Predictability of Precipitation from Continental Radar Images. Part IV: Limits to Prediction

2092 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 63

URS GERMANN MeteoSwiss, Locarno-Monti, Switzerland

ISZTAR ZAWADZKI AND BARRY TURNER Atmospheric and Oceanic Sciences, McGill University, Montréal, and J. S. Marshall Weather Radar Observatory, Sainte Anne de Bellevue, Québec, Canada

(Manuscript received 6 June 2005, in final form 9 December 2005)

ABSTRACT

Predictability of precipitation is examined from storm to synoptic scales through an experimental approach using continent-scale radar composite images. The lifetime of radar reflectivity patterns in Eulerian and Lagrangian coordinates is taken as a measure of predictability. The results are stratified according to scale, location, and time in order to determine how predictability depends on these parameters. Three companion papers give a detailed description of the methodology, and present results are obtained for 143 hours of North American warm season rainfall with emphasis on lifetime, scale dependence, optimum smoothing of forecast fields, and predictability in terms of probabilistic rainfall rates. This paper discusses the sources of forecast uncertainty and extends the analysis to a total of 1424 hours of rainfall. In a Lagrangian persistence framework the predictability problem can be separated into a component associated with growth of precipitation and a component associated with changes in the storm motion field. The role of changes in the motion field turned out to be small but not negligible. A stratification of lifetime according to location reveals the regions with high predictability and significant nonstationary storm motion. This work is of high practical significance for three reasons: First, Lagrangian persistence of radar patterns was proved to have skill for probabilistic precipitation nowcasting. The discussion of the sources of uncertainty provides a guideline for further improvements. Second, a scale- and location-dependent benchmark is obtained against which the progress of other precipitation forecasting techniques can be evaluated. And, third, the experimental approach to predictability presented in this paper is a valuable contribution to the fundamental question of predictability of precipitation.

1. Introduction dimensional system it can be investigated analytically. But, as soon as the system becomes complicated, as is From a purely dynamic point of view predictability is the case for the atmosphere, or even more pronounced related to the sensitive dependence of the trajectory of for precipitation, it is difficult to study its predictability. the atmospheric state in phase space to small perturba- Then, any approach can only give an estimate of the tions in the initial conditions. No matter how small the true predictability. difference between two initial states is, it will grow and There is a variety of studies that allow inferences of eventually be so large that the two atmospheric states the predictability of the atmosphere: some follow a are no more similar than any randomly chosen pair of purely analytical approach, others are more of experi- possible states. This sensitivity is an intrinsic and fun- mental character. Table 1 gives an overview of predict- damental property of any nonlinear system, and is re- ability studies including some selected references. An ferred to as the initial value problem. For a simple low- example of a purely analytical approach is the evaluation of the Liouville equation to examine perturbation Corresponding author address: Dr. Urs Germann, MeteoSwiss, growth of a dynamic system. The Liouville equation, CH-6605 Locarno-Monti, Switzerland. introduced in a meteorological context by Gleeson E-mail: [email protected] (1966) and Epstein (1969), describes the evolution of

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TABLE 1. Predictability studies.

Purely experimental approach Studies with observations: Autocorrelation and spectral analysis (Lorenz 1973; Zawadzki 1973) Analogs and periodicity (Lorenz 1993), interdependence (Zawadzki et al. 1981; Lilly 1986) Hovmöller diagram (Carbone et al. 2002; Ahijevych et al. 2004), composite analysis Precursor and predictor analysis (Bocquet 2002) Studies with statistical models: Forecast skill as evaluated against observations: Eulerian and Lagrangian persistence (Zawadzki et al. 1994; Germann and Zawadzki 2002) Dependence on scale (Lorenz 1969; Germann and Zawadzki 2002; Turner et al. 2004) Dependence on location (this paper) and weather Studies with numerical weather prediction models: Precursor analysis (Massacand et al. 1998) Forecast skill as evaluated against observations Sensitivity to initial conditions: singular vectors (Ehrendorfer et al. 1999), ensembles (Palmer 2002; Buizza et al. 1999; Walser et al. 2004) Studies with idealized systems of dynamic equations: Liouville equation (Gleeson 1966; Epstein 1969; Ehrendorfer 1994a,b) Lyapunov exponent (Hergarten 2002) Lorenz model, attractor (Lorenz 1963, 1993; Palmer 1993; Hergarten 2002) Purely analytical approach probability density of the atmospheric state in phase in the model structure, lack of resolution, errors in space and is the analog to the equation of mass conser- scale interactions, inadequate parameterization, param- vation in physical space. It states that realizations of the eter uncertainty, problems in boundary conditions if state of a system cannot spontaneously appear or dis- running a regional model, and numerical and compu- appear. Principally, it can be solved analytically for any tational errors that result, for instance, from finite dynamic system but, in practice, it is only applicable to difference schemes. In practice it is difficult to sepa- low-dimensional problems, such as a one-dimensional rate the problem of predictability into a component Riccati equation (Ehrendorfer 1994a). associated with initial error and a component associ- On the other end of the spectrum of predictability ated with model error (Palmer 2002). The validity of studies are those purely based on observations, such as the results of predictability studies using numerical autocorrelation analysis of the variable of interest weather prediction models can be very sensitive to (Lorenz 1973; Zawadzki 1973) or studies looking for model errors. precursors in the observation space (Bocquet 2002). Zängl (2004) found sensitivity of simulated oro- Between the purely analytical and purely experimental graphic rainfall to model components other than cloud approach there is a variety of studies employing statis- microphysics. He compared the sensitivity to param- tical and numerical weather prediction models that eterization for cloud microphysics with the sensitivity to combine analytical concepts with observations. In a the implementation of horizontal diffusion, the type of model approach predictability can be measured by the vertical coordinates, and boundary layer parameteriza- skill of the model to predict a certain phenomenon as tions. The experiment was conducted with the fifth- evaluated by statistical comparison with observations, generation Pennsylvania State University–National or by the growth of initially small perturbations in Center for Atmospheric Research (NCAR) Mesoscale model phase space using singular vector analysis or en- Model (MM5) and four nested domains with a horizon- sembles of model runs. tal mesh size being 37.8, 12.6, 4.2, and 1.4 km. Precipi- The analytical approach allows one to understand tation of the 1.4-km run accumulated over 36 h and basic concepts of nonlinearity, perturbation growth, averaged over a domain with a 200-km side length var- and sensitive dependence. But, as studies with idealized ied by a factor of 1.4 when using two different imple- systems of dynamic equations or with state-of-the-art mentations of horizontal diffusion. This and other re- numerical weather prediction models both assume the sults indicate that errors in simulated precipitation model to be representative of the true system, the crux fields are not only the result of weaknesses in the cloud lies in all sorts of model errors. These include errors microphysics scheme but may also arise from model

Unauthenticated | Downloaded 10/03/21 05:02 AM UTC 2094 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 63 components that are not directly related to precipita- weather in order to determine the dependence of pre- tion such as the diffusion scheme. dictability on these parameters. Another difficulty comes from the dimension and A similar stratification of predictability is not complexity of the weather system, which makes analyti- straightforward when using a numerical weather pre- cal approaches computationally expensive or impracti- diction model because design and implementation of a cal. To obtain a representative estimate of sensitive model both have a significant influence on the perfor- dependence we may need more than just a handful of mance at different scales, locations, and in different ensemble members calculated for a specific weather weather situations. Of course there are problems in the situation, location, and time. But this is often not pos- experimental approaches as well. First there are mea- sible because of limited computation time. surement errors, and, second, there is some difficulty in linking the results to the nonlinear and chaotic nature a. Predictability of precipitation of the system. There are several sources of errors when using radar The fact that precipitation is an indirect product of measurements to estimate precipitation rates. Strictly numerical modeling that relies on a relatively crude speaking we are looking at the predictability of radar parameterization of convection and microphysics reflectivity convoluted by the spherical scan geometry makes the problem even more complicated than illus- and the antenna pattern, and blocked by the horizon trated above. In fact, only very few results have been (Pellarin et al. 2002). Apart from convolution and presented on the predictability of precipitation. blocking also the transformation from radar reflectivity One way to obtain a quantitative scale-dependent Z to precipitation rates R introduces uncertainty. The estimate of the predictability of precipitation is to ex- Z Ϫ R relation depends on the type of hydrometeors amine the Eulerian and Lagrangian persistence of radar and its size distribution. Variability of the Z Ϫ R rela- precipitation patterns, introduced in a series of com- tion can lead to significant errors, in particular at small panion papers (Germann and Zawadzki 2002, 2004; scales (Lee and Zawadzki 2005). To a first approxima- Turner et al. 2004) and further explored here. tion we can neglect these inadequacies and assume that The basic idea is to take the skill of forecasts ob- the distribution of radar reflectivity represents the pre- tained from Eulerian and Lagrangian persistence of cipitation field. large-scale radar composite images as a measure of predictability. An Eulerian persistence forecast is obtained by keeping the image frozen: b. Benchmark concept The concept of taking the skill of a forecasting tech- ⌿ˆ ͑ ϩ ␶ ͒ ϭ ⌿͑ ͒ ͑ ͒ t0 , x t0, x , 1 nique, such as persistence of radar precipitation patterns, as a measure of predictability goes back to ⌿ where is the observed precipitation field, t0 is the Lorenz (1973) who said: “Regardless of what may be start time of the forecast, ␶ is the lead time, and indicated by theory, a conclusive proof that partial pre- ⌿ˆ ϩ ␶ ϩ ␶ (t0 , x) is the forecasted rate at time t0 and dictability exists at a given range would be afforded by position x. By advecting the precipitation patterns fol- any demonstration that at least one forecasting proce- lowing the field of storm motion we obtain a dure exhibits skill at that range.” Figure 1 illustrates this Lagrangian persistence forecast: approach. It conceptually shows the skill of different forecasting techniques as a function of lead time. By ⌿ˆ ͑ ϩ ␶ ͒ ϭ ⌿͑ Ϫ ␣͒ ͑ ͒ t0 , x t0, x , 2 picking for each lead time the best technique we obtain an envelope curve, which provides a conservative esti- where ␣ is the Lagrangian displacement vector. The mate of predictability. We say conservative because this forecast images are then compared to observations at estimate will never be above the exact predictability. the given lead times to calculate correlation, lifetime, This approach has high practical significance: First, it and skill scores such as the probability of detection, the tells what is achievable in the forecast office, and, sec- false alarm rate, and the equivalent threat score. For ond, it provides a benchmark against which the prog- more details the reader is referred to Germann and ress of any forecasting technique can be evaluated. Zawadzki (2002). Since this approach is conceptually The formation of precipitation in a numerical simple, we thus obtain an easy-to-interpret measure of weather prediction model may be closer to the real predictability of precipitation that can be applied to a atmospheric system than the simplistic concept of large sample of data. Calculation of lifetime and scores Lagrangian persistence of radar patterns. Yet, we can- can be stratified according to scale, location, time, and not say a priori which of the two approaches gives a

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Lagrangian persistence framework there are two or- thogonal factors that limit predictability: first, growth and dissipation of precipitation and, second, changes in the storm motion field. In other words, there is evolution in the precipitation field and evolution in the motion field that play a role. If both terms were precisely known ahead of time, we could combine them with Lagrangian persistence and would thus obtain a perfect forecast. We know from experience that the evolution of precipitation is important. But we do not know its relative importance compared to changes in the motion field nor do we know how this relation depends on

FIG. 1. Benchmark concept. Lorenz (1973) said: “Regardless of location and time. what may be indicated by theory, a conclusive proof that partial Instead of looking for sources of forecast uncertainty, predictability exists at a given range would be afforded by any we can also look for sources of certainty, and ask: What demonstration that at least one forecasting procedure exhibits are the factors that lead to predictability? First, there is skill at that range.” The dashed envelope picks for each lead time persistence of variables, atmospheric phenomena, and the forecast skill of the best technique at that time, and thus provides a first guess of predictability. processes that are related to precipitation. Examples are cloud liquid water content, instability, low pressure systems, and uplift of warm air in a warm front, which better estimate of predictability of precipitation. At all exhibit a certain degree of persistence and are some- present, any estimate is welcome. how linked to precipitation. Second, there is forcing with predictable or partly predictable amplitude, as for instance, the diurnal and annual cycle of net radiation, c. Limits to prediction orographic forcing, or Rossby wave dynamics. And, The methodology of Lagrangian persistence in the third, there are processes and feedback mechanisms context of predictability studies and nowcasting has that lead to convergence in phase space, such as latent been discussed in detail in previous papers: Germann heat release in a convective updraft, which at scales of and Zawadzki (2002, 2004) and Turner et al. (2004), a few tens of minutes and kilometers leads to stabiliza- hereafter referred to as GZ02, GZ04, and TZG04, re- tion. Certainly, this is not the case at microscales where spectively. GZ02 introduces the techniques and pre- latent heat release triggers turbulence, which is chaotic. sents first results. GZ04 extends the analysis to prob- From the perspective of looking for forecast certainty, ability space, and discusses the question of predictabil- the study of Eulerian and Lagrangian persistence of ity in terms of probability density of precipitation rates. radar precipitation patterns from the storm to synoptic Fourier and wavelet decomposition is applied in scales is an important step toward understanding pre- TZG04 to determine how predictability depends on dictability of precipitation. scale and to define optimum smoothing of forecast fields. A practical outcome is the McGill algorithm for d. Outline probabilistic precipitation nowcasting by Lagrangian The objective of this paper is to study the limits to extrapolation (MAPLE) including optimum smoothing prediction of precipitation in a Lagrangian persistence to reduce rms errors, which is operationally imple- framework and, in particular, to separate the predict- mented at the Meteorological Service of Canada. ability problem into a component associated with evo- The results presented so far provide a detailed pic- lution of precipitation and a component associated with ture of the predictability of precipitation in terms of evolution of storm motion. The practical significance of Eulerian and Lagrangian persistence. A logical next this work is threefold: (i) to provide a guideline for step is to identify the factors that limit predictability further improvements of precipitation nowcasting by and to quantify their relative importance. This question Lagrangian persistence, (ii) to set a benchmark against is central to predictability studies, both from a theoret- which the progress of precipitation forecasts from nu- ical and a practical point of view, and has not been merical models can be evaluated, and, (iii), to give an addressed yet. Generally speaking it is the sources of observationally based answer to the fundamental ques- growth of uncertainty with increasing forecast time that tion of predictability of precipitation. The strength of put an upper limit to prediction. In practice, the rel- this approach to predictability lies in its simplicity. evant factors depend on the forecasting technique. In a Section 2 summarizes the concepts of Lagrangian and

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Eulerian persistence of continent-scale radar composite The study is based on the methodology introduced in images as introduced in GZ02, GZ04, and TZG04. Sec- GZ02, GZ04, and TZG04. In particular, we make tion 3 briefly describes the input data. The results pre- use of sented in the previous three papers are based on 143 hours of North American warm season rainfall. In sec- 1) variational radar echo tracking (VET) to determine tions 4 and 5 we give an update of the estimate of storm motion fields at various scales. This tech- predictability by increasing the amount of data by a nique, first introduced by Laroche and Zawadzki factor of 10, and thus approaching a long-term average. (1994, 1995), was adapted to large-scale radar com- In section 6 we separate the problem of predictability posite images in GZ02. A cost function, composed into a component associated with evolution of precipi- of a reflectivity conservation constraint (Laroche tation and a component associated with evolution of and Zawadzki 1994) and a smoothing penalty func- storm motion. In section 7, predictability is stratified tion (Wahba and Wendelberger 1980), is minimized according to location. Section 8, finally, puts the results by means of the conjugate-gradient algorithm in a broader context and gives a short outlook. (Navon and Leger 1987). Figure 2 shows an example of the retrieved motion vectors and the corresponding radar reflectivity field at two times separated by 2. Lagrangian and Eulerian persistence 8h. Lagrangian persistence, as defined in Eq. (2), corre- 2) semi-Lagrangian advection (Sawyer 1963; Robert sponds to persistence of precipitation rates in storm 1981) of radar reflectivity patterns using four itera- coordinates. To assume persistence in storm coordi- tions; see GZ02. nates is the basis of many nowcasting techniques. In its 3) definition of lifetime, a summary statistic of the cor- most simplified form, Lagrangian persistence nowcast- relation function c(␶) between observed and fore- ing consists of two steps: first, determine the displace- casted precipitation fields (GZ02). The lifetime L is ment vector and, second, apply a simple translation to defined as the radar image. First attempts go back to the 1970s ϱ (Zawadzki 1973; Austin and Bellon 1974; Bellon and L ϭ ͵ c͑␶͒ d␶ ͑3͒ Austin 1978). More sophisticated implementations in- 0 clude a high-resolution storm motion field and curvi- ␶ linear advection (GZ02; Seed and Bowler 2003), near- with c( ) given by optimum filtering of nonpredictable scales (Bellon and c͑␶͒ ϭ Zawadzki 1994; Seed 2003; TZG04), estimates of initiation and growth of precipitation (Roberts and Rut- ͵͵ ⌿ˆ ͑ ϩ ␶ ͒⌿͑ ϩ ␶ ͒ t0 , x t0 , x dx ledge 2003; Mueller et al. 2003), schemes to produce ⍀ probabilistic output (Schmid et al. 2000; GZ04), and 0.5 . stochastic modeling of rainfall fields (Seed and Bowler ͫ͵͵ ⌿ˆ ͑ ϩ ␶ ͒2 ͵͵ ⌿͑ ϩ ␶ ͒2 ͬ t0 , x dx t0 , x dx 2003). ⍀ ⍀ For the predictability study presented here we deter- ͑4͒ mine Eulerian and Lagrangian persistence of precipitation patterns from North American radar composite If c(␶) follows an exponential law, L corresponds to the images for time windows from 0- to 8-h lead time. Given time when c(␶) falls below 1/e ϭ 0.37. With the above the time step of the input data of 15 min we thus get 32 definition of L data is aggregated in space over domain ⍀. forecast fields. A 0–8-h forecast is issued every hour. To increase the sample and thus get, statistically Finally, we compare forecasted and observed fields to speaking, more representative results we can extend obtain a measure of predictability of precipitation in the aggregation by adding time and integrate over start

terms of both Eulerian and Lagrangian persistence. time t0

tb ͵ ͵͵ ⌿ˆ ͑ ϩ ␶ ͒⌿͑ ϩ ␶ ͒ t0 , x t0 , x dx dt0 ⍀ ͑␶͒ ϭ ta ͑ ͒ c 0.5 , 5 tb tb ͫ͵ ͵͵ ⌿ˆ ͑ ϩ ␶ ͒2 ͵ ͵͵ ⌿͑ ϩ ␶ ͒2 ͬ t0 , x dx dt0 t0 , x dx dt0 ta ⍀ ta ⍀

Unauthenticated | Downloaded 10/03/21 05:02 AM UTC AUGUST 2006 G E R MANN ET AL. 2097 where ta and tb are the start times of the first and the last forecast run, respectively. Throughout the paper ⌿ is expressed in dBZ; see section 5 of GZ02.

What is Lagrangian persistence of precipitation patterns? The answer to this question lies in a careful analysis of the meaning of storm coordinates, that is, in the definition of storm motion. Storms move driven by environmental wind at some steering level. This motion is well captured by echo tracking, VET in our case. They also propagate through development and dissipation. When propagation occurs through new development of convection at the front of the system while there is dissipation at the rear, it results in an apparent displacement that is also well captured by VET. If this propagation is systematic and persistent, the resulting motion vector field will carry it as a skill in nowcasting. It has to be systematic in order to be captured during the period over which the motion vectors are determined. This is typically one hour in the operational setup. It has to be persistent in order to result in nowcasting skill. The steering level winds may have rotation, stretch- ing, and shearing; see advected boxes in the lower panel of Fig. 2. This results in changes in echo patterns that appear in the echo motion field. Systematic growth and/ or dissipation at the edges of the system may be captured, at least partially, as deformation of the pattern by the motion field. If persistent, this type of growth and dissipation is included in the skill of our nowcasting algorithm. What we do not capture is new development of convection in a region, separate from the old pattern, where none was present before; nor do we capture the dissipation of entire regions of precipitation. Thus, Lagrangian persistence, in the sense used here, could include a good deal of growth and decay, pro- FIG. 2. Radar composite image and corresponding storm mo- vided it is systematic and persistent. The degree that tion field at two times. Levels of gray shading correspond to re- this is effective depends on the skill of the echo tracking flectivity between 10 and 25, 25 and 40, and larger than 40 dBZ, algorithm. A compromise must be made between the respectively. The thick line in the upper panel indicates the radar coverage mask used for the calculation of correlation and life- desire of capturing as much of growth and dissipation as time. To illustrate the effect of differential motion discussed in possible and the stability of the motion field. Too much section 2 in the lower panel two small boxes are advected over sensitivity to growth and dissipation may result in de- 8 h using the implemented semi-Lagrangian backward scheme. crease of skill if the captured growth and dissipation is not persistent. Since most of the growth and dissipation is not persistent over time scales beyond few hours, too taken that severely limits the captured growth and dis- much sensitivity may improve the very short term now- sipation. This question will be explored further in fu- casting but spoil the longer term. The period over ture work. which the motion field is determined and the resolved When throughout the rest of the paper we talk of scales of the motion field determine the compromise. growth and dissipation we refer to that which is not Operationally both are fixed for all precipitation sys- captured by the Lagrangian persistence as discussed tems and consequently a conservative approach is above.

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TABLE 2. Statistics of the five warm season rainfall events and the three wet periods analyzed in this study. The extent is defined as the area with a precipitation rate larger than 0.1 mm hϪ1 (10 dBZ). The last two columns indicate the fraction of that area with rates larger than 1 mm hϪ1 (25 dBZ) and 10 mm hϪ1 (40 dBZ), respectively. The values are averages over the entire period. On an overall average, the fraction of the precipitation area with rates that exceed 1 mm hϪ1, e.g., is 41%.

Starting date Time (UTC) Duration (h) Extent (105 km2) Ͼ1mmhϪ1 (%) Ͼ10 mm hϪ1 (%) Rainfall events 30 Jul 1998 0400 16 6.0 42 6.4 16 May 2000 0000 22 3.9 34 2.0 25 May 2001 0400 20 6.1 33 2.6 1 Jun 1999 1600 25 3.6 47 12 26 May 2000 0000 60 7.8 45 7.0 Wet periods 15 Apr 1997 0000 144 2.7 22 1.7 21 Apr 1997 0000 321 6.4 36 5.2 24 May 1996 0000 816 4.3 46 9.5 All together 1424 4.9 41 7.3

3. Data hϪ1 for at least 32 out of totally 1424 h of data. If total duration of rainfall at a given location is Ͻ32 h, it is Input data of this study is six years of continent-scale most probably because that pixel is not visible from any U.S. radar composite images, a WSI Corporation of the radars used for the continent-scale composite. “NOWrad” product. A subdomain between 27° and Beyond the border with Canada and Mexico and over 51°N, 104° and 78°W was selected (Fig. 2). Resolution the Atlantic and the Gulf of Mexico we do not have in space, time, and reflectivity is 2 km, 15 min, and 5 radar observations, and thus cannot make inferences of dBZ. For the analyses presented in this paper the spa- predictability. These data voids must be borne in mind tial resolution has been reduced to 4 km. As the data when interpreting the geographical variability of pre- passed three stages of quality control, there is little re- dictability presented in section 7. sidual clutter in the images. For the previous paper, Pixels with no radar coverage are not taken into ac- GZ04, five events including a total of 143 h of warm count when calculating correlation and lifetime in Eqs. season rainfall were selected. Here, we largely extend (3)–(5). The integration domain ⍀ is the ensemble of the dataset by adding three wet periods, two from data pairs for which both ⌿ and ⌿ˆ are based on pixels spring 1997, and one from early summer 1996, of 144, within radar coverage. Generally speaking, the La- 321, and 816 h, respectively. Totally we have 1424 h of grangian displacement vector ␣ is a function of x, t , data, which corresponds to 5696 radar composite images. 0 and ␶. We thus obtain a different ⍀ for each pair of t A rough idea of the characteristics of the precipita- 0 and ␶. According to this definition ⍀ of Eulerian per- tion fields can be obtained from the summary statistics sistence is Ն⍀ of Lagrangian persistence. In practice, to listed in Table 2. The table indicates for each of the calculate correlation of Eulerian persistence ⍀ is set eight events and wet periods individually, as well as for equal to that of Lagrangian persistence. This way Eu- all the images pooled together, the average extent lerian and Lagrangian lifetimes are based on exactly the above 0.1 mm hϪ1 in km2, and its fraction with rates Ͼ1 same dataset and are thus fully comparable. and Ͼ10 mm hϪ1. On average, the extent of precipita- A similar treatment of missing data is applied to data tion above 0.1 mm hϪ1 in the selected subdomain is 490 gaps in time; any pair of ⌿ and ⌿ˆ that is anyhow af- thousand km2, which corresponds to 6.6% of the total fected by a missing radar image is eliminated before area. By multiplying the average extent by the number calculation of c(␶). of radar images we get a total of 2.8 ϫ 109 km2 of precipitation data. The fractions of precipitation rates that exceed 1 and 10 mm hϪ1 are, on average, 41% and 4. Predictability 7.3%, respectively. In GZ02, predictability was measured in terms of lifetime of radar patterns in Eulerian and Lagrangian Radar coverage and treatment of missing data coordinates. Results of four events with a total of 83 h The thick line in Fig. 2 delineates the area with radar of rainfall were presented. Here, we start with an up- coverage. It is the area with radar echoes above 0.1 mm date by extending the dataset to a total of 1424 h.

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FIG. 3. Correlation between observed and forecasted precipita- FIG. 4. Evolution of lifetime of Lagrangian persistence (solid) tion field using Lagrangian (solid) and Eulerian (dotted) persis- and precipitation area (dashed) during a 13-day period starting tence 1100 UTC 26 Apr 1997. Lifetime of Lagrangian and Eu- 21 Apr 1997. The precipitation area is defined as the area with lerian persistence is 8.3 and 4.4 h, respectively. rates Ͼ0.1 mm hϪ1. The two curves are 12-h running means using Eq. (5). Maximum lifetime is 12.3 h on 30 April around noon. First, we determine the correlation function c(␶) between observed and forecasted precipitation fields us- isolated showers (Fig. 7). In these cases advection as- ing Eulerian and Lagrangian persistence for one single suming stationary storm motion (Lagrangian persis- 0 to 8 h forecast run issued at 1100 UTC 26 April 1997 tence) does not double the lifetime with respect to no (Fig. 3). The lifetime L calculated for the two correla- advection (Eulerian persistence). tion functions is 4.4 and 8.3 h. By repeating this for other time steps we obtain a time series of lifetime, as 5. Scale dependence the one shown in Fig. 4. To make sure that we get meaningful values of L also at times with small samples As noted in the last sections, when the precipitation (e.g., 29 April) we apply a 12-h running mean using Eq. is dominated by small-scale features, lifetime values are (5). When the precipitation area is very small, the life- small. Usually precipitation patterns include features times are small. This is because small-scale showers and over a variety of scales. Predictability was studied as a isolated thunderstorms typically have much shorter life- function of scale in TZG04 using a wavelet spectrum times than organized mesoscale disturbances. However, S(m) and cospectrum Co(m) based on the Haar wave- it is interesting to note that above a certain threshold, let. The parameter m is used to label power-of-2 inter- say 0.6 ϫ 106 km2, there seems to be no dominant cor- vals of linear scale. Here S(m) is a measure of the im- relation between total precipitation area and lifetime, portance of each interval (or band) of scale in either an at least not during the selected 13-day period. Maxi- observation or forecast image. The Co(m) indicates the mum lifetime during this 13-day period is 12.3 h on 30 level of similarity between two images, here usually April around noon. There is a second maximum of 12.1 between a forecast ⌿ˆ and the verifying observations ⌿. h in the early morning of 2 May. The four radar com- 〈 full description of the wavelet transform and a deri- posite images of that morning reproduced in Fig. 5 vation of S(m) and Co(m) will not be repeated here; for show a large coherent precipitation system with high details, the reader is referred to section 2 of TZG04. persistence over several hours. Two scale-dependent variations of the Lagrangian To obtain an overall estimate of Eulerian and lifetime parameter were applied to the observations Lagrangian lifetime of North American warm season and forecasts of this study: low-pass lifetimes of spa- rainfall we apply Eq. (5) to all the eight rainfall events tially filtered observations and forecasts and bandpass and wet periods. See solid and dotted curve in Fig. 6. At lifetimes calculated from wavelet spectra and cospectra. several places in GZ02 and GZ04, we observed roughly For low-pass lifetimes, features smaller than a cutoff a factor of 2 between Eulerian and Lagrangian life- scale are eliminated from forecasts and the verifying times. After adding the three wet periods this factor is observations. Features below the cutoff scale are elimi- now slightly smaller than 2, namely 5.1 versus 2.9 h; see nated by block-averaged smoothing around each posi- Fig. 6. The reason is that by adding periods up to 34 tion, smoothing over squares with sides at different lin- consecutive days we also include situations with small ear cutoff scales. This removes details at scales consid-

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FIG. 5. Radar composite images at time of second lifetime peak in Fig. 4. Data are for 0500, 0700, 0900, and 1100 UTC 2 May 1997. In situations like this Lagrangian persistence produces a good forecast up to several hours and correspondingly we get long lifetimes. ered smaller by the two-dimensional Haar wavelet transform. Low-pass lifetimes are then calculated by integrating under the curve of correlation, as defined in Eqs. (3) and (4), between the set of low-pass filtered forecasts (from a particular forecast time) and the low- pass filtered versions of the verifying observations. Bandpass lifetimes are calculated using a measure of the coherence between forecasts and observations as a function of scale: ͑ ͒ Cofo m , ͑6͒ ͌ ͑ ͒ ͑ ͒ Sf m So m where Cofo(m) is the cospectrum between the forecast ⌿ˆ ⌿ and observations at scale m, Sf (m) is the spectrum of the forecast, and S (m) is the spectrum of the obser- FIG. 6. Correlation between observed and forecasted precipita- o tion field using Eulerian persistence (dotted), Lagrangian persis- vations. Integrating this expression over all forecast tence (solid), and Lagrangian persistence with nonstationary mo- lead times provides a measure of lifetime for each spa- tion (dash dot). Lifetime is 2.9, 5.1, and 6.2 h, respectively. Data tial scale. Note the analogy to Eqs. (3) and (4). are 1424 h of warm season rainfall.

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FIG. 7. Radar composite images at 1600, 1800, 2000, and 2200 UTC 24 Apr 1997.

Low-pass and bandpass lifetimes are calculated for spectively) and bandpass lifetime (upper and lower individual forecast runs issued every 15 min. Integrat- solid lines) as a function of scale are also shown. The ing up to forecast lead times of 16 h was found to pro- average bandpass lifetime clearly increases with scale vide stable values for the scales studied here. with an increase in variation, as indicated by the spread In Fig. 8, the evolution of bandpass lifetime of of the 10th and 90th percentiles, with increasing scale. Lagrangian persistence at various scales is shown for This is consistent with the time series of bandpass life- the same time period as Fig. 4 (13-day period starting times shown in Fig. 8. on 21 April 1997). There is a general pattern of shorter lifetimes at smaller scales and longer lifetimes at larger scales. Bandpass lifetimes for the largest scales shown 6. Limits to prediction (hundreds of kilometers) are similar to the Lagrangian There are two sources of forecast uncertainty when persistence values in Fig. 4 (solid curve). In addition, using Lagrangian persistence of precipitation patterns: there is a trend toward larger fluctuations in lifetime for growth and dissipation of precipitation and changes in larger scales. the storm motion field. To determine the relative im- General results for the entire dataset of 1424 h of portance of these two limiting factors, we propose the warm season rainfall are shown in Fig. 9. The average following experiment. lifetime is shown as a function of scale for low-pass (heavier dashed line) and bandpass (heavier solid line) 1) Stationary storm motion. Determine c(␶) and corre- calculations. Curves of the 90th and 10th percentiles of sponding L assuming stationary storm motion. Ra- low-pass lifetime (upper and lower dashed lines, re- dar patterns are advected along the trajectories de-

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FIG. 8. Evolution of bandpass lifetime of Lagrangian persistence at scales ranging from 4 to 512 km during a 13-day period starting 21 Apr 1997. A 12-h running mean was applied to lifetimes of individual forecast runs issued every 15 min. The vertical axis is logarithmic to allow clearer presentation of results for multiple scales.

rived from one single motion field that has been by growth and dissipation of precipitation not captured calculated at the time when the forecast is issued by the motion vectors. In other events the case is dif- using the radar images of the past hour (see forecast ferent. During the 25-h period of 1999, for instance, the mode; GZ02). (This is what we did so far.) lifetime decreases by about two hours when ignoring 2) Nonstationary storm motion. Determine c(␶) and changes in the motion field. corresponding L taking into account the evolution As discussed in the subsection of section 2, observed of the storm motion field. Instead of using one single storm motion is a combination of (i) steering level motion field, we use a time series of vector fields winds and (ii) apparent motion that results from sys- calculated a posteriori using all radar images of the tematic growth and dissipation. In the stationary mo- whole 1 ϩ 8 h period. Of course this cannot be done tion we assume persistence of this growth and dissipa- in real time in forecast mode because we do not tion; in the nonstationary motion some of the changes know ahead of time how storm motion will evolve. of growth and dissipation are considered. As a result, 3) Comparison. Take the difference of c(␶) and L be- Lagrangian persistence, as defined here, already incor- tween the two as a measure of the importance of changes in the storm motion field. Figure 6 shows the results pooling all events together. Comparing persistence without motion (Eulerian) with persistence with stationary motion (Lagrangian) we obtain an increase of lifetimes from 2.9 to 5.1 h. When taking into account the nonstationarities in the motion field the lifetime further increases to 6.2 h. We conclude that both storm motion and its evolution in time play a significant role in the “reshuffling” of precipitation. Of course the relative importance of nonstationarities of the motion field changes from event to event (Fig. 10). During the 16-h event of 1998 the evolution of the motion field is almost negligible with respect to FIG. 9. Bandpass (BP) and low-pass (LP) lifetime as a function growth and dissipation of precipitation. The decorrela- of scale for the 1424-h warm season rainfall dataset. The average tion with a time constant of roughly8hisfully driven lifetime is indicated, as well as 10th and 90th percentile.

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FIG. 10. As in Fig. 6 but for eight events and wet periods separately. porates a not yet quantified part of growth and dissi- The choice of space–time resolution of the motion pation. Pure advection driven by the wind field at some field defines the range of captured storm motion vari- steering level would therefore result in smaller life- ability. What are space and time scales with significant times. The fact that lifetime of Lagrangian persistence changes in the motion field? To answer to this question with nonstationary motion rarely exceeds 12 h, how- we performed a scale analysis for the 60-h event start- ever, clearly shows that a large part of growth and dis- ing on 26 May 2000. The lifetime of Lagrangian persis- sipation is not captured by our storm motion fields. tence with nonstationary motion is determined for a set of motion fields with different spatial and temporal Model errors resolutions. Figure 11 depicts the results for temporal resolutions of 8 h, 4 h, 2 h, 1 h, and 15 min. As a So far we are talking about two sources of uncertainty in Lagrangian persistence forecasting, that is, evolution of precipitation and evolution of storm motion. In practice there is a third one: model errors, the sum of inaccuracies caused by all sorts of approxima- tions in our implementation of Lagrangian persistence. The relevant components are (i) discretization in space, time, and reflectivity, as for instance, the limited resolution of the motion field, and (ii) numerical errors, such as numerical diffusion of the advection scheme (GZ02). Given the discretization of our input data of 5 dBZ, 15 min, and 4 km, features at or below the lower end of the mesogamma scale are not sampled at a sufficiently high resolution. Examples of such features are individual convective cells, turbulence in the melting layer, or drop sorting. Input data with higher resolution and a different setup would be required if emphasis is on FIG. 11. Scale analysis of temporal variation of storm motion (26 these small-scale phenomena. The focus of the imple- May 2000, 60 h). Lifetime of Lagrangian persistence is shown as a function of temporal resolution of the storm motion field for 8-h, mentation of Lagrangian persistence presented here is 4-h, 2-h, 1-h, and 15-min steps (stars). Lifetime of Eulerian per- on precipitation variation from the storm to the synop- sistence is plotted as reference (circle; its position on the abscissa tic scale. is arbitrary).

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FIG. 12. Storm motion field at 3 times (1100, 1500, and 1900 UTC 26 May). The vectors are most reliable in regions with precipitation echoes (black arrows), whereas far away from any precipitation they must be interpreted with care (light arrows). The precipitation field at 1100 and 1900 UTC is shown in Fig. 2. reference the lifetime in Eulerian coordinates is shown. changeable, and we would thus find similar variability Lifetime significantly increases from 8- to 4-h resolu- of predictability in space. When evaluating the spatial tion. There is a further slight increase when using 2-h variability of predictability on long time scales of 1000 resolution. Any higher temporal resolution does not h or more, however, we expect to find the influence on significantly increase the lifetime. This does not neces- predictability of geographical factors such as latitude, sarily mean that during the 60-h period there is no mountain ranges, landforms, vegetation, and lakes. variation at time scales smaller than 2 h. But if there is, For this purpose, the subdomain defined in section 3 it is not relevant compared to growth and dissipation of was divided into 13 ϫ 13 squares with a side length of precipitation. For illustration Fig. 12 shows an example 200 km. To avoid boundary problems trajectories that of motion fields at three times at 1100, 1500, and 1900 start in areas with no radar coverage are not consid- UTC 26 May 2000. ered; see the subsection of section 3. For some of the The same study is repeated for different spatial reso- areas there is no, or not enough, data to calculate life- lution in Figs. 13 and 14. Shortest (circle) and longest times; a missing flag is assigned to these areas. (unfilled star) lifetime of Fig. 13 are same as in Fig. 11. Here, the big step in terms of lifetime occurs between 520- and 104-km resolution. Although the results of this scale analysis are, strictly speaking, only valid for the selected data, techniques, and priorities they may give a rough idea of storm motion variability for other implementations as well. In practice numerical errors, such as numerical diffusion, limit the predictability. Numerical diffusion of the implemented semi-Lagrangian advection scheme is restricted to small scales that have short lifetimes. Its influence on our estimate of predictability can thus be neglected. For a detailed discussion of numerical diffusion the reader is referred to the literature (Ostiguy and Laprise 1990; GZ02). There is no other relevant source of numerical errors in our implementation. FIG. 13. Scale analysis of spatial variation of storm motion (26 May 2000, 60 h). Lifetime of Lagrangian persistence is shown as a 7. Geography of predictability function of spatial resolution of the storm motion field for 2600-, Figures 3, 4, 6, and 10 demonstrate how predictability 520-, 200-, and 104-km spacing (stars). Nonstationarities of motion field are sampled at a resolution of 15 min; thus, the unfilled evaluated at scales from a few hours up to several star is identical to that of Fig. 11. Lifetime of Eulerian persistence weeks varies in time. At small scales of a few hours and is plotted as reference (circle; its position on the abscissa is arbi- kilometers space and time are, to some extent, ex- trary).

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FIG. 14. Motion field at four scales (2000, 1900 UTC 26 May). The precipitation field is shown in Fig. 2, bottom.

Figure 15 depicts the lifetime evaluated for each of respect to growth and dissipation of precipitation (Figs. the 169 fields using all eight rainfall events and wet 6 and 10). There is some interesting signal though when periods. The spatial distribution shows a distinct corri- looking at the relative change of lifetimes. Figure 17 dor of maxima with lifetimes between 8 and 10 h that depicts the ratio between the lifetime with nonstation- extends from eastern Nebraska over Iowa, Wisconsin, ary motion and the lifetime with stationary motion, ex- and northern Illinois to Lake Michigan. If we add the pressed in dB. A first region of maximum influence of areas with lifetimes between 6 and 8 h, the corridor storm motion evolution has its center in the southern starts in northwestern Kansas and ends a few kilome- states (denoted by the letter A). A second maximum ters west of Lake Huron (denoted in Fig. 15 by LPC). can be delineated around North Dakota (denoted by One reason for this corridor of maxima of Lagrangian the letter B). Variation of storm motion seems to have persistence are long-living warm season mesoscale pre- mostly negligible influence along the border between cipitation systems that start in the lee of the Rocky South Dakota and Nebraska, in Minnesota, in the re- Mountain cordillera and travel eastward across the con- gion of the Great Lakes, and east of 84°W. In the cor- tinental United States (Carbone et al. 2002; Ahijevych ridor of maximum Lagrangian persistence, denoted by et al. 2004). More to the south, in the triangle formed by LPC in Fig. 15, the role of variation of storm motion is Texas, Florida, and Pennsylvania, convective activity is neither relatively large nor negligible. more pronounced, which largely reduces the predictability in terms of Lagrangian persistence. 8. Discussion The overall picture does not significantly change when introducing nonstationary storm motion (Fig. 16). Persistence of observed precipitation fields in moving This is not surprising because the evolution of motion storm coordinates is taken as a measure of predictabil- was found to be only of secondary importance with ity and for nowcasting. The methodology and first re-

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FIG. 15. Geographical distribution of lifetime of Lagrangian FIG. 17. Ratio between lifetimes with stationary (Fig. 15) vs persistence using stationary storm motion; all eight events (1424 h). nonstationary storm motion (Fig. 16). A value of 0.4 dB, e.g., LPC stand for Lagrangian persistence corridor. corresponds to a ratio of 1.10. That is, by taking into account the changes in the motion field the lifetime increases by 10%. Values of 0.8 and 1.2 dB correspond to an increase of 20% and 32%, sults were presented in three companion papers. The respectively. developed algorithms proved to be robust and fast, and are thus suitable to be applied on a large sample. The dataset was extended by a factor of 10 by adding three wet periods and now consists of 1424 h of North Ameri- can warm season rainfall. It is now large enough to make inferences of predictability on a larger scale, to look at the dependence of predictability on location and scale, and to investigate the sources of forecast uncertainty. Overall average lifetime in storm coordinates is 6.2 h (Fig. 6). When the storm motion is kept frozen, overall lifetime decreases to 5.1 h. When the precipitation patterns are kept frozen; that is, no storm motion at all, overall lifetime is 2.9 h. Decorrelation time is roughly doubled when introducing storm motion. These num- bers correspond to average lifetimes including 1424 h of warm season rainfall. Lifetimes can be much longer than this if evaluated for selected periods (Figs. 4 and 10), for selected areas (Figs. 15 and 16), or at larger scales (Figs. 8 and 9). In the context of operational nowcasting storm motion is usually kept constant, and we thus get an average lifetime of 5.1 h. If we can fully predict in real time the evolution of storm motion, lifetime of operational Lagrangian persistence nowcasting could be extended to 6.2 h. This improvement, although significant, does not overcome the dominant limit set by growth and FIG. 16. As in Fig. 15 but using nonstationary storm motion. dissipation not captured by storm motion. A big im-

Unauthenticated | Downloaded 10/03/21 05:02 AM UTC AUGUST 2006 G E R MANN ET AL. 2107 provement will only be possible if part of growth and Buizza, R., A. Hollingsworth, A. Lalaurette, and A. Ghelli, 1999: dissipation is successfully incorporated in the nowcast- Probabilistic predictions of precipitation using the ECMWF ing scheme. ensemble prediction system. Wea. Forecasting, 14, 168–189. Carbone, R. E., J. D. Tuttle, D. A. Ahijevych, and S. B. Trier, When evaluating the progress of numerical weather 2002: Inferences of predictability associated with warm sea- prediction one should use, as a benchmark, persistence son precipitation episodes. J. Atmos. Sci., 59, 2033–2056. in Lagrangian storm coordinates rather than persis- Ehrendorfer, M., 1994a: The Liouville equation and its potential tence in fixed Eulerian coordinates. Lagrangian persis- usefulness for the prediction of forecast skill. Part I: Theory. tence is easy to implement, easy to interpret, and can be Mon. Wea. Rev., 122, 703–713. ——, 1994b: The Liouville equation and its potential usefulness applied to a large sample. According to preliminary for the prediction of forecast skill. Part II: Applications. Mon. results presented in Lin et al. (2004), Lagrangian per- Wea. Rev., 122, 714–728. sistence of North American radar composite images ——, R. M. Errico, and K. D. Raeder, 1999: Singular-vector per- beats precipitation forecasts of the Canadian [Global turbation growth in a primitive equation model with moist Environmental Multiscale (GEM)] and U.S. (Eta) op- physics. J. Atmos. Sci., 56, 1627–1648. erational models up to plus 7 h. Epstein, E. S., 1969: Stochastic dynamic prediction. Tellus, 21, 739–759. The stratification according to location revealed for Germann, U., and I. Zawadzki, 2002: Scale-dependence of the the 1424 h of warm season rainfall a distinct corridor of predictability of precipitation from continental radar images. maximum lifetime ranging from northwestern Kansas Part I: Description of the methodology. Mon. Wea. Rev., 130, over Iowa to Lake Huron. This corresponds to the area 2859–2873. where persistence nowcasting works best. One reason ——, and ——, 2004: Scale-dependence of the predictability of precipitation from continental radar images. Part II: Prob- for high persistence is long-living eastward-moving ability forecasts. J. Appl. Meteor., 43, 74–89. warm season mesoscale precipitation systems that start Gleeson, T. A., 1966: A causal relation for probabilities in synop- in the lee of the Rocky Mountain cordillera (Carbone tic meteorology. J. Appl. Meteor., 5, 365–368. et al. 2002; Ahijevych et al. 2004). Hergarten, S., 2002: Self Organized Criticality in Earth Systems. The experimental approach presented in this series Springer Verlag, 272 pp. of papers provides a valuable contribution to the fun- Laroche, S., and I. Zawadzki, 1994: A variational analysis method damental question of predictability. Its strength lies in for retrieval of three-dimensional wind field from single- Doppler radar data. J. Atmos. Sci., 51, 2664–2682. its simplicity. ——, and ——, 1995: Retrievals of horizontal winds from single- Doppler clear-air data by methods of cross correlation and Acknowledgments. Examining the influence of evo- variational analysis. J. Atmos. Oceanic Technol., 12, 721–738. lution of the motion field goes back to an initial idea of Lee, G.-W., and I. Zawadzki, 2005: Variability of drop size distri- Dr. Gyu-Won Lee. The authors express their gratitude butions: Time-scale dependence of the variability and its ef- to the Global Hydrology and Climate Center (GHRC) fects on rain estimation. J. Appl. Meteor., 44, 241–255. Lilly, D. 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