One of the primary goals of ensemble NWP systems is to provide information about the evolution of possible errors over the forecast period. The sources of these errors are the uncertainties inherent in the system, whose origins include incomplete observations, assimilation hypotheses, discretization approximations and deficiencies in the forecast model. Accurately representing these uncertainties is essential for generating a reliable depiction of guidance quality. Sources of uncertainty within the forecast model are the focus of this two-part study, which investigates the utility of a recently proposed form of stochastic model error representation in the context of an operational global ensemble prediction system.
Two broad classes of strategies to represent model uncertainties have been adopted in operational ensembles: multiformulation and stochastic (Berner et al. 2017). Multiformulation approaches employ different algorithms for different ensemble members to generate a diversity of possible solutions, and include multimodel ensembles (Krishnamurti et al. 1999, 2016; Hagedorn et al. 2005) and multiphysics ensembles (Houtekamer et al. 1996; Stensrud et al. 2000; Berner et al. 2011, 2015). Stochastic strategies depend on distinct perturbations to individual members that promote the divergence of the ensemble in sensitive regions. Stochastically perturbed parameterization tendencies (Buizza et al. 1999; Palmer et al. 2009; Christensen et al. 2015) are an example of such a strategy that has been widely adopted in operational systems around the world (Palmer et al. 2009; Charron et al. 2010; Sanchez et al. 2016; Leutbecher et al. 2017). Stochastic uncertainty representation can also be implemented at the level of individual physical processes including deep convection (Lin and Neelin 2000; Teixeira and Reynolds 2008; Plant and Craig 2008), shallow convection (Sakradzija et al. 2015) and boundary layer turbulence (Rasp et al. 2018; Hirt et al. 2019; Clark et al. 2021). Such schemes depict only the sampling component of uncertainty (Plant et al. 2011); however, they benefit from the conceptual advantage of seeding ensemble diversity in close proximity to potential error sources and can improve ensemble predictions (Keane et al. 2014; Sakradzija et al. 2020; Wang et al. 2021).
This study focuses on a form of model uncertainty representation that has received increased recent attention: stochastically perturbed parameterizations (SPP). The distinguishing feature of this approach is that perturbations are made to uncertain internal model parameters rather than being applied in a post hoc manner to computed physics tendencies (Leutbecher et al. 2017). This incremental step toward process-level stochasticity allows the SPP scheme to depict uncertainty sources in a comprehensive way that ensures internal consistency and conservation (Lang et al. 2021). Unlike multiformulation approaches, the SPP technique also preserves the exchangeability of members, an attribute that promotes Gaussianity and is a prerequisite for fair evaluation techniques (Ferro 2014). Building on the work of Li et al. (2008) and Bowler et al. (2008), Ollinaho et al. (2017) describe the development of an SPP scheme in an experimental version of the ECMWF global ensemble. They conclude that despite improvements in short-range and precipitation predictions, the SPP scheme is unable to produce forecasts that are as skillful as those of the operational ensemble. Jankov et al. (2017) suggest that the SPP scheme is unable to generate sufficient ensemble spread because its sampling of potential error sources is incomplete, a finding that is consistent with the demonstration by Jankov et al. (2019) that combining SPP with other stochastic methods currently yields optimal ensemble performance. However, Lang et al. (2021) note that care must be taken in selecting these techniques to retain the conceptual benefits that the SPP scheme affords.
Spurred by the roadmap for stochastic algorithm development proposed by Leutbecher et al. (2017), refinement of the SPP scheme in the ECMWF ensemble has resulted in a configuration that yields medium-range forecasts that are generally as skillful as those generated by the operational system (Lang et al. 2021). Despite this rapid progress, important questions remain about the effects of model resolution and cycling within the ensemble data assimilation system. Similarly, Kalina et al. (2021) show that the SPP technique yields physically realistic solutions that improve the reliability of a convection-permitting ensemble; however, the perturbations worsen a dry bias in the system so that the resulting forecasts lack the sharpness required for severe-weather forecasting. These results suggest that continued development of the SPP approach has the potential to lead to a scheme that is capable of serving as the primary representation of model uncertainty in an operational context.
This study documents the development of an extended SPP scheme in the Global Ensemble Prediction System (GEPS; Houtekamer et al. 2014). In addition to free-parameter perturbations (Ollinaho et al. 2017), uncertainties in the model’s dynamical core and in the formulation of physical closures are sampled in this implementation. The resulting scheme attempts to represent a broader range of potential error sources than did its predecessors; however, there is a commensurate increase in the computational cost and the complexity of the system. As a result, careful examination of the impact of each individual perturbation (referred to as SPP “elements” hereinafter) is required to ensure an optimal balance between these competing imperatives. This element-level investigation builds on the findings of studies such as Reynolds et al. (2011) and Frogner et al. (2022) to provide insight into the physics and dynamics of model error growth that affects guidance quality. At the same time, it expands the catalog of well-documented sensitivities that will inform the development of future model uncertainty schemes.
The GEPS is introduced in section 2, with the implementation of the SPP scheme described in section 3. Section 4 contains a summary of results from both the full SPP scheme and a series of sensitivity tests in which individual elements are activated. For some of the leading elements, further investigations of processes underlying the observed sensitivities are presented in section 5. The first part of the study concludes with a discussion in section 6, which sets the stage for comparison between results from the SPP scheme and those obtained with the existing operational model uncertainty representations in McTaggart-Cowan et al. [2022, hereinafter Part II (of this investigation)].
2. Data and methods
The Global Environmental Multiscale (GEM) model is employed for all simulations discussed in this study. Its dynamical core uses implicit two-time-level temporal discretization and semi-Lagrangian advection to solve the Euler equations (Côté et al. 1998; Yeh et al. 2002; Girard et al. 2014). The global domain is represented using a pair of overlapping limited-area Arakawa C grids (Arakawa 1988) in a so-called yin–yang configuration (Qaddouri and Lee 2011) with 0.35° grid spacing. The model employs 84 terrain-following log-hydrostatic-pressure-based vertical levels (Girard et al. 2014) with a top at 0.1 hPa.
Diabatic and unresolved processes are represented using a physical parameterization suite that recently underwent a major update (McTaggart-Cowan et al. 2019b). The SPP-based GEPS ensemble employs a single physics configuration that is summarized in Table 1. Although operational GEPS forecasts include coupling to the NEMO ocean model (Gurvan et al. 2017; Smith et al. 2018), the simulations here are atmosphere only, with sea surface temperatures fixed to their analyzed values as in the GEPS data assimilation cycle (Houtekamer et al. 2014).
Configuration of the GEM physical parameterization suite used in all GEPS experiments.
The ensemble system employed in this study is a simplified form of the forecast component of the GEPS. It is initialized by subsampling analyses from the 256-member data assimilation ensemble over a 2-month boreal winter period (January–February 2020) using two different strategies. The system’s response to the full set of SPP elements is assessed in section 4 using a 20-member ensemble initialized at 36-h intervals. Forecasts for the 40 dates extend to 360-h lead times to duplicate the behavior of the operational system. To assess the isolated impacts of the individual SPP ingredients at a reasonable computational cost, a minimized system is employed in sections 4b and 5. Ensembles consisting of 10 perturbed members are initiated every 108 h (4.5 days) to promote independence (Hamill et al. 2004). This yields 15 initializations for each sensitivity test, with each forecast extending to only 120 h given the limited sample size (150 individual integrations). The leading-order signals obtained from the two ensemble configurations are qualitatively similar (section 4b), confirmation that the minimized system is sufficiently robust for SPP sensitivity analysis.
Guidance quality is assessed through comparisons with analyses produced by the Canadian Global Deterministic Prediction System (Buehner et al. 2015), interpolated onto a common 0.35° latitude–longitude grid. To minimize the volume of presented data and to improve the stability of the evaluation statistics, diagnostic results are pentad averaged for most assessments: pentad 1 (24–120-h forecasts), pentad 2 (144–240-h forecasts), and pentad 3 (264–360-h forecasts).
3. Stochastically perturbed parameterizations
The sources of error within numerical models are widespread, including problems with truncation, convergence, free parameter estimates, closures, and the depiction of physical processes. The SPP technique injects perturbations in close proximity to these underlying error sources, thereby representing the resulting uncertainty in a natural way. The conceptual framework for SPP is described by Ollinaho et al. (2017); however, each SPP implementation is unique because of the highly model-specific nature of the perturbed elements. This section begins with an introduction of the stochastic field generator used to create pseudorandom perturbation patterns (section 3a), followed by a description of the SPP scheme’s implementation in the GEPS (section 3b).
a. The stochastic field generator
An important ingredient in stochastic representations of model error is the structure of the perturbations. Although white noise (without temporal or spatial correlations) could be considered as a possible method for representing the seeds of uncertainty in a chaotic system (Lorenz 1969), Žagar (2017) and Christensen (2020) document the prevalence of large-scale errors in NWP forecasts. As a result, temporally and spatially correlated perturbations are needed to inject variability on scales that accurately reflect the uncertainties in the system.
The GEM model supports the representation of stochastic processes by creating a set of independent pseudorandom patterns using first-order autoregressive [AR(1)] processes. These fields [
The construction of
Summary of primary configuration parameters for the stochastic field generator. Complete definitions can be found in the appendix.
The demands of SPP for a large number of independent perturbation fields prompted an optimization effort within the stochastic field generator. As a result, addition of the full set of perturbation fields incurs a negligible computational cost. However, the use of a coarse-resolution global grid in the inverse transform of the spherical harmonics reduces the variance of
b. SPP implementation in GEM
The stochastic fields
Description of parameters and algorithms perturbed with the SPP scheme, split into subsections by grouping of physical processes. The elements that are employed in the full SPP configuration (section 4a) are identified with boldface type. The “Applications” column describes the role of
Candidate elements to be perturbed by the SPP scheme were identified by subject matter experts as those that were both uncertain and likely to have a significant impact on the model solution. The parameters for each
The use of 36-h relaxation τ for the majority of perturbations (Table 3) is consistent with the long time scales employed in other SPP implementations (Lang et al. 2021) and in the stochastic tendency perturbations used by the GEPS until December 2021 (Houtekamer et al. 2009). The SPP scheme’s impact depends on τ and other leading parameters that define
c. SPP error models
Using either Eq. (3) or Eq. (4), the error model amplifies perturbations under conditions in which differing closures yield divergent estimates of χ. The distributions of the perturbations are centered on the unperturbed value, a prerequisite for the development of an ensemble whose perturbed-member climates resemble that of the control. The SPP elements that employ error models are shown below to have significant impacts on the forecast skill of the ensemble.
4. Ensemble predictions with SPP
The success of the SPP scheme as a representation of model uncertainty will ultimately be determined by its ability to improve ensemble forecast skill. This implies that the spread generated by SPP should accurately reflect flow-dependent forecast error. The impact of the full SPP scheme on GEPS guidance is introduced in section 4a. The large number of perturbed parameters, however, makes it difficult to identify the sources of forecast improvements. This task is undertaken in section 4b, where the sensitivity to individual SPP elements is assessed.
a. Model uncertainty representation by SPP
The SPP scheme yields global spread growth that closely mirrors RMSEu in the middle and upper troposphere (Figs. 2a,b). Temperatures at the top of the boundary layer remain under-dispersed throughout the 15-day forecast as a result of insufficient spread in the initializations and a suppressed spread growth rate over the first 48 h of integration (Fig. 2c). The SPP scheme does not contribute to an increase in the error of the ensemble mean, a result that is consistent with the scheme’s objective to represent random errors within regions of enhanced uncertainty.
The SPP scheme’s fractional contribution Cf to ensemble spread [J = σe(x) in Eq. (2)] reaches a maximum of ∼18% in pentad-1 forecasts (Fig. 2, second column). A comparison of model uncertainty representations in section 3 of Part II shows that this is comparable to the contributions of schemes used in the operational GEPS. The peak Cf for upper-level winds (Fig. 2d) is delayed because few SPP elements directly affect this quantity; instead, spread appears to develop through balance adjustments to mass-field perturbations. The decay of SPP’s relative contribution at longer lead times is the result of saturation as the ensemble approaches its climatological dispersion.
A more complete overview of the ensemble’s response to the full SPP scheme is shown in Fig. 3. The first column suggests that the spread increases noted in Fig. 2 are largest in the tropics (Fig. 3b), although the scheme also contributes to diversity in the northern (winter) midlatitudes and to near-surface spread in pentad-1 forecasts in the southern (summer) midlatitudes. The greater longevity of significant SPP contributions in the tropics (Fig. 3b) is a reflection of the slower approach to climatological spread in areas where moist baroclinic growth is not a primary driver of variability (Judt 2020). These process-level distinctions in the GEPS response imply that all sensitivities must be assessed in the tropical and midlatitude regions independently.
The full SPP scheme reduces the fCRPS in all regions, indicative of forecast skill improvements (second column of Fig. 3). The largest impact of SPP is in the tropics, where significant forecast improvements are observed for all variables and lead times. In the midlatitudes, significant fCRPS reductions are generally restricted to lower-tropospheric temperatures and are most apparent in pentad 1.
Although these summary evaluations suggest that the SPP scheme improves ensemble behavior in the aggregate, a more refined assessment of the relationship between spread and error is possible using spread-reliability diagrams (Leutbecher et al. 2007). A well-balanced ensemble that reliably discriminates between high- and low-predictability conditions will have a distribution that falls along the diagonal of these plots; however, the limited GEPS ensemble size will tend to flatten this slope. An analysis of 72-h forecasts of 850-hPa temperature and 250-hPa zonal wind illustrates the impact of SPP on the ensemble from the spread-reliability perspective (Fig. 4). The control ensemble is underdispersed in all regions because it does not sufficiently account for uncertainties within the model. The slope of the distribution suggests that spread and error are well coupled in the midlatitudes (Figs. 4a,c); however, steepness in the tropics implies that large errors can develop in regions without increased spread (Fig. 4b). The SPP scheme shifts all distributions toward the diagonal, a result that is consistent across variables and lead times (not shown). This suggests that the SPP scheme implemented in GEPS improves the ensemble’s ability to depict the flow-dependent nature of spread and error growth.
b. Sensitivity to individual SPPs
The full SPP scheme comprises independent perturbations to numerous model components (Table 3), making it difficult to identify direct relationships between specific perturbations and the ensemble results described above. Such information, however, is extremely valuable to ensemble system designers, model developers and researchers interested in understanding how model uncertainties affect forecast error growth. Only candidate SPP elements that have a neutral or positive impact on GEPS performance from the spread or fCRPS perspective are considered for adoption in the full SPP configuration. Those with neutral impacts are included if they possess strong physical justification, often in the form of well-documented uncertainties, or if they are known to affect particular forms of high impact weather that are not well described by summary statistics.
In this section, sensitivity tests based on a minimized ensemble configuration (section 2; M = 10) are used to assess the impact of each SPP element in isolation. This framework reproduces the summary statistics of the full GEPS (Table 4), at a nearly 20-fold decrease in computational cost. Because of the potential for interactions between perturbations, the sensitivities identified by activating each SPP element in isolation will not sum to the total sensitivity shown in Fig. 3 (Posselt and Vukicevic 2010). However, this set of tests will make it possible to characterize the direct impact of each SPP element and to identify the most important perturbations so that they can be examined in more detail in section 5. The SPP elements are grouped by process as in Table 3 for this discussion to divide the large volume of information into physically relevant subsets.
Globally averaged pentad-1 contribution [Eq. (2); percentage computed as 100 × Cf] to the fCRPS by the SPP scheme using the full 20-member, 44-case GEPS ensemble (“Full Ensemble” column) and the minimized 10-member, 14-case experimental design (“Minimized Ensemble” column). Differences between the SPP-based experiments and the control that are statistically significant at the 99% level according to a 1000-member bootstrap test are shown in boldface type. A similar level of similarity is found for the tropics and midlatitudes individually (not shown).
1) SPP element sensitivity: Surface and boundary layer
Perturbations to elements related to turbulent transport in the planetary boundary layer are highly effective at generating spread and improving forecast skill (Fig. 5). The ensemble is particularly sensitive to the perturbation of surface exchange coefficients (fh_mult and fm_mult) and to the representation of the mixing length (ml_emod and longmel). Because of the large impact that these SPP elements have on the solution, they will be analyzed in more detail in sections 5a and 5b, respectively. In addition to their large amplitudes, these sensitivities are unique in their geographical scope, which extends through the midlatitudes in both hemispheres.
Other SPP elements related to the boundary layer have a limited impact on the ensemble metrics shown here; however, their intent is to sample uncertainties associated with specific conditions that are not well represented by summary statistics. The perturbations in ricmin sample uncertainty in the value of the critical Richardson number for turbulence transitions, a quantity that influences the model’s representation of freezing rain profiles (McTaggart-Cowan and Zadra 2015). Similarly, uncertainty in the transport of turbulence kinetic energy affects surface temperatures under very stable conditions but does not have a significant impact on hemispheric-scale statistics. Such elements are retained in the SPP configuration because their flow-dependent sensitivities may be particularly relevant to predictions of high impact weather events.
2) SPP element sensitivity: Moist convection
Despite the known limitations of convection parameterizations (Molinari and Dudek 1992; Arakawa 2004) and their impacts on forecast skill (Rodwell et al. 2013; Lillo and Parsons 2017), sensitivity to SPP elements related to moist convection is relatively small, particularly in the midlatitudes over the period of interest [Fig. 6; (Reynolds et al. 2011)]. Perturbation of the cloud updraft radius (crad_mult) yields significant improvements in the tropics, likely because this element samples uncertainties in the entrainment and detrainment rates that affect cloud depth and transport profiles (de Rooy et al. 2013). Perturbing the formulation of the trigger function for deep convection (deeptrig) introduces significant variability (Suhas and Zhang 2014); however, the worsening of a warm bias at 850 hPa (not shown) increases the fCRPS and precludes the use of this element in the full SPP configuration. Despite this problem, the limited sensitivity to perturbations within the existing trigger (kfctrig4, kfctrigwl and kfctrigwh) suggests that sampling formulation uncertainty has the potential to contribute to forecast skill improvements in the future (Li et al. 2008).
The other convective element not adopted in the proposed SPP configuration (bkf_evaps) also has a primarily positive impact on the fCRPS. Allowing a variable fraction of detrained shallow convective condensate to be evaporated within the scheme, as opposed to being handled by gridscale condensation, favorably cools the lower troposphere but worsens the 500-hPa tropical height bias to increase the fCRPS (Figs. 6e and 7). This outcome highlights the impact of both a tropical surface pressure bias (negative 1000-hPa height bias in Fig. 7b) and an indirect perturbation response, here in the form of reduced deep convective heating in the upper troposphere (Fig. 7a). These sensitivities also illustrate a problem with perturbations to quantities whose limiting values are employed in the control configuration. Assuming a linear response, centering the ensemble mean on the control member requires symmetric perturbations. In this case, all detrained condensate in the control member is passed to the gridscale scheme, such that perturbations can only act to reduce the transfer. This creates a conceptual problem for such SPP elements, which in this case manifests as an increased bias and fCRPS degradation.
3) SPP element sensitivity: Gridscale clouds and radiation
Perturbations to parameters within the gridscale condensation scheme and affecting cloud–radiation interactions generally have relatively little impact on the summary metrics shown in Fig. 8, although sampling uncertainty in the relative humidity threshold required for cloud formation in the lower troposphere [hu0max; (Li et al. 2008)] yields modest fCRPS improvements. Other SPP elements in this group are adopted for the full SPP configuration because of known uncertainties. For example, the concept of effective ice particle radius is ill-defined for current microphysical schemes given the range of cloud–radiation interactions possible for different crystal habits (McFarquhar and Heymsfield 1998). Sampling this uncertainty with the rei_mult element is physically justified and has a direct effect on high-cloud albedo estimates (Fig. 9a). Despite minimal sensitivity in the summary statistics (Fig. 8), this change in reflection can have a large impact on the local surface energy budget (±20%; Fig. 9b) and is known to affect both tropospheric stability and tropical cyclogenesis (Caron et al. 2012).
Perturbation of the highly parameterized subgrid-scale cloud variability estimate [hetero_mult; (Oreopoulos and Barker 1999)] induces systematic upper-level cooling that increases the fCRPS, rendering this element unsuitable for the full SPP configuration. Further efforts will be required to determine the optimal approach for sampling this uncertainty.
4) SPP element sensitivity: Momentum and dynamics
Sensitivities to SPP elements related to momentum transport and dry dynamics are shown in Fig. 10. Those affecting parameterized drag processes (sgo_phic and rmscon) have minimal impacts on the summary statistics; however, perturbing the velocity spectrum of nonorographic gravity waves promotes diversity in the wave-breaking layers of the upper stratosphere. Evaluations using satellite data show that the additional spread is beneficial for the assimilation component of the GEPS (these improvements will be documented in a forthcoming paper).
The two SPP elements that perturb quantities outside the physical-parameterizations suite are adv_rhsint and phycpl (Fig. 10). The former uses an error model to estimate uncertainty within the advection scheme and will be discussed in more detail in section 5c. The latter shows that the choice of dynamics-physics coupling strategy has a significant impact on forecast quality (Gross et al. 2018). Activating the phycpl SPP element replaces the split-explicit treatment of the model components with the implicit handling of a portion of the total physics tendencies. This decreases ensemble spread because the implicit tendency treatment damps variance over a broad range of wavenumbers (Fig. 11). Although arguably occurring at scales that are unpredictable by the end of pentad-1 forecasts, this spread reduction increases the fCRPS through the second term on the rhs of Eq. (6). This degradation precludes the adoption of the phycpl element in the full SPP configuration. Further investigation will be required to determine whether this sensitivity is indicative of true uncertainty in the system that should be sampled as a potential source of model error, or whether the current approach is simply optimal for the adopted model configuration.
5. Leading SPP parameters
The preceding analysis of individual SPP element sensitivities reveals a broad range of responses to the stochastic perturbations. Even if the level of uncertainty in two parameters is of the same order, perturbations may have significantly different impacts on the solution as a result of differing and possibly flow-dependent sensitivities within the model. In this section, the SPP elements that were found to have the largest impacts on the GEPS are analyzed in more detail to provide physical explanations for the behavior of the SPP-based ensemble.
a. Surface exchange coefficient perturbations
The multiplicative scaling used in Eq. (8) means that the direct impacts of these perturbations are maximized in regions where turbulent surface exchanges are typically large (Fig. 12; results shown only for fh_mult because of the large sensitivity shown in Fig. 5). Differences therefore peak along the western boundary currents (Munk 1950) and cold outbreak regions in the Northern Hemisphere winter (Fletcher et al. 2016; Smith and Sheridan 2020), and over landmasses in the tropics and Southern Hemisphere (summer). The enhanced spread in continental regions is accompanied by a significant decrease in the fCRPS, indicating that the broad perturbations range reflects underlying uncertainty in the forecasts.
In the North Atlantic Ocean, northerly flow in the wake of oceanic cyclones (Tilinina et al. 2018) and high winds associated with the Greenland tip jet (Doyle and Shapiro 1999) draw Arctic air across the extensive ice edge to create an environment conducive to frequent large-flux events (Papritz and Spengler 2017; Geerts et al. 2022). Despite symmetric perturbations to
b. Uncertainties in the boundary layer mixing length
The longmel SPP element makes direct use of the multiple λ estimates to sample uncertainty by adopting different closures through discrete perturbations (section 3b). Despite this introduction of spatiotemporal algorithmic variability, the longmel SPP element has limited impact on ensemble spread within the boundary layer (Fig. 15a) and leads to a bias-induced deterioration in the fCRPS at higher latitudes (Fig. 15b). Improved fCRPS values in the stratocumulus regions also originate from bias changes (not shown), suggesting that the Bougeault and Lacarrère (1989) λ introduced in this configuration may yield locally improved basic states. Overall, however, the longmel SPP element leads to significant deteriorations that preclude its adoption in the GEPS configuration (Fig. 5).
The mixing-length error model employed by the ml_emod SPP element proves to be more successful in sampling uncertainty in λ without introducing systematic errors (Figs. 15c,d). The Blackadar (1962) and Bougeault and Lacarrère (1989) estimates are used in Eq. (4) as χ° and χalt, respectively, for
The use of multiple λ estimates has the disadvantage of incurring additional computational cost, with model run time increased by 28%. However, restricting the Bougeault and Lacarrère (1989) calculations to lower levels (
Globally averaged pentad-1 fractional contribution [Eq. (2), in percentage as 100 × Cf] to the ensemble spread and fCRPS by the ml_emod SPP element and a sensitivity test using the “height restricted” form of the error model described in the text. Differences between the SPP-based experiments and the control that are statistically significant at the 99% level according to a 1000-member bootstrap test are shown in boldface type.
c. Advection error model
Although the large majority of SPP elements focus on components of the physical parameterization suite, uncertainties that exist within the dynamical core can have significant impacts on ensemble statistics (Fig. 10). The net effect of these potential error sources is represented in this implementation by the adv_rhsint SPP element, which uses an error model to sample uncertainty in the advection scheme. The semi-Lagrangian formulation begins by determining the departure point of the trajectory that arrives at each grid point over a time step. The advected quantities must then be interpolated to the departure point to give the upstream value: cubic polynomials are typically used for this operation in GEM (Fig. 18). The adv_rhsint error model samples uncertainties in these calculations by using cubic and linear interpolation estimates as χ° and χalt in Eq. (4) with
This error model generates perturbations where local gradients are highly spatially variable, such that the first- and third-order interpolation estimates are likely to differ (Fig. 18a). In such regions, the effects of discretization combine with uncertainties associated with the interpolants themselves to yield a potentially important error source. Conversely, the range of values computed using the different polynomials will be relatively small where the advected quantities vary smoothly in space (Fig. 18b). Unlike the mixing length estimates described above, this error model incurs no additional computational cost because the linearly interpolated value is precomputed within the advection scheme.
Despite the restriction of adv_rhsint perturbations to tropical regions [Eq. (11)], 500-hPa height forecasts improve at high latitudes in the North Pacific basin (Fig. 19b). This sensitivity originates primarily within the East Asian subtropical jet (Yang et al. 2002; Luo and Zhang 2015) in the form of enhanced ensemble spread (Figs. 19a and 20a). The perturbations appear to promote diversity in the initiation of Rossby wave packets (Röthlisberger et al. 2018), which propagate downstream at rates that fall between the expected phase speed (cp ≈ 10 m s−1) and group velocity (cg ≈ 20 m s−1; Fragkoulidis and Wirth 2020). The associated fCRPS improvements (Fig. 20b) suggest that the perturbations are effectively identifying uncertainty in this tropical–extratropical interaction and the resulting Rossby wave evolution (Wirth et al. 2018). Although perturbations deeper within the tropics are sometimes associated with Rossby wave initiation (Stan et al. 2017), an experiment in which ϕo = 30 [Eq. (11)] shows that this indirect mode of communication is relatively inefficient (Figs. 20c,d). The ability of adv_rhsint to represent the remote impact of tropical uncertainties, thereby reducing the fCRPS in all domains (Fig. 10), makes it an important component of the SPP configuration for the GEPS.
The SPP scheme implemented in GEM employs spatiotemporally correlated random fields as the basis for perturbations to free parameters, physical closures and the dynamical core. These perturbations represent inherent uncertainties in the model in an internally consistent way, yielding a diversity of solutions under conditions in which the associated errors adversely affect practical predictability.
The impacts of 27 SPP elements have been evaluated for potential inclusion in the GEPS. Together, the proposed configuration generates sufficient ensemble spread without increasing RMSEu. Individually, they trigger a broad range of responses in terms of contributions to ensemble spread and forecast skill. The spatial distributions of these sensiti