continuous rainfall simulation: estimation at ungauged locations

International Workshop ADVANCES IN STATISTICAL HYDROLOGY

May 23-25, 2010 Taormina, Italy

Westra et al., Continuous rainfall simulation: estimation at ungauged locations

1

CONTINUOUS RAINFALL SIMULATION: ESTIMATION AT UNGAUGED

LOCATIONS

by

Seth Westra(1)

, Rajeshwar Mehrotra(1)

, Ratnasingham Srikanthan(2)

, Ashish Sharma(1)

(1) School of Civil and Environmental Engineering, the University of New South Wales, Sydney, NSW, Australia 2052

([email protected]; [email protected]; [email protected]) (2) Water Division, Australian Bureau of Meteorology, GPO Box 1289, Melbourne, VIC, Australia 3001 ([email protected])

ABSTRACT

Continuous simulation of extended rainfall sequences is becoming an increasingly important tool in rainfall-runoff

modelling and design flood estimation. Recently, a nonparametric approach has been developed in which continuous

(sub-daily) rainfall fragments are conditioned to daily rainfall on the current day and the rainfall state (wet or dry) on

adjacent days using a k-nearest neighbour conditional resampling algorithm, with this method performing well in

simulating both Intensity-Frequency-Duration characteristics and antecedent moisture when tested in Sydney, Australia.

A limitation of this approach, however, is that extended pluviograph records at the location of interest are required,

limiting applicability to only a small number of locations across Australia where such records are available.

In this paper we propose an extension in which sub-daily fragments from nearby pluviograph records can be substituted

for at-site data, potentially widening the applicability of the method to all locations for which extended daily data is

available or for which such data can be stochastically generated. A preliminary analysis involving the substitution of

nearby stations for the Sydney Observatory Hill record suggests that the method performs well in preserving sub-daily

rainfall characteristics across the full range of historical exceedance probabilities. Furthermore, an analysis of the factors

which determine whether a daily/sub-daily record at one location is substitutable for that at another location suggests

that the two stations must be at similar latitudes, with differences in longitude, elevation and distance to coast found to

be less important.

The analysis therefore suggests that the non-parametric continuous simulation approach is likely to perform well in

representing the historical rainfall pattern at ungauged locations, and as such comprises a potentially viable approach in

the generation of extended synthetic sequences for use in rainfall-runoff modelling and design flood estimation..

Keywords: Continuous simulation, stochastic generation, sub-daily rainfall, regionalisation

1 INTRODUCTION

Continuous rainfall simulation, defined here as the stochastic (random) generation of extended rainfall

sequences at the sub-daily time scale such that the characteristics of historical rainfall variability are

accurately preserved, is becoming an increasingly important tool in rainfall-runoff modeling and design

flood estimation. In particular, continuous simulation has begun to be regarded as a viable alternative to the

design storm approach for estimating the design flood hydrograph in ungauged catchments, with the

advantage that it does not require the assumption of exceedance probability neutrality – that the flood event

at a given exceedance probability can be derived from a design rainfall event at the same probability – be

maintained (e.g. Kuczera et al, 2006). This assumption recently has been called into question, with

potentially significant dependencies between catchment antecedent moisture conditions and storm

recurrence interval (e.g. Hill et al, 1997; Sharma and Srikanthan, 2006) highlighting the benefits of

modelling the full joint probability between the flood-producing rainfall event and the rainfall occurring

over some sufficient period prior to the event.

Recently, a non-parametric approach for continuous rainfall simulation at a point location was proposed, in

which sub-daily rainfall fragments are randomly selected using a k-nearest neighbor approach by

conditioning to the daily rainfall amount for the current day and the rainfall state (wet or dry) on adjacent

days (Sharma and Srikanthan, 2006). This approach has been compared to a range of alternative approaches




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such as the random multiplicative cascades approach (e.g. Menabde et al, 1997, Molnar and Burlando,

2005) and the randomized Bartlett Lewis Rectangular Pulse model (e.g. Rodriguez-Iturbe et al, 1987), and

was found to perform well in maintaining a range of statistics of historical sub-daily rainfall, including

Intensity-Frequency-Duration (IFD) relationships and wet spell length (Pui et al, 2009).

A limitation of the non-parametric re-sampling approach is that it requires extended continuous rainfall

records to form the basis for generating the sub-daily fragments, such that the approach can be applied only

in regions where such records exist. In this paper, we explore the possibility of extending the approach by

substituting nearby continuous rainfall records for at-site continuous rainfall records, thereby allowing the

method to be applied anywhere that sufficient ‘nearby’ continuous rainfall records exist. To this end, this

paper seeks to answer the questions: how can the similarity between daily and sub-daily rainfall

characteristics at two sites be measured? To what extent are sub-daily fragments from nearby stations

substitutable? What are the key factors, such as geographic distance, elevation and distance from coast,

which determine the similarity of the daily/sub-daily relationship between two stations?

The remainder of this paper is structured as follows. In the next section we provide an overview of

Australia’s continuous rainfall record. This is followed by a description of the proposed methodology,

including the definition of the statistics used to determine the similarity between daily/sub-daily rainfall

records at any two locations. Results are then presented in Section 4, including a preliminary analysis of the

viability of the method for Observatory Hill in Sydney, Australia, together with an assessment of factors

which will determine whether the daily/sub-daily relationships at two locations are similar. Finally, a

discussion and conclusions are provided in Section 5.

2 DATA

Continuous (sub-daily) rainfall data was obtained from the Australian Bureau of Meteorology (pers. coms.,

Sri Srikanthan) at 1397 continuous gauging stations, in increments of 6 minutes. The location of each

gauging station is shown in Figure 1, together with an indication of the length of record. Of the 1397

available gauging stations, 101 locations having length greater than 40 years, and a further 331 locations

having length of between 20 and 40 years. The spatial distribution of the gauging stations is not

homogeneous, with a high density of gauges in the populated regions particularly along the eastern coastal

fringe of Australia and lower density elsewhere. As described in the Introduction, the comparatively small

number of extended continuous rainfall records highlights the benefits of pooling neighbouring continuous

rainfall data to form the basis for continuous simulation.

To derive a better understanding of the temporal distribution of the continuous rainfall dataset, the number

of gauging stations with continuous rainfall records is plotted against the year of record in Figure 2. As can

be seen, only a small number of gauging stations were available in the early 20th century (the longest record

is available from Melbourne Regional Office, gauge number 086071, with data available from 1873 to

2008), with significant increases in recording density apparent in the 1960s. To limit the effects of possible

temporal variability in the daily/sub-daily characteristics, the remainder of the paper only considers records

between 1970 and 2005 with less than 15% of the record classified as ‘missing’.




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3 METHODOLOGY

3.1 Defining sub-daily rainfall attributes

The basis of the method presented in Sharma and Srikanthan (2006) is that sub-daily rainfall fragments are

selected using a k-nearest neighbour

resampling algorithm to condition to daily

rainfall amounts. Thus, in this paper we

take as a starting point the assumption that

we are able to adequately reproduce daily

rainfall amounts (either by using an

extended daily rainfall record, or by

stochastic generation), and we wish to

know: what are the characteristics of sub-

daily rainfall conditional to daily rainfall

amount that must be preserved?

To this end for each wet day (defined as >

0.3mm rainfall) we calculate the following

sub-daily rainfall attributes:

1) 6-minute maximum rainfall intensity;

2) 1-hour maximum rainfall intensity;

3) 6-hour maximum rainfall intensity;

Figure 1 - Spatial coverage and record length of the Australian continuous rainfall record.

Figure 2 - Number of Australia-wide continuous rainfall

records against year of record.




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and

4) Fraction of day with no rainfall, estimated as the number of 6-minute intervals with no recorded rainfall

divided by the total number of 6-minute increments (i.e. 240).

To allow sub-daily fragments from nearby pluviograph stations to be substituted for sub-daily fragments at

the location of interest, we propose that the joint probability between daily rainfall and each of these sub-

daily attributes must be the same across the full distribution of wet day amounts, such that, conditional to a

given daily rainfall amount, the statistics of sub-daily rainfall will be correctly represented.

As an example, we plot the 6-minute maximum rainfall intensity against daily rainfall for three locations in

Australia: Hobart, Sydney and Darwin for the months January, February and March. This is shown in Figure

3, with both daily and sub-daily rainfall plotted on a logarithmic scale. The 1:1 line in the joint probability

plot represents the case where all the daily rainfall occurs within the maximum 6-minute rainfall burst, with

all points in the plot necessarily falling on or to the right of this line. The 1-hour and 6-hour plots (not

shown) behave similarly but are distributed closer to this 1:1 line, as clearly the maximum 6-hour burst

contains a greater fraction of the day’s rainfall than the 6-minute burst. A loess smoother is also applied, and

shows that the departure from the 1:1 line increases with increasing daily rainfall amount, suggesting that

the proportion of the daily rainfall falling as the maximum short-duration burst is conditional to the total

amount of rainfall for that day.

An interesting although not unexpected result in Figure 3 is the differing characteristics of Hobart, Sydney

and Darwin sub-daily rainfall fragments. Specifically, on average the 6-minute rainfall storm intensity for

Darwin (red line) is much closer to the 1:1 line, while the average 6-minute storm intensity for Hobart is

furthest from the 1:1 line, with the same conclusions derived for the 1-hour and 6-hour storm burst (not

shown). This suggests that in Darwin, a greater proportion of the daily rainfall falls as high-intensity short-

duration storm bursts, while in Hobart the rainfall is more evenly spread throughout the day.

A similar conclusion is derived by considering the relationship between daily rainfall and fraction of day

with no rainfall presented in Figure 4. Here, for all daily rainfall amounts Darwin rainfall shows a greater

proportion of the day as dry compared with the other locations, although once again there is a strong

conditional relationship between daily rainfall and this sub-daily attribute.

Based on this analysis it is clear that, conditional to the daily rainfall amount, the sub-daily fragments of

Hobart, Sydney and Darwin are not substitutable. Such a result is unsurprising, with Hobart and Darwin

deliberately selected for this example as they comprise the most southerly and northerly continuous rainfall

station, respectively, for which long continuous record are available. Nevertheless, this raises the question:

how does one measure the similarity in the daily/sub-daily relationship between different stations?

3.2 Measuring similarity

Although the preceding example visually highlighted distinct differences between the daily/sub-daily

relationships at three locations, it is necessary to develop a metric that will allow for a quantitative

comparison of the similarity between the daily/sub-daily attributes from a large number of continuous

records.

We propose a non-parametric statistic known as the Mean Integrated Squared Error (MISE; see Scott, 1992),

which provides a measure of the departure of the empirical joint probability density function of daily and

sub-daily rainfall attributes at any two locations. The empirical joint probability density function for each

daily/sub-daily attribute relationship is estimated using a histogram approach, in which the sample is divided

into equally spaced bins (with spacing in the logarithmic scale for all attributes except for fraction of day

with no rainfall), and the number of occurrences in each bin is then counted. The frequency histograms are

transformed to density histograms by dividing each bin by the total number of data points in all bins, such

that the joint density histogram integrates to 1. The marginal density histograms can be seen in Figures 3 and

4.

Letting vk denote the bin count of the kth bin, then the empirical histogram for daily rainfall and any one of

the sub-daily attributes listed above at a given location is defined as:




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(1)

where hx and hy are the bin widths in

the x and y dimensions, and n is the

total number of data points. The

MISE of the density histograms at

two locations can then be calculated

as the integration of the squared

difference of each histogram bin:

(2)

where the subscripts i and j refer to

any two locations.

In all cases the histograms are

constructed by pooling precipitation

from three consecutive months. For

example, Figures 3 and 4 were

derived using data from Jan-Feb-Mar.

This was done to maximise the

number of data points (rain days)

while simultaneously ensuring

seasonal effects did not unduly

affect the results. Using data from

1970 to 2005, and assuming that

about a third of the days in any

given month are wet, the average

number of histogram data points is

about 1000. Based on this number,

Sturges’ number-of-bins rule

suggests about 11 bins assuming a

Gaussian distribution, with a slightly

greater number recommended in the

case of non-Gaussian distributions

(Scott, 1992). We therefore selected

a bin width for the joint distribution

about double that used for the

marginal distributions indicated in

Figures 3 and 4.

The MISE can now be calculated for

any two station pair for each of the

four sub-daily attributes listed

above. Our final skill score is the

MISE for each two-station pair

averaged across all four attributes.

Figure 3 - Maximum 6-minute storm burst against daily rainfall for

each wet day, for Darwin (red), Sydney (green) and Hobart (blue),

plotted on a logarithmic scale. Mean response estimated using a

loess smoother fit to the log-transformed data.

Figure 4 - Fraction of each wet day with no rainfall against daily

rainfall amount, for Darwin (red), Sydney (green) and Hobart (blue),

plotted on a logarithmic scale. Mean response estimated using a

loess smoother fit to the log-transformed data.




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4 RESULTS

4.1 Sydney Observatory Hill

We commence by comparing the

daily/sub-daily rainfall characteristics

at Sydney Observatory Hill (gauge

number 066062) with the

characteristics at a number of

neighbouring continuous rainfall

stations within a 2° longitude/latitude

(approximately 200km) radius. Using a

consistent time period between 1970

and 2005 with the criteria that no more

than 15% of the record may be

classified as ‘missing’, we find a total

of 14 such stations, presented as brown

dots in Figure 5 and listed in Table I.

As can be seen from this figure, the

stations include a combination of

coastal and inland locations, and

therefore would be expected to yield a

variety of sub-daily characteristics.

The MISE statistic averaged across all

seasons and all the sub-daily rainfall

attributes described in the previous

section are listed in Table I. As can be

seen from these results, the station

with sub-daily characteristics most

similar to Sydney Observatory Hill is

Port Kembla BHP, located along the

coast about 70km south of Sydney.

This is followed closely by Sydney

Airport, which is located along the

coast slightly south of Observatory

Hill, and then three stations located

north of Sydney in the Hunter region.

Having identified the neighbouring gauges with daily/sub-daily relationships most similar to the Sydney

Observatory Hill record, we wish to know the extent to which these fragments can be substituted for the

Sydney Observatory Hill continuous rainfall record. To this end we take the five gauges with the lowest

MISE (based on the ranks shown in Table I) and pool fragments from all gauges together to form a single

record. We then use the k–nearest neighbor approach described further in Sharma and Srikanthan (2006) to

select sub-daily fragments from this pooled record conditional to the daily record at Sydney Observatory

Hill.

The re-sampling approach is based on daily rainfall data, such that given a certain daily rainfall amount, the

fragments from nearby gauges with a similar daily rainfall amounts are selected, with the selection of

surrounding locations based on the MISE statistic designed such that the characteristics of the sub-daily

rainfall burst will be preserved. To test this hypothesis, a quantile-quantile plot of the simulated maximum

6-minute intensity based on the re-sampling from nearby sub-daily rainfall fragments against the observed

maximum 6-minute rainfall intensity at Observatory Hill is shown in Figure 6. These results, although

preliminary, suggest a very close correspondence across the full range of probabilities, suggesting that at

least for Sydney Observatory Hill, the selected nearby stations are indeed substitutable.

Figure 5 - Sydney Observatory Hill gauge (red dot, centre), together

with all rain gauges within a 2° longitude/latitude radius. Larger

brown dots correspond to stations which comply with the admission

criterion of less than 15% missing between 1970 and 2005. For

station names, see Table 1.




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Table I: Stations used for Sydney Observatory Hill study.

Gauge number

(corresponding to

Figure 5)

Gauge number Gauge name Averaged

MISE

Rank

1 061151 Chichester Dam 0.0091 7

2 061158 Glendon Brook (Lilyvale) 0.0078 5

3 061211 Colo Heights (Mountain

Pines) 0.0098

8

4 061238 Pokolbin (Somerset) 0.0056 3

5 061287 Merriwa (Roscommon) 0.0104 10

6 061288 Lostock Dam 0.0072 4

7 061309 Milbrodale (Hillsdale) 0.0082 6

8 061334 Glen Alice 0.0111 12

9 063043 Kurrajong Heights 0.0137 14

10 066037 Sydney Airport 0.0032 2

11 068117 Robertson (St Anthony’s) 0.0107 11

12 068131 Port Kembla BHP 0.0027 1

13 069049 Nerriga Composite 0.0112 13

14 070012 Bungonia (Inverary Park) 0.0103 9

4.2 Australia-wide analysis

Although the preceding analysis is useful to illustrate the viability of the proposed approach, the ultimate

application of the method is for locations where no long continuous rainfall records are available, such that

one will not be able to calculate the MISE between the location of interest and a set of nearby stations to

determine the most similar continuous

records. As such, it is necessary to develop a

means to determine which factors influence

whether two locations will be sufficiently

similar to allow sub-daily fragments to be

used. These factors might include:

1) Distance metrics, including absolute

distance and the difference in

latitude and longitude between

stations;

2) Differences in elevation; and

3) Differences in the distance from the

coast.

To determine the relative importance of each

of these factors, we consider the full

Australian continuous rainfall record

between 1970 and 2005 for which less than

15% of the record is missing, totaling 167

locations Australia-wide. Similar to the

example for Sydney Observatory Hill, we

calculate the MISE as the similarity measure

for all 13861 possible station pairs, as well as

Figure 6 - Quantile-quantile plot of observed six-minute

rainfall intensity at Sydney Observatory Hill, and simulated

six minute intensity derived from conditional resampling

from the pool of continuous rainfall fragments obtained

from the five most proximate stations defined in Table 5.




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each of the above factors as possible predictors for station similarity.

A regression model was developed in which each of the predictors was regressed against a log-transformed

version of each of the MISE skill scores (i.e., 6-minute, 1-hour and 6-hour rainfall, and dry fraction), as well

as the average MISE for all four attributes. The use of a log-transformation ensures that the regression

residuals follow an approximately Gaussian distribution. The results from all four attributes were similar,

and as such the remainder of the paper will focus on the MISE averaged over all four attributes.

The coefficient of determination calculated based on a range of factors is shown in Table II. Considering the

distance metrics first, it is clear that the greatest coefficient of determination is for the differences in latitude

between respective station pairs, whereas very little influence could be observed for changes in longitude.

The absolute distance has a coefficient of determination greater than that for longitude and less than that for

latitude, suggesting that it is the difference in latitude which represents the most significant factor for

whether the daily/sub-daily fragments at two stations are similar.

A plot of the MISE against difference in latitude is provided in Figure 7. Although there is clearly a lot of

scatter, the general trend of increasing MISE with increasing difference in latitude is clear. The line of best

fit (red line) was developed through linear regression against the log-transformed the MISE, and therefore

appears here as an exponential curve. The implications of such a curve are significant; the skill scores are

relatively insensitive to small differences in latitude up to about 5° or 10°, whereas significant divergences

in the daily/sub-daily characteristics are apparent for greater latitude differences.

Finally, the results in Table II suggest that both the difference in distance from coast and difference in

elevation yield low coefficients of determination, with values of 0.04 and 0.02 respectively, suggesting that

the daily/sub-daily rainfall relationship is relatively insensitive to these factors. This is not to say that daily

rainfall does not change significantly as a result of changes in elevation or distance from the coast; rather,

conditional to a given daily rainfall amount, the sub-daily attributes used in this analysis do not appear to be

heavily influenced by these factors.

Table II - Log-transformed average MISE against a range of plausible predictors

Predictor R2

Distance between stations (km) 0.25

Difference in longitude 0.06

Difference in latitude 0.40

Difference in distance from coast (km) 0.04

Difference in elevation (m) 0.02




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5 CONCLUSIONS

In this paper we have presented an approach for the stochastic generation of continuous rainfall data for

locations where long continuous records do not exist, which can potentially significantly broaden the

applicability of the approach described in Sharma and Srikanthan (2006) to any location for which adequate

daily records are available. The basis of this approach is that sub-daily rainfall fragments derived from

‘nearby’ pluviograph records can be conditioned to daily rainfall amounts and Markov persistence attributes

at the location of interest, significantly extending the domain over which the method can be applied.

A preliminary analysis using the Sydney Observatory Hill continuous record shows that sub-daily rainfall

fragments from neighbouring stations are potentially substitutable, with neighbouring sub-daily fragments

behaving similarly to Observatory Hill sub-daily fragments across the full range of observed exceedance

probabilities. For the majority of potential applications, however, sufficient continuous rainfall data at the

location of interest will not be available to determine which nearby records are ‘similar’ and therefore

substitutable. For this reason, a metric was developed which would determine which stations will be similar

based on factors such as distance apart, differences in elevations and distance to coast. Of all the factors

assessed, the only factor which appeared to significantly influence the similarity of daily/sub-daily

relationship between two stations was the difference in latitude, with relatively low sensitivity to small

differences in latitude and much greater sensitivity to larger differences.

The results presented herein should be regarded as preliminary, with additional work required to confirm the

viability of the proposed technique. Specifically, future research in furthering this approach includes:

- Consideration of additional daily/sub-daily attributes such as the diurnal pattern in influencing the

MISE skill score;

- Consideration of additional predictors, such as total annual rainfall and seasonality in determining

whether two stations are likely to behave similarly; and

- Application of the approach for a range of locations around Australia to test the method, in particular

with regard to the reproducibility of IFD and antecedent moisture characteristics.

Finally, although this work has not attempted to account for the implications of the projected impacts of

Figure 7 - MISE averaged over all four sub-daily attributes for

all station pairs, against difference in latitude between the station

pairs (blue dots). Red line is the line of best fit after applying

linear regression to log-transformed MISE.




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anthropogenic climate change, it may be possible to adapt the method to achieve this by drawing fragments

from regions which have historical climates similar to the projected future climate in the region of interest.

Such an approach, which would involve conditioning the daily/sub-daily relationship to a range of climate

variables such as temperature and sea level pressure, and then using this as the basis for selecting fragments

based on future projections for these variables, represents an intriguing research direction worthy of further

exploration.

6 ACKNOWLEDGEMENTS We wish to gratefully acknowledge the Australian Research Council and Engineers Australia for partial

funding of this work. The rainfall data used was made available by the Bureau of Meteorology.

7 REFERENCES

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continuous rainfall simulation: estimation at ungauged locations

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