THE SCALING LAW RELATING WORLD POINT-PRECIPITATION RECORDS TO DURATION

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 24: (2004) Published online in Wiley InterScience ( DOI: /joc.1022 THE SCALING LAW RELATING WORLD POINT-PRECIPITATION RECORDS TO DURATION S. GALMARINI, a D. G. STEYN b, * and B. AINSLIE b a Institute for Environment and Sustainability, Joint Research Centre, Ispra (VA), Italy b Atmospheric Science Programme, The University of British Columbia, Vancouver, BC, Canada Received 25 April 2003 Revised 12 January 2004 Accepted 17 January 2004 ABSTRACT In order to understand the remarkable six-orders of magnitude scaling law underlying worldwide point-precipitation records, we analyse precipitation data from a wide range of stations worldwide. The analysis shows that single-exponent scaling laws exist only for single stations experiencing extremely high precipitation. This analysis, and a consideration of the sequence of earlier published precipitation records, leads us to conclude that record precipitation exists because of an optimization of all factors leading to precipitation. This idea is incorporated into a scheme for simulating the record duration curve for precipitation, which utilizes only the probability distribution function for precipitation amounts, the temporal autocorrelation of precipitation and a starting record duration point. The simulation suggests record precipitation is asymptotically independent of most underlying physical processes. Copyright 2004 Royal Meteorological Society. KEY WORDS: precipitation; records; scaling law; statistics 1. INTRODUCTION The existence of a scaling law relating world point-precipitation records to duration has been known for at least 50 years through published tabulations of data and the associated graphs (Jennings, 1950; Linsley, 1975; Paulhus, 1965; WMO, 1986, 1995). In more recent times, tabulations and graphs of such data have appeared on the World Wide Web (Commonwealth Bureau of Meteorology, 2003; National Weather Service, 2003), and have been published as parts of works dealing with fractal analysis of rainfall data (Hubert et al., 1993). In spite of the relatively wide knowledge of the existence of this scaling law, there appears to be no particular comment on it; although Hubert et al. (1993) do characterize rain as a multiplicative cascade of fluxes, which ultimately partitions water into highly concentrated cells, their work only suggests that the rainfall record is consistent with a multifractal analysis. Certainly, there is no explanation of the mechanism underlying the remarkably robust relationship p = αd β (1) where p(mm) is the observed record precipitation occurring in duration d(s), with α = 6.75 mm s 0.49 and β = That this scaling law holds over six orders of magnitude in duration, and that the data defining it come from different locations in remarkably diverse climatic zones subject to a wide range of precipitation regimes makes it even more noteworthy, and the lack of an explanation even more tantalizing. The objective of this work is to explore the scaling law in an attempt to illuminate mechanisms underlying extreme precipitation * Correspondence to: D. G. Steyn, Atmospheric Science Programme, Department of Earth and Ocean Sciences, The University of British Columbia, Vancouver, BC V6T 1Z4, Canada; dsteyn@eos.ubc.ca Copyright 2004 Royal Meteorological Society

2 534 S. GALMARINI, D. G. STEYN AND B. AINSLIE worldwide. Although the ultimate objective of providing a mechanistic understanding of the scaling law is not achieved in this work, we hope our analysis will point the way to such understanding. Figure 1 reproduces the plot of records and clearly displays the very close fit of the most recent data to Equation (1). Also plotted are older data sets compiled by Jennings (1950) and Paulhus (1965). An even earlier compilation of worldwide point-precipitation records by Visher (1941) adds nothing to this figure, as most of the data points are the same as those collected by Jennings (1950). As is to be expected, earlier data sets show smaller record amounts, reflecting an improvement in our estimate of record precipitation as more data became available. Clearly, continued monitoring will lead to successively higher records, but equally clearly there must be a limit to this process as available data come to include data representing the maximum physically possible precipitation for a given duration. In this study we will be viewing the record amount as occurring during an optimal combination of all atmospheric processes leading to precipitation. It is almost certain that even the most recent data set does not contain data representing the ultimate record for all (or even any) durations, but equally it is unlikely that the scaling law for the ultimate data set will have a different exponent than that in Equation (1). Table I contains the most recent data, as plotted in Figure 1, and is the basis for the fitted parameter values in Equation (1). A notable feature of this data set is the three sequences of data from Cherrapunji (India) and Commerson and Foc Foc (La Réunion). That these sequences are individually from the same very rainy spell in the three locations will become important in analyses presented in this work. Notable in Table I is that, for short durations, records come from locations over a wide range of latitudes (16, 47.5, 18 and 45 for durations 1 min, 8 min, 15 min and 20 min respectively). At these short durations, the records were almost certainly achieved during intense convective precipitation. For longer durations, the records occur at locations where topography and meteorological processes combine to produce long periods of intense precipitation. La Réunion is a very steep, tropical island at 21 S in the western Indian Ocean, and is frequently visited by intense tropical cyclones. The two sequences of records from La Réunion occurred during tropical cyclones minute 10 minutes 1 hour 1 day 1 month 1 year Record Precipitation (mm) Duration (s) Figure 1. Record point precipitation as a function of precipitation duration from worldwide data. Open circles are data contained in Table I, open squares are data from Paulhus (1965), and filled triangles are data from Jennings (1950). The solid line is Equation (1), which is the best fit to Table I data. Light lines above the best-fit line indicate sequences from single stations (as listed in Table I), starting at long durations with Cherrapunji (India), Commerson (La Réunion) and Foc Foc (La Réunion)

3 A PRECIPITATION SCALING LAW 535 Table I. Current world point-precipitation records and durations from National Weather Service (2003). a Precipitation amounts in parentheses have been converted from inches Duration Record (mm) Location Date 1 min 38 Barot, Guadeloupe 26 Nov min 126 Fussen, Bavaria 25 May min (198) Plumb Point, Jamaica 12 May min 206 Curtea-de-Arges, Romania 7 Jul min 280 Sikeshugou, Hebei, China 3 Jul min (305) Holt, MO, USA 22 Jun min 401 Shangdi, Inner Mongolia, China 3 Jul h 12 min 440 Gaoj, Gansu, China 12 Aug h 489 Yujiawanzi, Inner Mongolia, China 19 Jul h 30 min 550 Bainaobao, Hebei, China 25 Jun h 45 min (559) D Hanis, TX, USA 31 May h 600 Duan Jiazhuang, Hebei, China 28 Jun h 30 min (782) Smethport, PN, USA 18 Jul h 840 Muduocaidang, Inner Mongolia, China 1 Aug h 1400 Muduocaidang, Inner Mongolia, China 1 Aug h 1589 Foc Foc, La Réunion 7 8 Jan h 1697 Foc Foc, La Réunion 7 8 Jan h 1780 Foc Foc, La Réunion 7 8 Jan h 1825 Foc Foc, La Réunion 7 8 Jan days 2467 Aurere, La Réunion 8 10 Apr days 3240 Grand ilet, La Réunion Jan days 3721 Cherrapunji, India Sep days 3951 Commerson, La Réunion Jan days 4303 Commerson, La Réunion Jan days 4653 Commerson, La Réunion Jan days 4936 Commerson, La Réunion Jan days 5342 Commerson, La Réunion Jan days 5678 Commerson, La Réunion Jan days 5949 Commerson, La Réunion Jan days 6051 Commerson, La Réunion Jan days 6072 Commerson, La Réunion Jan days 6082 Commerson, La Réunion Jan days 6083 Commerson, La Réunion Jan days (9300) Cherrapunji, India 1 31 Jul months (12 767) Cherrapunji, India Jun Jul months (16 369) Cherrapunji, India May Jul months (18 738) Cherrapunji, India Apr Jul months (20 412) Cherrapunji, India Apr Aug months (22 454) Cherrapunji, India Apr Sep months (22 990) Cherrapunji, India Jan Nov year (26 461) Cherrapunji, India Aug 1860 Jul 61 2 years (40 768) Cherrapunji, India a Data on this site are extracted from WMO (1986, 1995) and updated by the National Weather Service, Office of Hydrology, Hydrometeorological Branch, Denise (7 8 January 1966) and Hyacinthe (18 28 January 1980) and cover durations 18 h to 15 days. For yet longer durations, a similar combination of topography and meteorological phenomenon is needed, but in these cases it is a monsoon circulation that drives precipitation processes. Records for durations of a month and longer all come from Cherrapunji, which is located on the windward flank of the quite modest Khasi

4 536 S. GALMARINI, D. G. STEYN AND B. AINSLIE Hills, which are the first topographic barrier encountered by the eastern branch of the Indian monsoon after it has crossed the Ganges Delta. In all three cases, the records are a result of an optimum combination of conditions lying behind precipitation production. These conditions include horizontal advection of moisture (particularly in long duration records), induced vertical motion and high temperatures. It is somewhat of a concern that the data from Cherrapunji are quite old, being from More extensive, and continuous precipitation monitoring in this region may well produce higher records for these long durations, thus slightly increasing the exponent in Equation (1). Record precipitation values are simply the most extreme of extreme precipitation amounts, whose statistics have been subjected to study from a number of distinct perspectives, and with a variety of objectives. The most richly developed approach to precipitation extremes attempts to estimate return period and related statistics for flood and water resource prediction. In this kind of analysis, represented by the work of Menabde et al. (1999), the statistics of extreme values of precipitation are fitted to extreme value distributions, the objectives being to generate tools for prediction of the statistical properties of extremes, including maxima for given durations. This class of approaches is unsuitable for elucidating mechanisms underlying the scaling law for records. Another approach to precipitation sequences involves building models for simulation of rainfall time series. This approach is exemplified by Foufoula-Georgiou and Lettenmaier (1986) and Menabde and Sivapalan (2000), who use very different approaches to achieve the same end, i.e. the generation of a synthetic precipitation sequence. Though such sequences are extremely useful in providing input to flood routing, water resource and other applied hydrological applications, they cannot, by their nature, be used to elucidate the mechanism underlying the scaling law in Equation (1). What does arise in these studies that is of crucial relevance to the present investigation is that the statistical properties of a precipitation sequence are contained in three frequency distributions: the distribution of lengths of sequences of measuring periods during which there was no measurable precipitation; the distribution of lengths of sequences of measuring periods during which there was measurable precipitation; and the distribution of precipitation amounts during these measurement periods. Peters and Christensen (2002) and Peters O, et al. (2002) approach rainfall sequences as an example of a phenomenon exhibiting self-organizing criticality. Their analyses emphasize the importance of understanding rainfall as sequences of measuring periods during which there was/was not measurable precipitation (which they call rainfall events and droughts respectively). They analyse radar-derived rainfall data (by contrast with most other analyses, which utilize pluviometer data) and thus are able to resolve extremely small rainfall rates. Their analysis is not directed towards an understanding of the scaling law, but it does reveal much about the stochastic nature of precipitation. Yet a fourth approach to the statistics of precipitation is that taken by Tessier et al. (1993), and Hubert et al. (1993), who apply the methods of fractal analysis to rainfall data. In this context, Hubert et al. (1993) examine precipitation records as a function of duration. They conclude that if a maximum order of singularity exists, then a scaling law (such as Equation (1)) must also exist. From analysis of rainfall data from a variety of locations, they arrive at fractal parameters that lead to an exponent in Equation (1) that is compatible within one standard deviation to 0.5. Their method is unfortunately unable to reveal a physical mechanism lying behind the scaling law, other than rather indirectly through the theory of multiplicative cascades. Based on this review of methods through which the statistics of extreme precipitation amounts are usually approached, we conclude that, though the scaling law is a remarkable and well-known phenomenon, an understanding of the processes underlying it are unlikely to come through further application of these methods. Therefore, we approach the question from a rather different perspective. Each of the points in Figure 1 is extracted from data measured at an individual station. In the most recent compilation, 19 stations in all are represented. It would be possible, were the data available, to compile record precipitation amounts for each of those 19 stations and plot a similar graph to Figure 1 for each station. The three sequences of records from Cherrapunji (India), Commerson (La Réunion) and Foc Foc (La Réunion) indicate that such an exercise might result in similar single-station scaling laws existing over short ranges of duration. It is extremely unlikely, however, that data from each station would contain a set of records that

5 A PRECIPITATION SCALING LAW 537 exactly matches the scaling law over the full range of durations represented in Figure 1. This idea suggests a host of questions regarding the scaling law for records from individual stations. The principal questions are: What is the relation between record precipitation and duration in data from single stations? Over what range of durations does the relation form a simple (single exponent) scaling law, and what is the exponent? Is this relation different for dry and wet locations? Does consideration of the relation as it ranges from dry to wet locations lead to an understanding of the scaling law for worldwide point precipitation (Figure 1)? We will address these questions by analysing data from a set of stations chosen to represent as wide a range of precipitation climatologies as possible. In addition, we will seek data from stations with very long records, and also stations with data at very high temporal resolution. In approaching the final question, we develop a stochastic model for simulating record precipitation as a function of duration, and thereby provide a process-based explanation of the scaling law for records Data to be analysed 2. POINT PRECIPITATION RECORDS AT A VARIETY OF LOCATIONS Precipitation data have been routinely collected by national meteorological services worldwide. In recent times, a number of these services have made their data available for free download (e.g. Meteorological Service of Canada), or for the cost of data extraction and transmission (e.g. Australian Bureau of Meteorology), or as a cost recovery exercise (e.g. National Climatic Data Center, USA). In addition to these data sources, there exist many precipitation data sets collected as part of research activities. From these sources, we have obtained the data listed in Table II, which indicates all data sets analysed, and gives details of locations, relevant measurement details and data-set sizes. Care has been taken to avoid data sets with substantial sequences of missing data, which would be interpreted as precipitation-free periods in certain types of analysis. It is worth pointing out that precipitation data include both rainfall and snowfall, but that record amounts for short durations will inevitably be as rainfall. In cooler climates, record amounts will include both snowfall and rainfall. Although record amounts for short durations will inevitably be warm-season events, and therefore rainfall, records for longer durations will, for some stations, include some snowfall. In all Canadian data, snowfall has been converted to rainfall equivalent Data analysis A continuous temporal sequence of precipitation data from a single station can be used to extract record amounts accumulated as a function of duration by passing a variable-length totalizing window through the data and searching for maximum totals for each duration. Records at durations equal to and a few times as long as the totalizing time will contain a small, downward bias because the durations occur at fixed clock times which, in general, will not correspond to the begin and end times of precipitation events (Von Storch and Zwiers, 1999). Similarly, as durations approach the full length of the data sequence, records will be underestimated due to small-sample limitations. We will not attempt to correct for these biases in the following analysis, but will avoid interpretations of records at the two duration extremes Data for a single station (Vancouver). Figure 2 shows the result of this analysis when applied to the Vancouver data. The records are calculated in durations spaced by a factor of 2 in the lower half of the curve and a factor of 2 in the upper half. The record minimum precipitation amounts for each duration are shown, as is the longest rain-free period (1304 h) for hourly measurements at a resolution of 0.1 mm. A number of features are notable:

6 538 S. GALMARINI, D. G. STEYN AND B. AINSLIE Table II. Stations providing precipitation data for analysis. Data are from water-collecting pluviometers except for MRR, which are from rain radar measurements. ntot and nzero are total number of data points and number of data points with non-zero precipitation respectively. Climatic regimes are those defined in Trewartha (1968) Station/country/ code/symbol Agency a Latitude, longitude Elevation (m ASL) Totalizing time Resolution (mm) Data period ntot nzero Annual mean (mm) Climatic regime b Henderson Lake/Canada/HeLk/ MSC N, W Commerson/La Réunion/Com/ MF S, E St Benoit/La Réunion/Ben/ MF S, S Estevan Point/Canada/EsPt/ MSC N, W Melbourne/Australia/BoM/ BoM S, E 4 1 day Do day Ar 10 1 h Ar 7 1 day Do min Do Vancouver International Airport/ MSC N, 4 1 h Do Canada/Van/ž W Zingst/Germany/MRR/ MPI N, 0 1 min mm h Dcb E Toronto/Canada/To/ MSC N, W Giles Met. Off./Australia/GiMO/ BoM S, E Bellenden Ker/Australia/BeKr/ BoM S, E Sachs Harbour Airport/Canada/SaHb/ MSC N, W day Dcb day BWh day Aw 86 1 day Ft a MSC: Meteorological Service of Canada; MF: Météo France; BoM: Commonwealth Bureau of Meteorology, Australia; MPI: Max Planck Institute for Meteorology, see Peters G, et al. (2002). b Climatic regime types are: Ar, tropical wet; Aw, tropical wet and dry; BWh, desert or steppe; Dcb, temperate continental; Do, temperate oceanic; Ft, tundra.

7 A PRECIPITATION SCALING LAW minute 10 minutes 1 hour 1 day 1 month 1 year 10 years 10 4 Record Precipitation (mm) Duration (s) Figure 2. Record maximum ž and minimum precipitation as a function of duration for the Vancouver data set and synthetic record duration curve constructed from the Vancouver non-zero precipitation amounts. Also plotted are worldwide point-precipitation records, represented by the best-fit line in Figure 1. The small vertical bar on the lower horizontal axis indicates the record longest rain-free period of 1304 h Record precipitation amounts at this modestly rainy location are, as expected, far below those for worldwide records, for all durations. A simple scaling law for precipitation records does not exist for this station, as indicated by the quite substantial upwardly concave curvature of the plotted line. Moreover, records for short durations show a slope somewhat less than that of the worldwide scaling law, and for long durations somewhat greater. Both biases mentioned earlier would make the differences larger than are evident in this graph. The record precipitation for 1 h duration is simply the largest precipitation amount in the raw data. The position of this point determines in a powerful way the shape of the lower part of the curve. This precipitation amount (23.3 mm) is part of the first three data points in Figure 2 as it occurred during a particularly rainy spell that contributed 1, 2 and 4 h precipitation records. Record minimum precipitation amounts approach record maximum amounts as duration approaches the length of the data sequence. At these durations, the apparent record maxima/minima will be smaller/larger than true record amounts at this location due to small sample sizes. The graphed region below record maximum and above record minimum precipitation gives the range covered by precipitation amounts at each duration. For data measured by the type of instrument used at this station, having a minimum detection limit of 0.1 mm, the frequency distribution of measured precipitation amounts is closely exponential for short durations, contains zeros for all durations less than the record longest rain-free period (1304 h in this case), and becomes progressively more nearly Gaussian for long durations. This behaviour readily leads to modelling by a three-parameter Weibull distribution Data for all stations. Data for all stations listed in Table II are analysed in a similar way, and plotted in Figure 3. Records have been calculated for durations roughly uniformly spaced in the logarithm of duration. Also plotted are the records for Cherrapunji for the years 1903 to 1959 extracted from Dhar and Farooqui (1973). Notable features on this plot are:

8 540 S. GALMARINI, D. G. STEYN AND B. AINSLIE minute 10 minutes 1 hour 1 day 1 month 1 year 10 years 10 4 Record Precipitation (mm) Duration (s) Figure 3. Record precipitation for all stations in Table II. Solid line is worldwide point-precipitation records, represented by the best-fit line from Figure 1. Station symbols (with codes from Table II) are: BeKr, ;Ben, ;BoM, ; Cher, ;Com, ; EsPt, ; GiMO, ; HeLk, ; MRR, ; SaHb, ;To, ; Van,ž The absence of a simple scaling law noted for the Vancouver data is evident in data from all stations shown here, to a greater or lesser extent. The curvature is most marked for stations subjected to lesser precipitation, and becomes progressively less marked as one moves to wetter locations. The curvature is all but absent for Cherrapunji, which is closely parallel to the line representing worldwide records, as noted by Dhar and Farooqui (1973). The ranking of records between stations varies with duration. This is clearly seen in the curves representing Bellenden Ker and Henderson Lake. For short durations, Bellenden Ker records are considerably higher than those at Henderson Lake, whereas the ranking reverses for durations greater than about 2 months. From Table II one notes that the annual mean (not record) precipitation at Henderson Lake is more than 50 times that at Bellenden Ker. This reversal is, of course, due to the latter station being in a subtropical location, and experiencing infrequent but intense convective precipitation (which contributes to high short-term records), whereas the former station experiences a very moist, marine west-coast climate, with frequent synoptic disturbances dominating its precipitation regime. A similar relationship is seen for Giles Meteorological Office, whose records curve crosses that for BoM, Vancouver and Toronto. Record curves for very dry stations (such as Sachs Harbour and Giles Meteorological Office) exhibit stronger curvature, and are more complex than those for other stations. This is caused by higher intermittency at dry locations whose precipitation regimes are dominated by a small number of large, intermittently spaced events. This curve complexity is also evident for stations for which very high temporal resolution data are available (Melbourne and Zingst). Such data contain long sequences of zero precipitation, punctuated intermittently by periods of precipitation. Although intermittency is not the objective of this work, it appears that intermittency, and the associated persistence of periods of non-precipitation (Peters and Christensen, 2002) is indicated, but not necessarily quantified by strong curvature in the duration record relation. There is markedly more local curvature in the shorter duration parts of all curves compared with that at longer durations. This is presumably due to greater intermittence in statistics of short-duration precipitation

9 A PRECIPITATION SCALING LAW 541 than in that of longer durations. This is not surprising, since the durations range over times characteristic of atmospheric microscale processes to those of climate, and the processes dominating at these (and the intermediate synoptic scale) have quite different statistical properties. It was noted that an individual, particularly rainy spell contributed to the first three points in the Vancouver data. This is also true of data presented in Figure 3, though the fact is not evident in this figure. For example, the first 10 points in the curve for Melbourne come from 10 consecutive 6 min data points. This feature is evident in all stations, and not only for the shortest durations, but more obviously so. These observations lead to a model for the scaling law evident in worldwide rainfall records. 3. A MODEL FOR WORLD POINT-PRECIPITATION RECORDS Insights into record precipitation gleaned from the preceding analysis lead to the following set of work-points on which to build a model for the scaling law in worldwide rainfall records. Record precipitation is achieved when processes are in a state of saturation, with all necessary conditions being at their optimum. This implies a multiplicative set of preconditions for a record to occur. The persistence of periods of non-precipitation in the data appears to play a key role in the making of a record. These droughts, as referred to by Peters and Christensen (2002), must be taken account of if the statistics of records are to be modelled. Sequential occurrences of extreme precipitation dominateinmakingupmanyoftherecords. The maximum amount of precipitation that occurred in the data sampling time provides a crucial starting point for the record duration curve, since it fixes the record for the shortest resolved duration. If precipitation were constant in time, then the record duration curve would be a simple power law with an exponent of 1.0. Clearly, the sequences of precipitation amounts making up successive records along the curve are not constant, since the exponent is less than 1.0. Given the starting point (shortest duration record), the record duration curve can be simulated if subsequent precipitation amounts are known (assuming that the record shortest duration precipitation occurred first in a sequence that makes up all longer duration records). As a first (though rather artificial) approach to this, we take a ranked sequence of all non-zero precipitation amounts from a single station (Vancouver data at hourly resolution in this illustrative case) and construct the record duration curve. This, in effect, means that the 1 h record is the highest hourly precipitation amount in the data (23.3 mm in this case), the 2 h record is the sum of highest and second highest ranked ( mm) amounts, and so on. The data set contains roughly h of non-zero precipitation, and thus produces records at durations from 1 h to about 4 years. The resulting record duration curve is shown in Figure 2, which demonstrates that this process generates a simple scaling law for all durations between 1 h and 1 year, with very slight sigmoid curvature affecting only the lowest two and highest three points. The exponent in the scaling law is 0.75 (determined by least-squares fitting in log space to avoid bias due to logarithmically spaced durations), rather than 0.5 as in the worldwide records curve. The weakness of this approach is that it demands the extremely unlikely sequential occurrence of the highest precipitation amounts. In physical terms, the atmosphere cannot support this sequence of precipitation because of inadequate supply of precipitable water. At short durations, the major limitations are in vertical advection at the cloud/microscale; for longer durations, limited horizontal advection of moisture at synoptic scale intervenes; at the longest durations, limited supply of atmospheric moisture at the climatological scale is operative. A statistically more reasonable approach would be to generate a sequence of values making up a record at a given duration by sub-sampling the ranked sequence of precipitation amounts. The most natural scheme would be to use the serial autocorrelation function of precipitation amounts to sub-sample the actual amounts. Rather than continue with simulations based on observed data from a particular station, it seems appropriate to proceed using known statistical properties of precipitation amounts in general. What is

10 542 S. GALMARINI, D. G. STEYN AND B. AINSLIE needed is the maximum amount of precipitation that occurred in the sampling time, the probability distribution function (PDF) of precipitation amounts, and the temporal autocorrelation function of precipitation amounts. The purpose of this exercise is to show that the observed scaling law (with exponent close to 0.5) can be reproduced from a very simple statistical description of precipitation. The statistics needed are a frequency distribution (which describes the occurrence of precipitation amounts) and an autocorrelation function (which describes the statistical relation between successive precipitation events). This statistical reconstruction is far simpler than the multifractal approach of Hubert et al. (1993) or the critical theory approach of Peters O. et al. (2002). It is clear, therefore, that the starting point for a model of record precipitation must be the maximum short-term precipitation. This amount will be determined by the maximum precipitation production during intense convective events. A simple way to estimate this maximum is to utilize cumulus parameterization schemes commonly employed in atmospheric mesoscale models. The widely used mesoscale model RAMS (Pielke et al., 1992) employs a scheme that has a maximum precipitation (in one model time-step) of p = w c r c ρ a (2) where p(kg m 2 s 1 ) is the precipitated liquid water flux rate, w c is the vertical velocity at cloud base, r c is the cloud base mixing ratio and ρ a is air density. Maximum observed vertical velocities at cloud top (in tropical cyclone Hilda, reported by Ebert and Holland (1992)) are m s 1, though at most half this would be appropriate for cloud base. Pruppacher (1982) reports a maximum measured mixing ratio of 27 g kg 1, and specifications of the US Standard Atmosphere (Anon., 1976) give a record high surface mixing ratio of 35 g kg 1. The time step of a mesoscale model is typically of the order of minutes. Using these values gives a maximum 1 min precipitation amount in the range 29 to 63 mm. The 1 min rainfall record of 38 mm in Table I is neatly bracketed by these values, and will be used as the starting point for our simulated record duration curve. It is worthwhile noting that this amount of water is far in excess of the total atmospheric water column, thus implying that record precipitation amounts are supplied by considerable horizontal advection. In the case of short durations, the advection is being driven by feeder flows in individual convective cells; for intermediate durations it is local advection in mesoscale convective complexes or cyclonic systems; for the longest duration, the advection is driven by persistent winds in global, cyclonic or monsoon circulations. Lettenmaier (1944) reviews the probabilistic characteristics of precipitation and reports the use of exponential, mixed exponential, gamma and kappa distributions for daily or shorter-term precipitation amounts. An analysis of hourly data used in this work shows a Weibull distribution to provide the best fit, particularly in the distribution of extremes. High intermittency complicates the temporal autocorrelation function for shortduration precipitation, whereas seasonality of precipitation dominates the autocorrelation function for long lags. An analysis of data used in this study shows short-term autocorrelation to be closely exponential. This form will be used in the following analysis. The autocorrelation-conditioned sub-sampling scheme is explained schematically in Figure 4. The idea behind this sampling scheme is that the sequence of precipitation amounts making up the record at a given duration is selected from the distribution of precipitation amounts based on the autocorrelation function of precipitation events. We first assume that the frequency distribution of precipitation amounts is Weibull: f(p)= Ap (γ 1) e pγ (3) where A and γ are given in Table III. We further assume that the autocorrelation function is exponential: r(d) = e αd (4) where α is given in Table III and d (s) is the duration. Equation (4) expresses the relation between the magnitude of precipitation events and the time scale of their separation. In order to use the autocorrelation

11 A PRECIPITATION SCALING LAW 543 Figure 4. Schematic representation of autocorrelation conditioned sub-sampling scheme used to generate individual record duration curves. f is the Weibull PDF (of precipitation amount p). r is the exponential temporal autocorrelation (of time lag d). Links between r(d) and f(p) are explained in the text. p 0 is the precipitation record amount for the sampling time under consideration Table III. Values of parameters in Weibull PDF and exponential autocorrelation functions used to generate synthetic record duration curve in Figure 5. The functions are f(p)= A(p µ) γ 1 e (p µ)γ and r(d) = e αd respectively. All quantities are dimensionless except α α (s 1 ) γ A µ Barot, Guadaloupe Muduocaidang, Inner Mongolia, China Foc Foc, La Réunion function to draw samples from the full distribution of precipitation events, we consider R = (1 r) (5) whichindiscreteformis i = 1 R i = (1 e iαd 0 ) (6) where d 0 now represents the sampling frequency. This function is used to sample the frequency distribution in order to construct the precipitation record for a given duration. The use of R, rather than r, is necessary since the records for the shortest durations (small multiples of d 0 ) must be sampled from adjacent (or at least closely spaced) large precipitation events, which are necessarily infrequently occurring, and therefore associated with very small values of r. By contrast, records at large durations will contain many small precipitation events that occur very frequently, and are associated with large values of r. The precipitation record is obtained through record = N f 1 (R i ) (7) i=2

12 544 S. GALMARINI, D. G. STEYN AND B. AINSLIE minute 10 minutes 1 hour 1 day 1 month 1 year 10 4 Record Precipitation (mm) Duration (s) Figure 5. Record duration curve generated by three applications of the autocorrelation conditioned sub-sampling scheme. The three starting points are record precipitations from Table I at: 1 min (38 mm observed at Barot, Guadeloupe); 6 h (840 mm at Muduocaidang, Inner Mongolia, China) and 22 h (1780 mm at Foc Foc, La Réunion). Solid line is worldwide point-precipitation records, represented by the best-fit line from Figure 1 where f 1 is the inverse of f. Since the Weibull function f cannot be inverted, the record is obtained numerically. The process whereby records are constructed is depicted schematically in Figure 4. For the shortest resolved duration d 0, the record is p 0, which is the precipitation amount at which the PDF is truncated. The record at duration 2d 0 is p 0 + p 1, where p 1 is the precipitation at probability equal to [1 r(d 1 )], where d 1 = 2d 0. The record at duration 3d 0 is p 0 + p 1 + p 2,wherep 2 is the precipitation at probability equal to [1 r(d 2 )], where d 2 = 3d 0. This process continues until p n becomes effectively zero. Note that not all precipitation amounts between p 0 and zero are incorporated into the records, and that sampling is sparser for larger precipitation amounts. Clearly, this process will generate a scaling law with an exponent smaller than that generated without sub-sampling of the ranked set of precipitation amounts. One should notice that the use of r(d) rather than R(d) for sub-sampling the frequency distribution at a given duration would contradict the hypothesis that the atmosphere can produce a record value at short time scales. In such a case, in fact, the values of precipitation sampled for small durations would be obtained from frequent ordinary events and we would take into account the infrequent intense events only when sampling for long durations. The result of this process, applied three times, is displayed in Figure 5. The simulation procedurewas started at 38 mm for a 1 min total (the record 1 min precipitation observed at Barot, Guadeloupe) and terminated at about 1 day. The simulation was restarted at 840 mm for a 6 h total (the record 6 h precipitation observed at Muduocaidang, Inner Mongolia, China) and again at 1780 mm for a 22 h total (the record 22 h precipitation observed at Foc Foc, La Réunion). The restart points were chosen at roughly the duration/record that the previous simulated curve began to show substantial downward curvature. The three sequences of points in Figure 5 were generated using different parameters in the PDF and autocorrelation function. These values

13 A PRECIPITATION SCALING LAW 545 are given in Table III. As is evident, the three sequences of simulated records closely match the fitted curve representing worldwide point-rainfall records. The need to restart the simulation and to continue with a piecewise reconstruction of the scaling should not be surprising, since, as argued earlier, precipitation records are limited by processes that are scale dependent. From a statistical point of view, we have implicitly imposed a scale, through the selection of d 0, whereas the record appears to be scale free. It is worthwhile noting that the worldwide record duration curve exhibits features similar to these three segments in the three sequences of records from Cherrapunji, Commerson and Foc Foc pointed out in Figure 1. This strongly suggests that records at these stations show scale dependence, which is almost surely driven by scale dependence of horizontal moisture advection processes. 4. DISCUSSION AND CONCLUSIONS The record duration curve for point precipitation is a fascinating example of a scaling law in geophysical phenomena. This examination of the scaling law and data sets that lie behind it lead to a number of insights into record precipitation, and the processes that lie behind them. We have extracted record duration curves from precipitation data for a number of stations representative of a wide range of climate regimes. The data have sampling times from 1 min to 1 day, and total lengths up to 140 years. The stations range in annual average precipitation from 48 to 6577 mm. The analysis shows that a simple (single exponent) scaling law only exists for the very wettest stations. It is evident that intermittency associated with a substantial number of zero precipitation intervals in data from the driest locations causes this deviation from a single scaling law, independent of duration. This analysis leads to the idea that record precipitation amounts are achieved by the coincidence of optimum states in a set of underlying factors or processes. That these processes are scale dependent is self-evident, but what is most remarkable about the scaling law is that it holds over a wide enough set of durations to involve the full range of dynamical processes underlying precipitation processes and yet remains a single scaling law. Because of the scale invariance of the power law, it would be surprising if the simulation required inclusion of detailed precipitation processes, since these processes are scale dependent. It must be pointed out that all record precipitation amounts exceed the maximum amount of water in the atmospheric column, implying that considerable horizontal advection must occur to supply the precipitated water. It also seems reasonable to hypothesize that advection is the record-limiting process. While the process that leads to this advection may range from convective cells to monsoon circulations, the record rates show no jumps at the meso-, synopticor planetary-scale. It remains unclear exactly what factors determine the magnitude of the scaling exponent. It does seem reasonable that a value near 0.5 is robust, and will not change as more data become available. The strong dependence of record rates on a small number (or even a single) intense rainfall incident and the sequence of records created by including the neighbouring values, forces the exponent in Equation (1) to be less than 1.0. Extracting records from ordered rainfall sequences produces an exponent of This exponent appears to be robust as it occurs at all stations investigated, and is found for both hourly and daily totals. This then provides an upper limit for the scaling law exponent. Also notable is the idea that particularly intense precipitation events often contribute to records at a sequence of durations, and that this phenomenon occurs for precipitation sampled at minute, hourly and daily resolution. These observations have led us to a means of simulating the scaling law that is based only on very general statistical properties of precipitation. The simulation involves sub-sampling a ranked sequence of precipitation amounts using the autocorrelation function as a sampling scheme. Simulation of the full scaling law over the full six orders of magnitude requires three separate applications of the scheme, each started at different observed record precipitation amounts. The three sequences of records each show a droop towards high durations that is remarkably suggestive of a similar feature seen with the observed scaling law. Because the simulation is based on only very simple statistical properties of precipitation data, and contains no physically based variables, we are led to conclude that record precipitation is achieved by all underlying processes being in a state of optimization in which the amounts become independent of most atmospheric and microphysical

14 546 S. GALMARINI, D. G. STEYN AND B. AINSLIE variables. We have argued that the ordered rainfall sequence is unrealistic, since the real atmosphere cannot support this amount of precipitation sequentially because of limited precipitable water supply. Since horizontal advection limits this supply, and the autocorrelation-conditioned scaling law has the correct exponent, we can infer that the autocorrelation of precipitation amounts captures the essence of advection at a wide range of scales. This will have important implications for estimating maximum precipitation amounts. ACKNOWLEDGEMENTS Data from the Micro Rain Radar (operated by the Joint Remote Sensing Working Group of the Meteorological Institute, University Hamburg, and Max-Planck-Institute for Meteorology in Hamburg) were provided by Gerhard Peters. Data from Melbourne were provided by the Commonwealth Bureau of Meteorology, by kind favour of Alan Seed. Data from La Réunion (collected by Météo France) were provided by Bruno Haie and Hubert Quetelard. Many colleagues and organizations willingly gave data, which were analysed but did not eventually appear in this work. We are grateful to Brian Doherty for untangling some extremely complicated data sets. DS was supported as a visiting scientist by JRC during the execution of this study, and is supported by NSERC and CFCAS. BA is supported by a University Graduate Fellowship from The University of British Columbia. REFERENCES Anon US Standard Atmosphere. US Government Printing Office: Washington, DC: 44, table 20. Commonwealth Bureau of Meteorology Notable point rainfall events. [1 February 2003]. Dhar ON, Farooqui SMT Study of rainfalls recorded at the Cherrapunji Observatory. Hydrological Sciences Bulletin 18(4): Ebert EE, Holland GJ Observations of record cold cloud-top temperatures in tropical cyclone Hilda (1990). Monthly Weather Review 120: Hubert P, Tessier Y, Lovejoy S, Schertzer D, Schmitt F, Ladoy P, Carbonnel JP, Violette S Multifractals and extreme rainfall events. Geophysical Research Letters 20(10): Jennings AH World s greatest observed point rainfalls. Monthly Weather Review 78(1): 4 5. Foufoula-Georgiou E, Lettenmaier DP Continuous-time versus discrete-time point process models for rainfall occurrence series. Water Resources Research 22(4): Lettenmaier DP Stochastic modeling of precipitation with applications for climate model downscaling. In Analysis of Climate Variability, von Storch H, Navarra A (eds). Springer-Verlag: Berlin; Linsley Jr RK, Kohler MA, Paulhus JLH Hydrology for Engineers, 2nd edn. McGraw-Hill: New York. Menabde M, Sivapalan M Modelling of rainfall time series using bounded random cascades and Levy-stable distributions. Water Resources Research 36(11): Menabde M, Seed A, Pegram G A simple scaling model for extreme rainfall. Water Resources Research 35(1): National Weather Service Worldwide rainfall extremes chart. precip/maxprecp.htm [1 February 2003]. Paulhus JLH Indian Ocean and Taiwan rainfalls set new records. Monthly Weather Review 93(5): Peters G, Fischer B, Andersson T Rain observations with a vertically looking Micro Rain Radar (MRR). Boreal Environment Research 7: Peters O, Christensen K Rain: relaxations in the sky. Physical Review E 66: Peters O, Hertlein C, Christensen K A complexity view of rainfall. Physical Review Letters 88(1): Pielke RE, Cotton WR, Walko RL, Tremback CJ, Lyons WA, Grasso LD, Nicholls ME, Moran MD, Wesley DA, Lee TJ, Copeland JH A comprehensive meteorological modeling system RAMS. Meterology and Atmospheric Physics 49: Pruppacher HR In Engineering Meteorology, Plate EJ (ed.). Elsevier: Amsterdam. Tessier Y, Lovejoy S, Schertzer D Universal multifractals: theory and observations for rain and clouds. Journal of Applied Meteorology, 32: Trewartha GT An Introduction to Climate. McGraw-Hill. Visher SS Rainfalls of 10 inches or more, during 24 hours observed in the United States. Monthly Weather Review 69(12): Von Storch H, Zwiers F Statistical Analysis in Climate Research. Cambridge University Press. World Meteorological Organization Manual for Estimation of Probable Maximum Precipitation, 2nd edn. Operational Hydrology Report Number 1, WMO Number 332. WMO: Geneva, Switzerland. World Meteorological Organization Guide to Hydrological Practices, 5th edn. WMO No WMO: Geneva, Switzerland.