Storm-scale ensemble forecast based on breeding of growth modes

Feng Gao , JinZhong Min , FanYou Kong

1. Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China

2. School of Atmospheric Sciences, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China

3. Center for Analysis and Prediction of Storms/The University of Oklahoma, Norman OK 73072, USA

*Correspondence to: Feng Gao, School of Atmospheric Sciences, Nanjing University of Information Science and Technology, No.219, Ningliu Road, Nanjing, Jiangsu, 210044, China. Email: gf_nuist@hotmail.com

Storm-scale ensemble forecast based on breeding of growth modes

Feng Gao1,2*, JinZhong Min1,2, FanYou Kong3

1. Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China

2. School of Atmospheric Sciences, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China

3. Center for Analysis and Prediction of Storms/The University of Oklahoma, Norman OK 73072, USA

*Correspondence to: Feng Gao, School of Atmospheric Sciences, Nanjing University of Information Science and Technology, No.219, Ningliu Road, Nanjing, Jiangsu, 210044, China. Email: gf_nuist@hotmail.com

How to obtain fast-growth errors, which is comparable to the actual forecast growth error, is a crucial problem in ensemble forecast (EF). The method, Breeding of Growth Modes (BGM), which has been used to generate perturbations for medium-range EF at NCEP, simulates the development of fast-growth errors in the analysis cycle, and is a reasonable choice in capturing growing errors modes, especially for extreme weather by BGM. An ideal supercell storm, simulated by Weather Research Forecast model(WRF), occurred in central Oklahoma on 20 May 1977. This simulation was used to study the application of BGM methods in the meso-scale strong convective Ensemble Prediction System (EPS). We compared the forecasting skills of EPS by different pertubation methods, like Monte-Carlo and BGM. The results show that the ensemble average forecast based on Monte-Carlo with statistics meaning is superior to the single-deterministic prediction, but a less dynamic process of the method leads to a smaller spread than expected. The fast-growth errors of BGM are comparable to the actual short-range forecast error and a more appropriate ensemble spread. Considering evaluation indexes and scores, the forecast skills of EPS by BGM is higher than Monte-Carlo’s.Furthermore, various breeding cycles have different effects on precipitation and non-precipitation fields, confirmation of reasonable cycles need consider balance between variables.

storm scale; ensemble forecast; Monte-Carlo; breeding of growth modes

1. Introduction

The prediction ability and accuracy of convective-scale(or storm-scale) hazardous weather is very significant in both meteorological and public service aspects. However, it continues to be a major challenge.

Significant limitations in conventional deterministic numerical weather predictions behave as follows: (1) the development process of meso- and micro-scale systems is sensitive to physical and boundary layer processes, which are responsible for forecast uncertainty (Crook, 1996; Martin and Xue, 2006); (2) the chaotic characteristics and nonlinear action of the atmosphere results in predictability limits (Lorenz, 1963). Considering both limitations mentioned above and the successful application of a ensemble forecast technique for global medium-term (Toth and Kalnay, 1993; 1997)and regional short-term forecasts (Chenet al., 2004; Zhonget al., 2007; Zhang and Niu, 2008), it is reasonable to choose an ensemble forecast (EF) so as to enhance prediction skills of meso- and micro-scale extreme weather events and express prediction uncertainty.

From 2007-2010, the Center for Analysis and Prediction of Storms (CAPS), NCEP, NOAA ran and supported real-time storm-scale ensemble forecast (SSEF) experiments with 4 km high resolution and WRF-NMM/ARPS models.Preliminary results of the 2007-2008 experiments against a tornadic thunderstorm system (Konget al., 2007a; 2007b)show that EF with 4 km resolution has the ability to describe storm cell structures in detail and predicts low probability and high impact storm activities, except for positive deviation in precipitation forecast and lower ensemble spread than expected which is important for evaluation and significance of EF.

The term "spread" represents the average distance between perturbated forecast and control forecast, which is the index measuring uncertainty of ensemble prediction system(EPS) (Huang-fu, 2002). Ensemble spread lower than normal is related to initial perturbation methods which have less dynamic meaning so as not to express short-range forecast error growth. Toth and Kalnay (1993) developed a new perturbation method, called Breeding of the Growing Mode(BGM) that generates perturbation containing fast-growing modes and the growth error process is adequately related to baroclinic instability in a medium-range forecast. Toth and Kalnay (1997) demonstrated that BGM could obtain a reasonable spread and dynamic. Cheung (2001) proved that BGM can obtain a reasonably larger spread, and improve track forecast and skill scores by comparing random and BGM perturbation methods.

To obtain perfect spread, a significant attempt is required to apply BGM in SSEF. However, the evolving characteristic of bred modes in convective instability is different from baroclinic instability. There are some crucial problems to be solved when generating initial perturbation with BGM, such as superposed patterns of bred modes, breeding cycles and time span. In view of the facts mentioned, and by simulating the supercell storm (Rayet al., 1981) event in the Oklahoma,USA on May 20 in 1977 using WRF model (version 2.2),this paper will analyze and discuss the application value of BGM in SSEF and how to handle crucial questions, with the aim of providing a reference for meso-scale strong convective weather forecast.

2. Model parametric and case overview

Weather Research Forecast model (WRF, version 2.2) is used in this study. Dimension of the domain is 160 km long(in the east-west, orxdirection) by 160 km wide (in the north-south, orydirection) and a height of 20 km, with horizontal and vertical resolution of 1km/0.5km respectively.The microphysics scheme of Linet al. (1983) using six kinds of water quality properties and icing processes is chosen. Lateral boundary conditions were kept open, and a rigid upper and lower boundary was considered.

A supercell storm occurred in Del, Oklahoma, USA on May 20th 1977. The CAPE of storm environment field is up to 2,750 J/kg, wind shear from 0.75 km to 6.75 km is 0.0067/s and the maximum relative humidity approached 80% (Figure 1), which together provided unstable conditions and water vapor for the development of a strong storm.

Figure 1 Air sounding profiles of Del, Oklahoma on 20 May 1977

The initial field in this study consists of the horizontal and homogeneity field from air sounding and thermal which is used to stir up convection. Symmetrical potential temperature perturbation, with maximum amplitude of +4K,triggers convection in uniformitarian environments. Thermal style refers to equation(1). The thermal center is defined by location (x=115 km,y=41 km,z=1.5 km), and the radius is 8km/1.5km in a horizontal and vertical direction. The simulated storm is termed as the true storm. Control forecast(single deterministic forecast) is constructed by perturbation of the true storm’s initial field, which models the error between analysis and true state. The perturbation variablesTandUexhibit gaussian distribution with a standard deviation 1 K and 1 m/s.

where (xc,yc,zc) is the thermal center location, (xr,yr,zr) is thermal radius of (x,y,z) direction; Δθmax=Δθ(xc,yc,zc), is maximum thermal amplitude.

From the cloud water mixing ratio (Figure 2), the true storm at 60 minutes developed up to the tropopause and formed a self-organized supercell. A 4-hour cumulative precipitation was 110 mm (not shown). The process of storm development is similar to that of Rayet al. (1981), illustrating that WRF (version 2.2) has the ability to model a supercell storm.

Figure 2 The cloud-water mix-ratio of the model observation at 60 minutes (＞0.1g/kg)

3. Experimental design

3.1. Members generation

Perturbation method refers to Zhang and Yu (2007).Figure 3 is a simplified flow chart of BGM, from a detailed account in Toth and Kalnay (1993; 1997). Adding (subtracting) perturbation breeding is defined as adding (subtracting)bred modes to analysis fields in a cycle process, short for 'A'(adding) pattern ['S' (subtracting) pattern]. To this analogizes,'A-A' (A-S) is defined as adding random perturbation in initial time, and then adopt 'A' (S) breeding pattern in every cycle,'S-A', 'S-S' is likewise. Thus, every pattern can generate one member, with four members in all. Thus, eight members can be generated by free-breeding and rescale-breeding defined as follows.

Free-breeding is according to the above definition. Rescale-breeding is adding (subtracting) the "controlled" bred modes to analysis field where "control" represents the amplitude of bred modes that is adjusted to that of the first bred modes, but the distribution pattern is not manually controlled. The purpose of rescale-breeding is to control the scope and spread of bred modes to reasonable range.

Figure 3 The simplified flow chart of breeding of growing mode (BGM)

3.2. Temporal and spatial distribution of bred modes

Bred modes, controlled by baroclinic instability, represent a fast-growth error distribution pattern in a short-range forecast, where the growth rate reaches saturation value around 1.5/d in 2-3 days (Yuet al., 2007), but the development of error is related to convective instability in a meso-scale system. This led’s to bred modes without evident saturation characteristics.

We first begin by examining the evolution of temperature bred modes’ amplitude and its growth rate (5 minutes per breeding cycle, Figure 4). Growth rate of bred mode is defined asr=E(t)/E(t-1), whereE(t) expresses amplitude of bred modes inttime. In figure 4a, we can see that the smooth effect which numerical model makes on ageostrophic component in initial random perturbation, and suppression effect which stability area beyond convective area makes on growth error. The effects cause free-breeding modes promptly to decrease and the amplitude of bred mode contributes to ensemble spread in a reasonable range. Figure 4b shows that wave amplitude of growth rate decreases and fluctuates around 1 after 25 minutes. The bred modes with stable change will be considered as the initial perturbation of ensemble member.

Figure 4 The time series of the amplitude (a) and growth rate (b) of temperature bred modes

Figure 5 shows the horizontal distribution of temperature bred modes with 12 km altitude in the event that breeding cycle and breeding time span is 5 min and 30 min respectively. Compared to radar reflectivity (not shown), error can develop rapidly in unstable areas such that high value area of bred mode mainly appears in convective areas. When comparing free-breeding to rescale-breeding (Figure 5a, b, c,d), similar distribution pattern are evident when correlation coefficient is above 0.9. However, amplitude difference shows that rescale-breeding can effectively control amplitude of bred modes which has a positive effect on ensemble spread. Thus, rescale-breeding is used to run breeding cycle in the following study.

3.3. Ensemble forecast system

Breeding various variables makes different effects on a changing weather system, for example, breeding temperature mainly affects storm intensity through environmental instability, and breeding wind will affect storm track through environment average wind. Based on the above conclusions,three experiment groups are used to state how to generate ensemble members: testα: to simultaneously breed temperature and wind fields; test β: to breed temperature field only; testγ: to breed wind field only. Considering the advantage of various tests, we choose 3, 3, 4 rescale-breeding modes inα,β,γtest respectively as initial perturbations of members so that 10 members are generated.

According to the above outline, EPS’s with 10 members are constructed according to the following methods: Two EPS’s, called breeding system A and B are constructed, with a run breeding time span of 30 minutes, with 5 and 10 minutes as breeding cycle respectively. To compare the advantage of BGM with dynamic meaning, we use the Monte-Carlo with statistic meaning to construct EPS with 10 members, short for MC system. Control forecast also is a reasonable member, so all EPS’s in this paper possess 11 members.

The following section will verify and evaluate three EPS’s, checking the practical and application value of BGM in SSEF.

Figure 5 The 12 km temperature bred modes (breeding cycle is 5 minutes, breeding time span is 30 minutes)(a) 'A-A' rescaling breeding experiment (b) 'A-A' free breeding experiment(c) 'S-A' rescaling breeding experiment (d) 'S-A' free breeding experiment

4. Experiment results

4.1. Relative skill score

EPS performance can be expressed quantitatively in Root-Mean-Square (RMS) error by Relative Skill Score(RSS) which is defined as follows:

whereEcontrol,Emeanare errors of control forecast (single determinate forecast) and ensemble average forecast respectively.RSSis a dimensionless quantity where a positive ensemble average forecast can improve forecast skill and vice versa.

Figure 6 presents a temperature time series of three EPS’s,which shows thatRSSis greater than zero in all EFS’s. Thus,for forecast error, ensemble technology can improve the accuracy of the control forecast. Figure 6b shows that BGM has an evident advantage to MC system, with increased improvement for system B with a 10 minute cycle.

RSSchecks EFS only from the standpoint of forecast error. However, the research goal for ensemble forecast is the reflection of forecast uncertainty, so occurrence probability and divergence of members are important indexes in evaluating the performance of EFS.

4.2. Talagrand distribution

Ensemble members in reasonable EPS are equiprobability, which can be evaluated by talagrand distribution whose principle can be found in Huang-fu (2002).

Figure 7 presents a temperature talagrand distribution.Clearly, talagrand distribution of MC system expresses a curved pattern in the shape of a "U". The probability of omitting forecast is up to 28% which is above average probability, while ensemble spread is below average probability. Comparatively speaking, breeding systems A and B are closer to average probability, where probability of omitting forecast is lower than 20%. Thus, EPS’s based on BGM method are more reasonable, where system B has a smaller slope factor of linear trend (-0.007), meaning that talagrand distribution is closer to average probability. This illustrates that EPS with a 10 minute cycle is preferable,accordance withRSS.

Figure 6 The temperature RSS of Monte-Carlo system (a) and breeding system A, B (b)

Figure 7 The temperature talagrand distribution of Monte-Carlo system (a) and breeding system A, B (b)

4.3. Spread

Spread is defined as the average distance between perturbated forecasts and control forecast, which reflects uncertainty of ensemble members, and can be expressed as follows:

whereNis the number of ensemble members,di=fi(t)-f0(t),fi(t) andf0(t) are perturbated forecasts and control forecast respectively, — represents spatial average.

Slow-growth spread corresponds to higher omission probability, leading to less chance to reflect the true nature of the atmosphere in future forecast. In theory, spread of reasonable EPS should be close to the RMS (root-mean-square) error.

Figure 8 presents a time series of temperature spread and RMS error. Overall, spread of breeding systems is closer to RMS error compared to MC system. Using system A as an example, error decreases from 2.5 °C to 2.2 °C, while spread increases from 1.7 °C to 2.3 °C. BGM improves obviously the situation that Monte-Carlo methods with statistic meaning make lower spread and slower divergence speed due to few dynamical meaning. Average distance is herein used to compare behavior of breeding systems A and B which is defined as the time average of absolute value of difference between spread and RMS error. The average distance of breeding systems A and B is 0.111 °C and 0.074 °C respectively. Thus, definite dynamic meanings of BGM improve Monte-Carlo’s smaller spread, reflecting the application value of BGM in SSEF.

Based onRSSand talagrand distribution and spread, we note that scores of ensemble temperature forecast is higher with 10 minute cycle than 5 minute cycle (the results of wind field accords with temperature’s qualitatively, not shown), which may be related to smoothing effect that the geostrophic adjustment of numerical model make on non-growth perturbation in domain. Generally speaking, the

shorter the cycle time span, the weaker the smooth effect.Thus, the stronger smooth effect restraining nonlinear interaction between errors results in higher scores for the 10 minute cycle than that of the 5 minute cycle.

Figure 8 The temperature spread and RMS error of Monte-Carlo system (a), breeding system A, B (b,c)

4.4. Rainfall forecast scores

Meso-scale strong convective systems frequently cause rainstorms. Also, rainfall forecast scores are important indicators for the performance of EPS. Bias score and equitable threat scores (ETS) are used herein.

(1) Bias score: defined as the ratio of the grid number of precipitation forecast above a given value in the model area to that of "precipitation observation". The indicator only accounts for the amount and scope of precipitation, and hasn’t guidance meaning for rainfall area forecast. Bias=1 for a perfect forecast, and bias ＞ (or ＜) 1 when the forecast presents a wet (or dry) deviation.

(2) Equitable Threat Scores (Wang and Yan, 2007): defined as

where the terms are defined as table 1. Erroneous data will create a bias in ETS while a perfect forecast receives a value of 1.

Table 1 Contingency table used to obtain the skill scores

Figure 9 presents a 4-hour surface accumulated precipitation scores. From bias scores (Figure 9a), breeding systems A and B with a larger spread shows a wider scope of precipitation and a smaller rank peak (≥10 mm) and a larger wet deviation than MC system with a smaller spread.In the higher precipitation rank, breeding systems have an evident advantage compared to the control forecast and MC system. From ETS (Figure 9b), the scores of all EPS’s

are higher than control forecast in moderate rain (≥10 mm)rank such that the breeding and Monte-Carlo systems have similar percentages. However, the advantage of BGM is mainly express in storm rain (≥50 mm) rank in which breeding systems A and B improve ETS from 20%-30%that Monte-Carlo system makes to around 80%. The application of BGM in SSEF is highly significant in extreme precipitation.

Figure 9 The bias score (a) and equitable threat score (b) of 4 hours accumulated precipitation moderate rain (≥10mm), heavy rain (≥25mm), storm rain (≥50 mm) forecast

For comparing the effect of different breeding time spans on forecast skill, we use the same scheme to redo the tests with a breeding time span of 50 minutes. Because of the different development patterns and breeding modes in different breeding time spans, the study can be considered as the ensemble test in different style storm systems. The experiments produce similar results as compare to the tests with the breeding time span of 30 minutes, meaning that BGM can improve SSEF stably.

5. Conclusions

This study uses the Monte-Carlo method with statistics meaning and breeding of growth modes methods with dynamic meaning to construct SSEF, and by contrast analyze the application value of BGM in SSEF.

From the standpoint of forecast error (RSS), ensemble technology can significantly improve deterministic forecast of strong convective systems. But statistics meaning of Monte-Carlo method leads to a lower spread than expected,which restricts the ability of EPS expressing forecast uncertainty. As an improvement, BGM with dynamic sense considers fast-growth forecast error and increase ensemble spread in reasonable range.

Breeding cycle has various effects on different style of variables. Overall, breeding systems A and B has an advantage over MC system, but breeding systems A and B have advantage over each other. In heavy rain and storm rain rank,the ETS of breeding system with 5 minute cycle is higher than 10 minute cycle, which is opposite to the conclusion of temperature and wind field.

As mentioned, non-growth perturbation noise affects forecast stability by nonlinear interaction with growth perturbation, so for temperature, breeding system with 10 minute cycles has a stronger smooth effect and higher forecast skill. However, for precipitation forecast, the instability derived from nonlinear interaction of perturbations has important significance to extreme precipitation forecast. Different breeding cycles have different effects on extreme precipitation and non-precipitation fields. Thus, reasonable breeding cycles need consider balance between variables in the future.

The supercell storm simulated in this paper has very strong convective characteristics via the meso-scale system.Simulation results have reference value for real forecast.However, as an ideal experiment, our conclusions need to be researched and verified by further experimentation if it is suitable for operational forecast.

This research was supported jointly by the Nature Science Foundation of China (Project No: 40875068) and Public-Welfare Meteorological Research Foundation (Project No: GYHY200806029).

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10.3724/SP.J.1226.2011.00061

10 May 2010 Accepted: 25 July 2010