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Prerequisites

A total of 11 diverse cases of midlatitude convection in the central United States are used to systematically address the three scientific goals mentioned above. A case study is also used to qualitatively understand the features seen in the systematic evaluation. Section 2 describes the OSSE design, model, and experiment configurations; the IC perturbation methods; and the verification methods. Results are presented in section 3 while section 4 contains a summary and discussion.

This newly extended GSI-EnKF system was shown to accurately analyze features across multiple scales for convection-permitting forecasts of midlatitude convection Johnson et al. Given the multiscale emphasis of this study, the multiscale GSI-based EnKF system is adopted, following the configuration of Johnson et al. The analysis ensembles are used to construct IC perturbations for the experiments in this study. While greater detail can be found in Johnson et al. The outer domain analyses provide initial and lateral boundary conditions LBCs for the inner domain DA ensemble.

Simulated storm-scale NEXRAD radar observations are then assimilated on the inner domain every 5 min during a 3-h period preceding the final analysis time Fig.

Perturbation methods

The different cycling intervals for the mesoscale and storm-scale DA are chosen based on the different approximate error growth rates on the different spatial scales. Adapted from Johnson et al. Observations of wind, temperature, water vapor, and sea level pressure are then simulated by sampling the nature run at observation locations representative of the actual observation networks e. The simulated observations are then assimilated into the experiment forecasts in order to try to recover the true state of the nature run, using the GSI-based multiscale DA system.

An advantage of the OSSE framework is that the observations are perfectly known at the model grid points. For this study of IC perturbations, the OSSE framework has the additional advantage of eliminating model and physics uncertainty as a source of forecast error by using identical model configurations for the nature run and experiment forecasts i. The outer domain analyses do contain model error arising from the coarser resolution and convection parameterization.


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However, such errors only enter the convection-permitting forecasts through the ICs provided by the inner domain DA and the LBCs from the outer domain. The same 10 cases used in Johnson et al. Like the real data cases Johnson et al.

Background

Another advantage of the OSSE framework is that the nature run provides the exact truth values for verification of all forecast variables on the same grid as the forecast variables. For the hourly accumulated precipitation forecasts, rectangular verification domains for each case are chosen to include the areas of active convection at all lead times while excluding large areas where convection is neither observed nor forecast.

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For the 2-h lead time reflectivity forecasts, smaller rectangular verification domains are used to encompass each subjectively identified mesoscale area of organized convection during the first two forecast hours. Some of the forecast cases contained multiple areas of mesoscale organized convection, resulting in a total of 18 unique verification domains for the reflectivity verification. Since different MCSs on the same case occur within the same larger-scale environment, such MCSs are not treated as independent samples for statistical significance testing.

Since the verification domain for each case was a slightly different size, the main results were also recalculated using same-sized verification domains and the conclusions were not changed not shown. The actual evolution of the 20 May case study, including upscale growth of initially cellular convection into a long-lived MCS in central Oklahoma OK , has been described in Johnson et al. The nature run for this case also shows similar upscale growth of convection into a long-lived MCS, as seen in the observation contours in Fig.

This case is chosen for an initial investigation into multiscale IC perturbation methods because of the multiple scales of motion influencing such upscale growing MCSs e. This case is also selected to qualitatively demonstrate the impact of the IC perturbation methods because it is found to be representative of the systematic results in this study.


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  • In all experiments, the ensemble forecasts have the same mean analysis, provided by the ensemble mean analysis of the multiscale GSI-based EnKF DA system. The only difference among the experiments is the IC perturbations added to the ensemble mean to generate the initial ensemble. The LARGE IC perturbations are obtained by adding to the inner domain ensemble mean analysis the difference between each outer domain ensemble member and the outer domain ensemble mean both interpolated to the inner domain. SSEFs have proven useful for users interested in convective precipitation forecast applications on space and time scales ranging from very short-term warn-on-forecast applications e.

    Forecasts on such different scales may show different sensitivities to the multiscale IC perturbation methods. To provide a robust understanding of the impacts of IC perturbation methods, the convection forecasts are here evaluated in terms of both instantaneous reflectivity during the first two forecast hours and mesoscale hourly precipitation accumulation out to 9 h.

    The NEP is defined as the percentage of grid points from all ensemble member forecasts within a search radius that exceed the threshold being forecast. The use of the NEP reduces the sensitivity to errors on scales smaller than the search radius Roberts and Lean A radius of 48 km is chosen for the mesoscale hourly accumulated precipitation forecasts in order to eliminate the impact of smaller-scale and less predictable details Johnson and Wang The observation hourly accumulated precipitation in the nature run simulation is not converted to a neighborhood probability, following Johnson and Wang The reflectivity forecasts are evaluated across a range of different spatial scales i.

    The BSS provides a simple way to verify the ensemble probabilistic forecasts that is sensitive to both the reliability and resolution of the forecasts Murphy For both precipitation and reflectivity, the reference forecast for calculating BSS is the domain-averaged observed event frequency, as also used in Johnson and Wang Differences in skill among experiments are not sensitive to the choice of reference forecast since it is the same for all experiments.

    In addition to the objective verification, subjective verification is also conducted to qualitatively understand the physical processes behind the objective skill metrics for the selected, representative case. The statistical significance of differences in BSS is determined using permutation resampling of the 11 cases Hamill For reflectivity forecasts with multiple MCSs on the same day, results are first aggregated for that day and treated as a single sample since the different MCSs may not be statistically independent.

    The relatively low confidence level is chosen because an sample dataset is rather small to expect very high levels of confidence. Since the nonprecipitation variables are the directly perturbed IC variables, results for wind, temperature, and water vapor are considered first. One-dimensional detrended Fourier spectra for these variables are calculated along east—west grid lines, and averaged over all possible such grid lines Skamarock Ensemble mean error is also calculated, using the nature run as truth i.

    The lack of initial small-scale spread is a result of the coarser resolution of the outer domain ensemble used to generate the LARGE perturbations. Although the small scales are initially very underdispersive for LARGE, downscale energy propagation results in rapid perturbation growth on such scales, consistent with the results of Durran and Gingrich However, it is not clear what impacts the small-scale IC perturbations during the first hour have on the convective precipitation forecasts both during and after the first hour and on larger scales.

    The ensemble spread i. The systematic underdispersion cannot be attributed to insufficient sampling of model errors in the ensemble design because of the perfect-model OSSE framework. It also likely is not attributable to the LBC perturbations, generated on the outer domain, which does contain model error, since the underdispersion is present from the beginning of the forecasts. The systematic results here are representative of the results for the 20 May case study, and therefore not shown again for the case study.

    Ensemble spread i. The MULTI skill advantages are significant at the 1-h lead time for all thresholds, the 7-h lead for the 6. The 7- and 9-h lead times are nearly significant at the 2. The correspondence between the resolution differences and the BSS differences suggest that the same impact of the different IC perturbations cannot be achieved by simple calibration of the forecasts to improve their reliability.

    Basic perturbation theory: Differential Equation, Regular Perturbation Part I

    Brier skill score BSS of the neighborhood ensemble probability NEP forecasts from all 11 cases for hourly accumulated precipitation thresholds of a 2. Also shown are the resolution component of the Brier score at the d 2. Strictly speaking, given only one realization i. However, subjective evaluation of this case study provides physical understanding of the causes of the more systematic differences in forecast skill noted above. However, there are subtle errors such as a slight westward displacement of the axis of maximum NEP, relative to the observed MCS in the nature run, at the southern end of the MCS at later lead times Fig.

    As in Figs. The purpose is to estimate the uncertainty in the verification statistic resulting from sampling errors due to the finite ensemble size. The blue circle in a highlights the subtle area of nonzero forecast probability that is reduced in Fig. The corresponding observation of hourly accumulated precipitation is contoured in black at the same threshold. Blue and red circles highlight areas referred to in the text. The reflectivity forecasts, verified on smaller scales than the accumulated precipitation forecasts, are not considered beyond the 2-h forecast range because of the intrinsic lack of predictability at longer lead times for storm-scale features Cintineo and Stensrud Like the hourly accumulated precipitation forecasts, the differences in reflectivity forecast skill also correspond to the differences in the resolution component, rather than the reliability component Figs.

    Whereas the resolution component of the Brier score is generally better i. The vertical axis on each panel is the reflectivity threshold dB Z and the horizontal axis is the neighborhood radius km. As in Fig. Significance tests were not repeated separately for the resolution and reliability components.

    The reflectivity forecast skill differences on the 20 May case study are again representative of the systematic results Fig. Second, MULTI enhances the forecast probability outside of the observation contour at the southern and eastern edges of the observed MCS, negatively impacting the forecast skill. The causes of these qualitative differences are discussed further in the following subsections. The color scale is the same as in Fig.

    This result shows that the small-scale IC perturbations, omitted from MULTI48, also play an important role as further discussed in section 3b 3.

    Perturbation methods in optics: application to the interferometric measurement of surfaces.

    Since the mesoscale differences in the IC perturbation methods have similar qualitative impacts on precipitation and reflectivity forecasts, the qualitative explanation of these forecast differences are explained in the following subsection. One possible explanation is that this is a result of the diurnal cycle of convective precipitation. Many of the cases show more convection over larger areas during the evening hours i.

    Such perturbations may be consistent with the poorly resolved and poorly analyzed cold pools in the outer domain analysis. However, they are inconsistent with the errors of the inner domain analysis of this feature after radar DA Figs. The above discussion was also found to apply to the other cases in this study as well not shown. Generating just the mesoscale part of the IC perturbations while assimilating radar observations on the convection-permitting grid, and allowing upscale and downscale interactions with the convective scales, is therefore advantageous for storm-scale reflectivity forecasts, in addition to the mesoscale precipitation forecasts.

    As described in greater detail in Dey et al. The observed NP is calculated the same way as the forecast NP, instead of using a binary verification field as in the other NEP skill scores. Smaller values of dFSS indicate greater spread. The eFSS is calculated as the average FSS between all member-observation pairs and is a measure of the deterministic forecast accuracy of the ensemble members.

    Smaller values of eFSS indicate greater error of the individual deterministic forecasts comprising the ensemble. An advantage of this method is that it can be calculated over a range of radii to understand the scale dependence of the ensemble characteristics. It therefore takes some time for the directly perturbed variables to generate reflectivity spread. Therefore, Fig. This result shows that it is important to explicitly include such perturbations in the IC perturbation design, rather than rely on the downscale propagation of perturbation energy indicated by Fig. Figure 4 also shows that at forecast hours 2—5, MULTI is slightly less skillful than MULTI48 at all thresholds, although the difference is only significant at the 2-h lead time for the This negative impact of the small-scale IC perturbations may be related to an initial enhancement of disorganized weak convection surrounding the observed convective systems Fig.

    For the 20 May case study, the small-scale IC perturbations contribute to the overall NEP difference at some locations, especially at later lead times. The impact of the small-scale IC perturbations in this localized area although not over the entire domain is nearly as large, and at some times and places larger than, the impact of the differences in mesoscale IC perturbations. Therefore, while the mesoscale component of the IC perturbations dominates the ensemble forecast skill for this case, the small-scale IC perturbations are not entirely unimportant for the mesoscale hourly accumulated precipitation forecasts.

    Subjective evaluation of the differences between individual members of the MULTI and MULTI48 ensembles for the 20 May case study demonstrates how the small-scale IC perturbations can directly affect the development of new convection during the early forecast hours e. Given the smaller spatial scale of newly developing convection, it is not surprising that it is particularly sensitive to the small-scale IC perturbations. Such convection can then grow upscale during the forecast period, influencing the mesoscale precipitation forecast at later lead times.

    The continued development of new convection during the early forecast period thus provides a mechanism for the small-scale IC perturbations to impact the mesoscale precipitation forecasts at later lead times. The small-scale perturbation energy that rapidly develops through downscale energy propagation i. An example of this mechanism is demonstrated by ensemble member 18 in Fig. However, the corresponding MULTI member, which also includes the small-scale component of the IC perturbation, does forecast such a convective cell Fig.

    The observation contour at the 6. Although small-scale IC perturbations lead to some small but statistically significant disadvantages e. As demonstrated in the following paragraph, the small-scale early advantage corresponds to subjectively smoother probability gradients where grid-scale details of the observation contour at a particular threshold are not well forecast.

    This contrasts with the mesoscale precipitation forecasts that are improved by upscale growth of the explicitly added flow-dependent small-scale IC perturbations.

    Perturbation Methods for Saddle Point Computation

    For storm-scale reflectivity forecasts, the impact of the small-scale IC perturbations is limited to the very small scales and the time period before downscale propagation generates sufficient perturbation energy on such scales Fig. The slight degradation of skill at some lead times likely results from an increase in weak spurious convection away from the convective system caused by the small-scale perturbations.

    Subjectively, there are two clear impacts of the small-scale IC perturbations in the 20 May case study. CO;2-G Publication status Published - International journal for numerical methods in engineering , 40 2 , Jonker, Jan B. In: International journal for numerical methods in engineering. In: International journal for numerical methods in engineering , Vol. AU - van Essen, T. PY - Y1 - N2 - A finite element based method has been developed for computing time-averaged fluid-induced radial excitation forces and rotor dynamic forces on a two-dimensional centrifugal impeller rotating and whirling in a volute casing.

    CO;2-G DO - International journal for numerical methods in engineering. Access to Document London , Ltd. Reprints and Permissions. International Journal of Bifurcation and Chaos By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. Advanced search. Skip to main content.

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