ALGORITHMS

Table of ContentsAcknowledgments References

- Introduction
- Development Data
- Candidate Predictors
- Probability Equations
- Adjusting Probability Fields To Account For Missing Radar Coverage
- Verification Scores For Probabilistic Forecasts
- Categorical Forecasts of Rainfall
- Example: Forecasts and Verification For 2100-0000 UTC, 2-3 May 1997
- Extrapolative Forecasts and Operational Numerical Model Forecasts
- Summary and Future Work

Extrapolative techniques have been successfully used over the years to make short-range (0-6 h) forecasts of precipitation from remote-sensor observations. These techniques usually infer the current precipitation intensity pattern from gridded maps of digital radar and/or satellite data, and then predict the movement and possibly the evolution of the pattern in time.

We have refined this approach by determining statistical relationships between the occurrence of rainfall exceeding various thresholds and extrapolative forecasts of radar reflectivity, satellite-estimated cloud-top temperature, and lightning strike rates for a large number of historical cases. The statistical relationships are applied to real-time extrapolation forecasts to produce probabilistic forecasts of rainfall amount. This approach is similar to the Model Output Statistics (MOS) method (Glahn and Lowry 1972) which has been used to create precipitation probability equations based on an optimal combination of precipitation and humidity predictors from numerical weather prediction models. The application of MOS methods to a radar extrapolation model was first tested by Saffle and Elvander (1981).

The extrapolative-statistical algorithm yields probabilistic and categorical forecasts as follows:

probabilities that 0-3 h rainfall will reach or exceed 0.1, 0.5, 1, and 2 inches;

a categorical rainfall amount forecast for the 0-3 h period;

the probability of two or more cloud-to-ground (CG) lightning strikes during the 0-3 h period.

The rainfall forecasts refer to the highest amounts at within boxes of a 40-km grid, and the lightning forecasts are for the entire region within the grid boxes. The rainfall thresholds correspond to 2.5, 12.5, 25.4, and 50.8 mm of rainfall.

Input includes a 7-level radar reflectivity mosaic of 10-km resolution, analyses of lightning strikes and satellite-derived 11-µ cloud-top temperature on the same grid, and output of the Nested Grid Model (NGM) run by the National Centers for Environmental Prediction (Hoke et al. 1989). These input fields are extrapolated at the velocity of the NGM-forecasted 700-mb wind field. Experiments are underway to test improvements that might be realized by deriving the extrapolation vectors from observed radar echo motions.

The algorithm is slated for operational implementation in the National Weather Service's Advanced Weather Interactive Processing System (AWIPS) as part of the System for Convection Analysis and Nowcasting (SCAN; Smith et al. 1998). This note describes the logic and development of the algorithms, presents an example of its output for a convective weather event, and documents their skill in terms of several forecast performance measures.

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The statistical development sample described here was created from data during the warm seasons (May-September) of 1996-1998. A total of eight "initial times" were considered, starting at 0230 UTC and continuing at 3-hour intervals through 2330 UTC; the corresponding valid periods were 0300-0600 UTC through 0000-0300 UTC. In practice, the equations derived in this manner can be applied at any time near the nominal initial time.

Historic radar reflectivity analyses were derived from a 2-km national reflectivity mosaic reduced to 10-km resolution. In real time, the analysis is derived from Radar Coded Messages (RCM's) transmitted from individual WSR-88D sites. The reflectivity observations are automatically quality-controlled by comparison with satellite data, to identify and remove anomalous propagation and ground clutter. Within any grid box, the analysis contains the highest observed reflectivity, expressed as one of 7 categories:

0: < 15 dBZ 4: 45-49 dBZ

1: 15-29 dBZ 5: 50-54 dBZ

2: 30-39 dBZ 6: > 54 dBZ

3: 40-44 dBZ

Cloud-to-ground (CG) lightning data were incorporated by determining strike rates over a 15-minute period (00:05-00:20) within the 10-km grid boxes. The strike rates were reduced to categories by dividing nonzero strike counts by 10 and adding 1. Within the development sample, the highest strike rate observed was 60-69 (category 7).

Historic radar data were provided by the Global Hydrology Resource Center (GHRC) at the Global Hydrology and Climate Center, Huntsville, Alabama. Historic lightning observations were provided by NASA Marshall Space Flight Center through GHRC.

Satellite data from the eastern Geostationary Operational Environmental Satellite (GOES) were objectively interpolated to the same grid; the resulting analysis contains the coldest cloud-top temperature indicated for each grid box. The nominal image time for these observations was 00:15 within the hour.

Forecasts of humidity, stability indices, moisture divergence, and precipitation from the operational NGM were also submitted as candidate rainfall predictors. These fields are regularly archived on an 80-km grid at 6-h intervals by the Techniques Development Laboratory.

The statistical rainfall predictands were derived from WSR-88D Stage III precipitation estimates produced operationally by National Weather Service River Forecast Centers (Breidenbach et al. 1998). These 1-h analyses are based on a combination of radar estimates and remote-reporting gauge observations, and are generally quality controlled by analysts. The data represent average rainfall over 4-km grid boxes. Predictands were derived from the highest 3-hour rainfall value observed within contiguous 10x10 grid box subsections. These contiguous subsections comprise the Manually-Digitized Radar (MDR) 40-km grid. The four predictands are binary, being zero when the threshold rainfall amount was not met, and unity when the threshold was met or exceeded.

The lightning predictand was derived from the historic archive by recording the number of strikes within each box of the MDR grid during the 3-h valid period. A minimum of two strikes was used to define an event to insure that misregistered strikes and isolated single strikes were not included as significant convective events.

To construct the statistical development sample, predictor values and the corresponding predictands were drawn from every fourth MDR box in both the horizontal and vertical directions, for all available days. This procedure yielded between 69,000 and 89,000 rainfall cases, and approximately 100,000 lightning cases, for each of the 8 starting times. The sample sizes are not identical because if any one predictor or predictand was missing, the case was eliminated. Significant spatial coverage gaps exist in the present Stage III archive, thus there are more lightning cases than rainfall.

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Each of the remotely-sensed fields (reflectivity, cloud-top temperature, and lightning strike rate) was treated as an indicator of current rainfall rate, and was extrapolated forward in time by advection with the NGM-forecasted 700-500 mb mean wind vector, which proved to be a robust estimator of radar echo motion over the United States during the warm season. Extrapolation was performed at 15-minute intervals. All extrapolation processes were carried out on the 10-km radar analysis grid.

Predictors were generally defined as the highest reflectivity and lightning strike rate categories within each 40-km MDR box, that is, a 4x4 subsection of the 10-km grid. An extrapolative rainfall amount forecast was also made by converting reflectivity categories to rainrates through the relationship:

R = (Z/300)^{0.71429} (1)

where R is rainrate in mm h^{-1} and Z is reflectivity in mm^{6} m^{-3}. Obviously, this forecast is generally unrealistically
large, since it is based on the assumption that the entire RCM grid box is covered by the highest reflectivity value
within it. However, the rainfall forecast did prove to be a statistically useful predictor.

Predictors were derived from data at the initial time, during each 1-h interval within the forecast period, and over the entire 3-h forecast period. Additional predictors were derived by finding the largest value within a square region of 120x120 km centered on the MDR grid box in question; these predictors reflect the influence of convective features expected to pass near the grid box.

Satellite-based predictors were treated in a similar manner by finding the maximum difference between the 700-mb
temperature and the satellite-observed temperature (hereafter referred to as TDIFF_{700}).

One useful radar-based predictor was derived by summing the number of 10-km grid boxes within a 3x3 region of MDR grid boxes that were forecasted to have a particular echo level during the three hour period; such predictors account for both the presence of intense echoes and their areal coverage. One of these, called NLVL456, was calculated by summing the number of boxes with level 4 or higher echoes, the number with level 5 or higher echoes, and the number with level 6 echoes. The value of NLVL456 could thus range from 0 to 432 (all boxes forecasted to have level 6 echoes), but it is divided by 14.4 to scale it to the range 0-30.

Predictors derived from NGM forecasts and analyses included mean relative humidity, precipitable water, K index, lifted index, moisture divergence, and 6-h model-generated precipitation totals.

We found that the individual predictors most highly correlated with rainfall were the 0-3 h maximum values, indicating that the extrapolation procedure added significantly to the initial-time rainrate information. The maximum radar reflectivity level and NLVL456 were those most highly correlated with all thresholds of rainfall. Of the remaining predictors, the ones most strongly correlated with rainfall were satellite-based, lightning-based, and NGM-based, in that order. Among the NGM predictors, the ones most highly correlated with rainfall were the model-generated precipitation, 850-mb lifted index, and K index.

Likewise, the predictors most highly correlated with lightning occurrence were NLVL456 and maximum strike rate as forecasted by extrapolation. The reflectivity-based predictors were actually more highly correlated with the predictand than were the lightning-based predictors; this could be because radar reflectivity represents a somewhat more conservative and less time-sensitive indication of the convective nature of precipitation than does CG lightning. Among the NGM-based predictors, the one most highly correlated with lightning was the 850-mb lifted index.

A number of "interactive" predictors were derived by finding the rainfall relative frequency for each element of a
table defined by two predictors simultaneously. One of the most useful (referred to as RADSAT) is illustrated in
Fig. 1, where the observed relative frequency of 1 inch is shown as a function of TDIFF_{700} and NLVL456. The data
shown are from the 2030-UTC initial time (2100-0000 UTC verification period). The enhancement of the
probabilities by either higher echo intensity or colder cloud tops is evident. The relative frequencies derived from
the table range from 0 to 43%, while the overall observed relative frequency was 3%.

The effects of diurnal variations in precipitation climatology are evident in Fig. 2, where the same predictor/predictand combination is shown but for the morning 1200-1500 UTC valid period. The probabilities over much of the predictor space are about one third lower than the probabilities for the afternoon valid period, a result of the fact that convective activity is generally intensifying during the afternoon, while in the morning it is either decaying or being maintained by synoptic-scale or larger mesoscale systems. The relative frequency of 1-inch rainfall during the 1200-1500 UTC period was 0.6%.

The relative predictive value of radar reflectivity vs. satellite- and lightning-based information is illustrated in Fig. 3,
where the relative frequency of 0.1 inch rainfall is shown as a function of NLVL456 and TDIFF_{700} (Fig. 3a) and of
maximum 15-minute strike rate and TDIFF_{700} (Fig. 3b), the latter predictor combination hereafter being referred to
as LTSAT. The valid period was 2100-0000 UTC, and the relative frequency of 0.1-inch rainfall in the sample was
14.8%. As might be expected, given that the predictand is largely radar-defined, the RADSAT predictors identify
high-probability situations more precisely than does LTSAT, with probabilities > 70% being forecasted for a larger
portion of the cases in which rainfall was actually observed. Still, the LTSAT predictor combination is highly useful
in identifying potential precipitation areas where radar data are absent.

The most powerful predictor combination for lightning was NLVL456 and lightning strike rate, as illustrated in Fig. 4. It is evident that much of the information on lightning potential is contained in the reflectivity data. However, particularly for cases where NLVL456 is < 5, indicating little coverage by echoes of 45 dBZ or higher, the presence or absence of CG lightning itself is important. In Fig. 4, the valid period was 2100-0000 UTC, and the relative frequency of two or more CG strikes was 9.5%.

Probability tables based on these combinations of predictors were derived for all rainfall thresholds and lightning, for all start times. The probability values determined from the tables were added as new predictors to the basic set of predictors derived directly from remote sensor or NGM data.

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We submitted various combinations of the predictors described above to a forward-selection linear regression procedure to obtain equations for probability of rainfall exceeding each of the four thresholds. The following equation was obtained for rainfall during the 2100-0000 UTC period:

P(0.1) = -0.896 + 0.491 RADSAT0.1 + 4.512 MXREF_{3H} + 3.510 MXREF3x3_{INIT} (2)

where P(0.1) is the probability (%) of 0.1 inch or more occuring at some place within the 40-km MDR grid box,
RADSAT0.1 is the probability of 0.1 inch (%) as derived from an interactive predictor table similar in form to that
in Fig. 1, MXREF_{3H} is the maximum reflectivity level forecasted over the grid box over the next 3 h, and
MXREF3X3_{INIT} is the maximum reflectivity level over the surrounding 3x3 MDR box region at initial time. In the
dependent data sample, the event relative frequency was 14.8% and the equation explained 38.3% of the predictand variance.

A similar equation was derived for the probability of 1 inch or more:

P(1) = -0.110 + 0.637 RADSAT1 + 1.716 LTG3X3_{INIT} + 0.015 MAXRAIN_{3H} +

0.526 MXREF3X3_{INIT} (3)

where P(1) is the probability (%) of 1 inch of rainfall, RADSAT1 is the probability (%) of 1 inch of rainfall as derived
from the table shown in Fig.1, LTG3X3_{INIT} is the maximum 15-minute lightning strike rate category over the
surround 3x3 MDR box region at initial time, and MAXRAIN_{3H} is the maximum value of rainfall (.01 inch) within
the MDR box as forecasted by the extrapolation model. In the development sample, the event relative frquency
was 3% and the regression equation explained 14% of the predictand variance.

For the probability of CG lightning during the same period, the following equation was derived:

P(LTG) = 4.716 + 0.594 RADLTG_L + 2.488 MXREF3X3_{INIT } - 8.39 NGMRAIN + 7.686 MAXSTRK

- 0.319 MDIV_{850} + 0.099 TDIFF_{0-1H} (4)

where P(LTG) is the probability of two or more strikes within the 40-km square grid box, RADLTG_L is the
probability (%) derived from the table shown in Fig. 3, NGMRAIN is the 1800-0000 UTC rainfall as forecasted by
the NGM, in mm, MAXSTRK is the maximum 15-minute strike rate category as forecasted by the lightning
extrapolation model, MDIV_{850} is the moisture flux divergence at 850 mb at 0000 UTC as forecasted by the NGM
(s^{-1} 10^{7}), and TDIFF_{0-1H} is the 0-1 hour maximum value of TDIFF_{700} as forecasted by extrapolation, in C. The
climatic frequency of lightning events in this sample was 9.5%, and the expression in (4) explained 32% of the
predictand variance.

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**5. ADJUSTING PROBABILITY FIELDS TO ACCOUNT FOR MISSING RADAR COVERAGE**

In operations, it is necessary to account for missing radar data in the reflectivity mosaic, since both permanent and temporary coverage gaps exist over the United States. Forecasts derived from only satellite, NGM, and lightning data must be used within such coverage gaps. Therefore, equations containing no radar-based predictors provide forecasts for areas where radar data are lacking at initial time or for areas downstream of coverage gaps.

First-guess probability fields are derived under the assumption that missing RCM grid boxes contain reflectivity that is 15 dBZ, which is the most likely value. Then another radar forecast is created by extrapolation, in which missing indicators are tracked in the same manner as radar echoes. Two missing-indicator fields are derived from this forecast. One contains the percentage of each MDR grid box that is covered by missing indicators at initial time. The other field contains the percentage of the space-time domain over each MDR box that is occupied by missing indicators during the forecast period. In practice, this domain is 16 RCM grid boxes times 14 time steps, or 224.

If either of these fields has more than 20% coverage by missing indicators, the first-guess probability is compared to the initial-time only and satellite-only probabilities, and the final guess is taken to be the highest of the three. Our aim is to present the user with the greatest possible rainfall threat, whether from radar-detected precipitation already in the immediate area, or from a satellite-detected system that can be expected to affect the area.

Grid boxes in which the first-guess probability has been replaced are flagged, and at the conclusion of the checking process the probability field is locally smoothed where necessary to blend in the new values with first-guess field.

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**6. VERIFICATION SCORES FOR PROBABILISTIC FORECASTS**

To obtain estimates of the skill achievable through the use of these probabilities, we converted the probability values in the dependent data sample to categorical (yes/no) forecasts by applying a range of threshold values. For each threshold, several forecast scores were computed by comparing the forecasts to the verifying values. These scores include the probability of detection (POD), false alarm ratio (FAR), critical success index (CSI), and bias (Donaldson et al. 1975; Schaefer 1990).

A brief explanation of the results for categorical forecasts for rainfall 0.1 inch during the 2100-0000 UTC period (Fig. 5) will illustrate the practical meaning of these scores. Considering the condition where the rain/no-rain threshold probability is 30% (ie, all cases in which P(0.1) 30% are forecasted to have 0.1 inch of rain). The yes/no forecasts detected approximately 71% of the cases in which 0.1 inch was observed (POD=0.71), 44% of the yes forecasts featured observations < 0.1 inch (FAR=0.44), and there were about 25% more yes forecasts than there were observations of 0.1-inch rainfall (bias was 1.25). Lowering the yes/no threshold increases the POD, FAR, and bias, while raising it decreases them. The CSI is highest near a threshold of 32%; this represents a very rough estimate of the probability at which the an acceptably high POD and acceptably low FAR are reached, under the assumption that missed events and false alarms are equally penalized.

For the remaining rainfall thresholds (Figs. 6-8) it is apparent that lower skill is generally achieved. As the rainfall amount increases and the percentage of cases above the rainfall threshold decreases, the FAR and bias associated with any one POD increase (ie, it is necessary to issue a higher percentage of false alarms in order to capture the same percentage of all events). The CSI reaches its peak at lower probability threshold values, approximately 23%, 15%, and 8% for the 0.5-, 1-, and 2-inch rainfall events, respectively.

For the same scoring tests as applied to forecasts for 1200-0000 UTC, when the climatological relative frequency is significantly lower than in the afternoon, there is also a decrease in the skill level for all thresholds (Figs. 9-12). The probability thresholds yielding the highest CSI for this valid period are 28%, 23%, 17%, 7% for the 0.1-, 0.5-, 1-, and 2-inch thresholds, respectively.

For yes/no forecasts of lightning during the 2100-0000 UTC period, the peak CSI is reached at a threshold probability of 24%, where the POD is 0.67, the FAR 0.53, and the bias 1.43 (Fig. 13). Similar results were obtained for the other verification periods.

These results, obtained from the dependent data sample, almost certainly overestimate the skill that will be realized within a sample of independent data. However, they do provide guidelines on gauging the threat of rainfall and lightning based on the probabilities.

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**7. CATEGORICAL FORECASTS OF RAINFALL**

It is often desirable to obtain an estimate of the rainfall amount associated with the probabilities. This is essentially a categorical forecast of the peak rainfall amount. A simple approach to this problem is to select a set of thresholds for the probability values, and to forecast the highest rainfall category whose threshold has been exceeded.

We adapted the method used by Charba (1998), and selected probabilistic-to-categorical conversion thresholds by finding the probability at which the bias of the forecasts was approximately 1.5. These thresholds were close to those that maximize the CSI, and they allow the categorical forecasts to capture a very significant portion of rainfall events without greatly overforecasting areal coverage. Two sets of thresholds were selected, for the morning (0700-1700 UTC) and afternoon-nighttime (1800-0600 UTC) periods. The threshold sets are 22%, 20%, 16%, and 7% for P(0.1), P(0.5), P(1), and P(2) for the morning period, and 25%, 23%, 18%, and 9% for the afternoon-night period.

Verification results for the dependent data sample appear in contingency tables in Fig. 14, where forecast categories define the columns and observed categories, the rows. Results for the 2100-0000 UTC period (Fig. 14a) indicate that, for all forecasts 0.1 inch, 21% fall into the correct category and 75% fall within one category of the correct one. For all forecasts of 1 inch, 43% fall within one category of the correct one.

In terms of detection, 46% of the 2-inch observed events fall within the 1-inch forecast isohyet, and 63% of the 1-inch plus events fall within the 0.5-inch forecast isohyet.

Verification of forecasts for the 1200-1500 UTC period (Fig. 14b) indicate that for all forecasts of 0.1 inch, 23% fall within the correct category and 83% fall within one category of correct. For all forecasts of 1 inch, 47% fall within one category of the correct one. For all observed 2-inch events, 39% fall within the 1-inch forecast isohyet, and for all observed 1-inch plus events, 62% fall within the 0.5-inch forecast isohyet.

Again it should be noted that these results, derived from the dependent data, are unlikely to be reproduced at the same skill level in an independent sample. However, it should also be noted that currently-operational numerical models run by the NWS rarely forecast amounts in excess of 0.5 inches in three hours. The extrapolative-statistical method was specifically developed to forecast higher point amounts.

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**8. EXAMPLE: FORECASTS AND VERIFICATION FOR 2100-0000 UTC, 2-3 MAY 1997**

An example is presented to illustrate the relationship between initial-time observations and forecast probability fields, and between the forecasts and verifying observations. In the figures following, observations and verifying data are presented over light-colored backgrounds, and forecasts over dark backgrounds.

Initial-time fields for this case, observed between 2000 and 2030 UTC on 2 May 1997, appear in Figs. 15-17. Radar (Fig. 15) and lightning (Fig. 16) observations indicated instability lines from northern Louisiana through Tennessee and Kentucky, other convective precipitation over Arkansas and Missouri, and lighter precipitation over Alabama. A small but intense storm was located over extreme south Florida. Satellite (Fig. 17) indicated an extensive -50 C cloud shield over the areas of active convection.

The results of extrapolating the radar field for 3.5 h with the 700-500 mb mean wind vector appears in Fig. 18, where the highest reflectivity value forecasted for each MDR box is shown. The field is displaced east and southeast of the initial-time field, given the prevailing east-west motion of the system.

By applying the probability equations to this and the other extrapolated data fields, probability fields for 2100-0000 UTC rainfall were derived. The maximum values are displaced east of the highest initial-time reflectivity and lightning activity, given the prevailing west-east motion of the overall pattern. Values for P(0.1) exceeded 90% from northern Louisiana through south-central Tennessee, with values over 80% in Missouri and extreme western Kentucky (Fig. 19). In the area immediately downstream of the most intense squall line, P(0.5) exceeded 70% (Fig. 20), P(1) reached 40% (Fig. 21), and P(2) exceeded 10% (Fig. 22). The categorical forecast field (Fig. 23) featured amounts above 2 inches in extreme western Kentucky, in a band from northern Louisiana through northern Alabama and southern Tennessee, and over extreme south Florida. A rather large area with peak amounts forecasted to exceed 1 inch surrounded the 2-inch area.

The verifying Stage III analysis (Fig. 24) indicated peak amounts in excess of 2 inches over extreme southern Arkansas, northern Louisiana, northern Mississippi, and northern Alabama. Additionally, 1-inch amounts were observed over Missouri, extreme northeastern Texas, Tennessee, and Kentucky. The rainfall was overforecasted over the southern halves of Alabama and Mississippi and over south Florida, where the largest rainfall amounts were under 0.5 inch.

The probability forecast (Fig. 25) and verification (Fig. 26) for lightning follow the basic pattern evident in the precipitation forecast fields, as would be expected. Most areas with probabilities of 30% or more experienced lightning, except south Florida. A gap in the lightning-affected area from central Arkansas through western Tennessee was reflected in lower probabilities between the two largest lines of convective storms.

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**9. EXTRAPOLATIVE FORECASTS AND OPERATIONAL NUMERICAL MODEL FORECASTS**

Forecasters can refer to precipitation forecasts from operational numerical models including the NGM and the Rapid Update Cycle 2 (RUC-2) model (Benjamin et al. 1999). These models explicitly predict precipitation from both large-scale vertical motion and from parameterized convective processes, and thus can generate rainfall even in areas featuring no precipitation or deep clouds at initial time, while the extrapolation model cannot. On the other hand, these models generally do not explicitly simulate convective features, and sometimes fail to forecast rain from rather intense mesoscale systems even when these systems are fully developed at the model initial time. Thus it is not immediately clear which approach, explicit numerical models or extrapolation, is superior for forecasting rainfall in the 0-3 hour timeframe.

To compare the relative value of forecasts from the NGM, the RUC-2, and the radar extrapolation algorithm, we correlated output from each with the peak 40-km rainfall observations used as the predictand in our model development. The verification periods used were 1200-1500 and 2100-0000 UTC, during the period 7 July - 9 September, 1998. For the NGM, rainfall was taken from 12-18 h forecasts of the 0000-UTC operational run for the morning verification period, and from 6-12 h forecasts of the 1200-UTC run for the afternoon verification period. For the RUC-2, both 0-3 h and 3-6 h forecasts were tested (though the operational 0-3 h forecasts are generally not available until some time into the valid period). The NLVL456 radar-based predictor was used as a representative of the extrapolative-statistical model, though it is not an actual rainfall forecast. The complete probability forecast algorithm includes some input other than wind vectors from the NGM and thus was unsuitable for this comparison.

To describe the statistical correlation between each set of forecasts and the verifying rainfall events, we used the correlation ratio, a nonparametric measure of the fraction of predictand variance explained by a predictor (Panofsky and Brier 1968). When the predictor-predictand relationship is nearly linear, the correlation ratio approaches the square of the linear correlation coefficient. Correlation statistics were derived separately for each of five rainfall amount thresholds, with a 0.01-inch (.25 mm) threshold being added to the others mentioned previously.

We found that, during this test period, the radar-based predictor generally explained more than twice as much of the predictand variance as did any of the numerical models for the 2100-0000 UTC verification period (Fig. 27a), and almost three times as much during the 1200-1500 UTC verification period (Fig. 27b). The improvement due to the extrapolation model might be greater for the morning verification period because of the longer lead time imposed on the NGM and on the model driving the boundary conditions of the RUC-2 (12-15 h vs. 9-12 h for the afternoon verification period). Differences among the scores for the numerical models were much smaller than the differences between the numerical models and the radar extrapolation algorithm.

The lower correlation ratios for the NGM and RUC precipitation forecasts would be reflected as relatively high false alarm rates and biases for categorical (yes/no) rainfall forecasts. For example, when the radar-based NLVL456 predictor was used to produce yes/no forecasts of rainfall 1 inch within this comparison sample, the threshold value that produced a POD of 0.82 produced an FAR of 0.85 and a bias of 5.6. The threshold value of the RUC precipitation forecast that produced the same POD yielded an FAR of 0.92 and a bias of 9.5.

Of course, some of the improvement is due to the fact that the radar system itself was used to verify the forecasts. It is likely that all of the correlation ratios would be smaller if gauge data were used in verification. However, since the radar rainfall estimates are fairly reliable, we conclude that extrapolation does provide an operationally significant improvement over numerical models for this short verification period and lead time. It is certain that at some point beyond three hours' lead time the numerical models would improve on extrapolation.

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Real-time testing of the 0-3 h rainfall forecast algorithm is ongoing within the Techniques Development Laboratory. Work is underway to extend the methodology to the cool season; initial results suggest that skill is significantly higher during that time of year, when convective activity is more strongly controlled by synoptic-scale systems and is more tightly organized than it is during the late spring and summer. The method will be applied only to areas where the atmosphere is warm enough to support rainfall at the surface. Experiments in using a regionalized pattern-matching method to estimate the extrapolation velocity are also underway.

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We are indebted to Robert Saffle and Robert Elvander for their suggestions and inspiration during the early stages of this work. Invaluable programming support for archiving procedures was provided by Sun Junyuan of the China Meteorological Administration during his exchange tour of duty with TDL .

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Benjamin, S. G., J. M. Brown, K. J. Brundage, D. Kim, B. Schwartz,
T. Smirnova, and T. L. Smith, 1999: Aviation forecasts of the RUC-2. *Preprints, 8th
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Figure 1. Probability of 1 inch (25mm) or more of rain during the period 2100-0000 UTC, as a function of rainfall predictors NLVL456 and TDIFF700 derived from radar observations at 2030 and satellite observations at 2015 UTC, respectively.

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Figure 2. As in Fig. 1, except the valid period is 1200-1500 UTC, with NLVL456 and TDIFF700 derived from radar observations at 1130 UTC and satellite observations at 1115 UTC, respectively.

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Figure 3. Probability of 0.1 inch (2.5 mm) of rainnfall during the period 2100-0000 UTC, as a fuction of (a) NLVL456 and TDIFF700 and (b) 15-minute lightning strike rate and TDIFF700.

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Figure 4. Probability of 2+ CG lightning strikes within a 40-km square grid box, as a function of NLVL456 and maximum 15-minute strike rate as forecasted by extrapolation. Valid period is 2100- 0000 UTC.

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Figure 5. Categorical forecast scores for P(0.1) forecasts valid 2100-0000 UTC. Traces are for POD (red), FAR (green), CSI (blue), bias (dashed).

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Figure 6. As in Fig. 5, except for P(0.5).

Figure 7. As in Fig. 5, except for P(1.0).

Figure 8. As in Fig. 5, except for P(2.0).

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Figure 9. As in Fig. 5, except for the 1200-1500 UTC valid period.

Figure 10. As in Fig. 6, except for the 1200-1500 UTC valid period.

Figure 11. As in Fig. 7, except for the 1200-1500 UTC valid period.

Figure 12. As in Fig. 8, except for the 1200-1500 UTC valid period.

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Figure 13. Categorical forecast scores for P(LTG) during the period 2100-0000 UTC. Scores are as in Fig. 5.

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Figure 14. Verification contingency tables for categorical rainfall amount forecasts for (a) 2100-0000 UTC and (b) 1200- 1500 UTC. Forecast categories are in columns, observed categories in rows. Marginal totals appear outside the boxes.

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Figure 15. Radar reflectivity at 2030 UTC, 2 May 1997.

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Figure 16. RCM grid boxes with CG lightning during the period 2005-2025 UTC, 2 May 1997.

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Figure 17. Cloud-top temperatures from GOES at 2015 UTC, 2 May 1997. Blue indicates -10 to -29 green -30 to -49^{o}C,^{o}C, red -50^{o}C and colder.

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Figure 18. Maximum reflectivity within each 40-km MDR grid box as forecasted by extrapolation, valid 2100- 0000 UTC, 2-3 May 1997.

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Figure 19. Probability of 0.1 inch (2.5 mm) or more of rain during the period 2100-0000 UTC, 2- 3 May 1997, as forecasted by extrapolative-statistical algorithm.

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Figure 20. As in Fig. 19, except for probability of 0.5 inch (12.5 mm) or more.

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Figure 21. As in Fig. 19, except for probability of 1 inch (25 mm).

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Figure 22. As in Fig. 19, except for probability of 2 inch (50 mm). Note change in color scale.

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Figure 23. Categorical rainfall amount forecast valid 2100-0000 UTC, 2-3 May 1997. The forecast is for the highest amount with 40-km grid boxes. The 2-inch category is open-ended.

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Figure 24. Rainfall from Stage III estimates, during the period 2100-0000 UTC, 2-3 May 1997. Values indicated are the highest amounts within each grid box. The limits of radar coverage in the rainfall analysis are shown by gray shading over water areas.

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Figure 25. Probability of 2 or more CG lightning strikes, during the period 2100-0000 UTC, 2-3 May 1997.

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Figure 26. Grid boxes with CG lightning during the period of 2100-2359 UTC, 2 May 1997. Boxes with one strike are colored blue, those with two or more cyan.

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Figure 27. Correlation ratios for radar extrapolation predictor NLVL456 ("RADAR"), NGM precipitation, and 3-h RUC-2 precipitation forecasts, with respect to rainfall events of varying magnitudes. Verification periods are (a) 2100-0000 UTC 1200-1500 UTC. Return to Document.