km and from −2.78 to 0.25 min, respectively) and thus smaller average C errors (from 48.0 to 17.5 km). This is analogous to the duality of error described by Doswell (2004), where increasing the POD for a fixed forecast accuracy will require an increase in FAR as false negatives become false positives.

Also of note, we find that for all cases the matched model CI objects on average are larger than their observed counterparts in areal coverage by about 38 km2. These results coincide with those of Davis et al. (2006b), who found convective rainfall objects were too numerous and large in areal coverage in a 4-km convection-permitting study, although it should be noted that early versions of the Method for Object-based Diagnostic Evaluation (MODE) model tended to overestimate object sizes (e.g., Fowler et al. 2013). It is interesting to see such common tendencies within convection-permitting model studies with horizontal grid-spacing differences of one full order of magnitude, suggesting that such biases simply do not go away with the use of finer horizontal grid spacings and are likely an artifact of a more general issue with the explicit handling of deep, moist convection.

In addition, cases of CI are sorted by the strength of larger-scale ascent prior to and during a CI event, to investigate any correlation between model performance and synoptic-scale forcing. Forcing for each case is quantified using Interim European Centre for Medium-Range Weather Forecasts (ECMWF) Reanalysis (ERA-Interim; Dee et al. 2011) 500-hPa pressure vertical velocities (Pa s−1) at 6-hourly analysis times. The horizontal grid spacing of ERA-Interim is approximately 70 km, meaning the spatial scale of resolved vertical velocity structures is roughly 300–500 km. The minimum vertical pressure velocity (which ranges from −0.17 to −1.8 Pa s−1 in all cases) between all ERA-Interim grid points within our inner-model domain is averaged between the two 6-hourly analyses whose times are closest to that of the first simulated CI event for each of the 25 cases. A linear correlation between C errors (Fig. 5, top) and vertical pressure velocity values for all cases in which CI was observed reveals that synoptic forcing has a very weak positive correlation (0.146) with C errors. Stronger synoptic ascent cases on average have a slightly higher positive correlation (0.238) with CSI (Fig. 5, bottom) scores in all 25 cases. This positive correlation suggests that less negative values of ascent (weaker synoptic-scale forcing) are associated with more positive values of CSI. With a sample set of only 23 observed CI cases, however, these weak correlations are statistically insignificant to ≥50% confidence. Note that this result is insensitive to the method (averaging only negative values or using only the minimum value between the two analysis times) and reanalysis product [ERA-Interim versus the National Centers for Environmental Prediction (NCEP) Climate Forecast System Reanalysis; Saha et al. (2010)] used to obtain the estimated 500-hPa vertical velocity (not shown).

Fig. 5.
Fig. 5.

Linear fit between domain-averaged vertical velocity (Pa s−1; vertical axis) and (top) average case-object C error and (bottom) CSI.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-13-00089.1

These results are similar to those of Duda and Gallus (2013), who found no identifiable relationship between large-scale forcing and forecast model skill of CI in their 3-km convection-permitting model simulations. However, the weak relationship we find between model performance (CSI occurrence score) and synoptic forcing compared to Duda and Gallus (2013), who use different model domain placements across the United States to sample CI events, may be an artifact of our inner domain comparatively including more topographically induced circulations. In general, it should be noted that although results here may be representative of CI predictability in other regions outside of the west-central high plains of the United States, they are likely in part regionally dependent given the topographic features and climatological characteristics of the storm environments in the region.

To give additional insight into the operational utility of such high-resolution simulations, we further stratify the results to only the first CI objects that are forecast and observed for each case. For the 23 cases in which CI was observed, model simulations in four cases had the first CI object within 50 km of the first observed CI object with a timing error that did not exceed one hour. However, in nine cases the first forecast CI object was displaced more than 300 km from the first observed CI object. Although a majority of the first forecast CI objects are within 2 h of first observed CI, similar to results of Kain et al. (2013), four cases do have a first forecast CI object timing error of two or more hours. On average, the first forecast CI object is 46.6 min too early and displaced by 213 km, with a tendency to be displaced to the northeast of the first observed CI objects. Note that no spatiotemporal-matching thresholds are applied for analyzing only the first CI objects, leaving the errors unconstrained and larger in comparison to all matched average CI object errors.

Although very high-resolution model simulations here show some modest skill in occurrence and characteristics of CI, the number of simulated CI objects that contain errors on the order of 1 h and/or 100 km is rather sobering. This is in contrast to other recent predictability studies, albeit idealized, that have shown that modern numerical predication models with Δx = 1 km demonstrate rather high skill (~60%–80% success rates) in developing and maintaining certain storm modes, such as supercells (Cintineo and Stensrud 2013) and mesoscale convective systems (Wandishin et al. 2010), in idealized environments in which convection initiation is prescribed or guaranteed. Because these works utilize simplified, idealized environments, including an artificially prescribed initiation of convection, however, it further motivates the need for quantifying the errors produced by PBL schemes that influence (or may influence) CI forecasts so that they can be addressed (e.g., Coniglio et al. 2013).

b. Model tendencies and biases of CI

As is evident in the forecast bias, the model overproduces CI objects by a factor of about 2.6. Although matched CI objects on average demonstrate good model performance, they are “buried,” in a sense, by numerous falsely produced model CI objects. From an operational forecasting standpoint, the utility of a model that produces CI objects that closely match what is observed is significantly diminished if those objects are surrounded in space and time by many other false modeled CI objects. Below, we show that these overproduced model CI objects have certain spatial and temporal attributes, which suggest some possible reasons for the overproduction.

To illustrate the bias and tendencies of all model CI objects, we analyze their temporal and spatial distributions in comparison to the observations. For the temporal distribution, the time of all—and not just matched—CI objects are binned in 5-min intervals spanning the range of all CI times from 1330 to 0100 UTC (Fig. 6, top). We fit a high-order polynomial to the CI time distributions to better illustrate the difference, or spread, in CI frequency per time between the model simulations and observations (Fig. 6, middle). We also fit a high-order polynomial to the ratio of model CI objects to observed CI objects to visually demonstrate the times of greatest overproduction relative to the observed CI count (Fig. 6, bottom). This shows that the greatest model overproduction of CI occurs during the peak times of CI occurrence. On average, model forecast CI objects (both matched and unmatched) form 43 min too early compared to observed CI objects (Fig. 6, top).

Fig. 6.
Fig. 6.

(top) Temporal distribution of all CI objects in 5-min bin intervals, (middle) its fitted polynomial distribution, and (bottom) the fitted polynomial of the ratio of modeled to observed object count.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-13-00089.1

To best illustrate the spatial distribution of all CI objects, a weighted CI density field is created. We first sum the CI count at each grid point from all 25 cases on the 1.29-km common verification grid. Then, similar to Davis et al. (2006a,b), we apply a Gaussian-like weighting function that is summed at each grid cell to smooth the total CI count:
e6
where c is the number of CI events in any grid cell and d is the distance from any grid cell with c number of CI objects to the cell in which the weighting is being applied. The numerator in the exponential function represents the decaying scale factor, which we subjectively choose as 10 interpolated grid lengths (similar to Davis et al. 2006a). The differenced spatial CI density between the model and observations helps to show areas in which the model produces too many, and too few, CI objects.

Two widespread biases are apparent in the model minus observed spatial density image (Fig. 7). While a slight model underproduction of from one to two CI objects (blue shading) is seen across much of western Nebraska and in several other smaller areas, the most noticeable difference is the spatial coverage of overproduction (red shading) of model CI. The overproduction of CI objects in the model relative to the observations is most notable near the western inner-domain boundary extending east to the Front Range of central and eastern Wyoming and Colorado. Several reasons for CI object overproduction in the model are possible. One is that the observed radar sampling is more limited across far western portions of the inner-model domain near the peaks of the Front Range. In these areas, radar data from the Denver and Cheyenne Weather Surveillance Radar-1988 Doppler (WSR-88D) sites is partially obscured by terrain, thus likely limiting to (some extent) the number of observed CI objects there (not shown). Another potential reason for this overproduction is that the location of the western inner-domain boundary is near steep terrain gradients. We attempt to mitigate this issue by placing the western lateral boundary away from the steepest gradients within the Rocky Mountains (e.g., along the Continental Divide rather than to its east or west). Further, outer–inner-domain interface blending of orography in ARW that helps to smoothly ramp up or down topography height within the first five grid points of the inner domain from domain interface (Skamarock et al. 2008) is hypothesized to mitigate most, if not all, possible spurious, short-wavelength energy development from strong topography gradients near the domain interface.

Fig. 7.
Fig. 7.

Weighted spatial density difference in CI occurrence between model and observed, where positive (negative) values denote more modeled (observed) CI objects.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-13-00089.1

The third potential reason for significant model CI overproduction is the propagation of explicitly resolved deep, moist convection from the outer domain into the inner domain. Although CI objects that move into the inner-domain area, whether modeled or observed, are discounted, it is speculated that such convective objects may promote low-frequency spurious wave growth near the lateral boundary that may aid in forcing additional CI events. However, we speculate that this should be mitigated with the use of an odd outer–inner-domain grid-spacing ratio (7:1) on the ARW default Arakawa C grid, which allows for variables to have the same physical point at the interface between the outer and inner domains (Skamarock et al. 2008). Conditions from the outer domain moving into the inner domain then only need to be interpolated to fit the first row or column of inner-domain grid cells. This simple downscaling mitigates the amplification of gradients in meteorological variables in the outer domain as they move into the inner domain. Additionally, we note that there are no significant visual signs of such wave activity influencing deep, moist convection near the lateral boundaries (not shown).

Another possible reason for model overproduction of CI is the use of physical parameterizations that are not ideally suited for high-resolution applications and may resultantly degrade model forecasts. Many recent studies have suggested that commonly used physical parameterizations are not ideally suited for use with horizontal grid spacings that are capable of explicitly resolving deep, moist convection (e.g., Mass et al. 2002; Done et al. 2004, Skamarock 2004; Davis et al. 2006b; Stensrud 2007; Weisman et al. 2008; Schwartz et al. 2009; Coniglio et al. 2010; Coniglio et al. 2013; Kain et al. 2013). This has been most often suggested with regard to PBL schemes, where convection-permitting models are capable of resolving some of the larger-scale PBL circulations that may superpose with those parameterized in a PBL scheme (Stensrud 2007). This may be particularly relevant for the simulations reported upon herein, which have a horizontal grid spacing of 429 m. Further research is necessary, however, to identify if and at which grid spacings a PBL parameterization may no longer be necessary and to tune or develop PBL schemes better suited for convection-permitting model applications.

To address this speculation, we perform a very preliminary and small sample set of model integrations using a different PBL scheme. We select three cases to examine, selected from the simulations exhibiting the largest spatiotemporal errors in forecasts of the “first” CI occurrence for a given case. Three simulations are performed with the nonlocal-mixing Yonsei University (YSU) scheme (Hong et al. 2006) instead of the local-mixing MYJ scheme (Janjić 2001). The MYJ scheme that is originally used has been shown to create daytime boundary layers that are often too shallow and moist (Hu et al. 2010; Coniglio et al. 2013), while the YSU scheme, typically producing smaller biases, tends to be slightly too warm and dry in the boundary layer. Because such biases will impact the thermodynamic state prior to and during CI, we wish to investigate if such issues lead to differences at least in the statistics of CI. Verification results of CI between the MYJ and YSU PBL schemes are similar; however, in two out of the three simulations, the MYJ PBL runs produce a higher CSI and a lower object average C error (Table 3). The differences in verification scores between these two cases, however, are at or below one order of magnitude, making them statistically insignificant. Of note, visual analysis of near-surface dewpoint temperature in and around the timing of CI in these cases shows values to be, on average, 0.5°–1°C higher in the MYJ PBL runs than in the YSU PBL runs (not shown), following what has been shown recently by Hu et al. (2010) and Coniglio et al. (2013). Although these results are preliminary in nature, it is worth noting that despite biases present in forecast near-surface fields that presumably result from PBL parameterization errors, the CI forecasts (insomuch as the chosen error statistics are appropriate) do not appear to be substantially influenced by the choice of PBL parameterization.

Table 3.

CI occurrence and spatiotemporal statistics with varied PBL schemes.

Table 3.

4. Summary

ARW simulations with horizontal grid spacings of 429 m are run for 25 CI cases across the west-central high plains from 15 May to 15 August 2010. This resulted in a sample of CI events that were triggered by a diverse set of atmospheric and orographic triggering mechanisms. We investigate the ability of very high–resolution model simulations to replicate observed CI by evaluating forecast skill within the context of that documented in previous CI-focused modeling studies. This work assesses model performance using an object-based approach, classifying CI as spatially and temporally sustained radar reflectivity-derived objects. The occurrence of CI is verified with traditional statistical scores.

Objective matching of modeled and observed CI objects demonstrates an average early bias of 2.78 min per matched model object. The average distance error is 38.4 km, with model objects tending to be too large in areal coverage. The average spatiotemporal, flow-dependent C error is 48.0 km per object. Matching is only considered for model objects that are within 1 h and 100 km of an observed object. The model’s ability to produce matched CI objects is given by a POD of 0.506. However, a significant overproduction of CI objects compared to observations contributes to a FAR of 0.765 and a bias score of 2.16. The overall CSI score for all cases is 0.191. For all CI objects, matched and unmatched, between model simulations and observations, model simulations show a 43-min early bias due to CI overproduction. Model overproduction in CI is most spatially prevalent along and west of the Front Range of the Rocky Mountains and most temporally prevalent during peak heating and peak observed CI frequency. Model performance is found to be slightly higher in regard to CSI occurrence scores, albeit statistically insignificantly so, in CI cases with weaker synoptic forcing. This constrains the present operational utility of such a model configuration with matched CI objects being “buried” in time and space by the model’s overproduced CI objects. Contrasting these results with those of predictability studies of state-of-the-art model simulations of specific storm modes suggests that further improvements to the predictability of deep, moist convection remain strongly contingent upon accurately predicting CI.

Although a limited return in forecast skill improvement is expected for model simulations utilizing 429-m horizontal grid spacing compared to the more typical 3–4-km spacing in convection-permitting models (Kain et al. 2008; Schwartz et al. 2009; Clark et al. 2011), there are a couple of specific reasons as to why such improvements are inherently limited in nature. One is the interaction and superposition of explicitly resolved large-scale PBL turbulence with physically parameterized turbulent energy. Although the existence of this issue has been speculated by many (e.g., Skamarock 2004; Stensrud 2007) in regard to convection-permitting models with horizontal grid spacings of 3–4 km, how much this may be further amplified in simulations with grid spacing one full order of magnitude smaller is unknown. A second potential limiting factor is an insufficient density of observations contributing to the model initial conditions. Although mesoscale-resolving models are capable of developing mesoscale circulations with only large-scale atmospheric details provided, a “spinup” time is required to resolve such features, limiting forecast performance in the first few hours of model integration. How the inclusion of nonstandard observations impacts regional convection-permitting models beyond very short-range forecast times [0–9 h; beyond the time period of CI-focused studies from the International H2O Project (IHOP); e.g., Liu and Xue (2008); Sun et al. (2012)] warrants further study.

Given that the model performance results here are rather similar to those of other coarser (3–4-km grid spacing) model CI studies (e.g., Duda and Gallus 2013), an important question, which has been posed by Clark et al. (2011), among others, arises: How are computer resources best utilized for convective-scale model forecasts? The answer to this question is complex and likely domain and flow dependent, involving the choice of ensemble system member diversity, forecast length, grid spacing, etc. However, the given results here, operational model forecasts for convection at lead times of 6 h or greater would appear to be more practical using horizontal grid spacings of 2–4 km rather than below 1 km, given the current state of PBL parameterization schemes.

Our results, while robust, are explicitly valid only for the specific model configuration utilized and, to a lesser extent, cases examined in this research. While sensitivity to the choice of planetary boundary layer schemes appears to be minimal in preliminary testing, we cannot rule out the possibility that different findings may be obtained for different model configuration selections and different domain sizes and locations. Consequently, further research is necessary to examine the sensitivity to these choices and, by extension, further illuminate the practical and intrinsic predictability of CI under current observational and computational constraints.

Acknowledgments

We thank Ken Howard and Carrie Langston of the National Severe Storms Laboratory for providing archived 3D NMQ radar reflectivity for this project. This research benefited from discussions with Adam Clark, Chris Davis, John Peters, and Russ Schumacher. This research also benefited from constructive feedback provided by three anonymous reviewers. Partial support for this work was provided by a UWM Graduate School Research Committee award to C. Evans.

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