# Using Probable Maximum Precipitation to Bound the Disaggregation of Rainfall

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Theory

_{1}has a time-step of one day and R

_{8}has a time-step of 11.25 min. Assuming that volumes are preserved from intervals to sub-intervals, 0 ≤ W

_{L}≤ 1 and the proportion of the interval’s rainfall falling in the second sub-interval equals 1 − W

_{L}.

_{01}, is estimated. Then 1 − P

_{01}is the probability that 0 < W < 1, and, in this range, W is modelled using a probability distribution function, P

_{x}.

_{L}. If an available estimate of PMP is taken to be the maximum possible rainfall, then, if the rainfall volume R

_{L}is greater than PMP

_{L}

_{+1}, it is impossible for it all to fall within only one of the two sub-intervals, so P

_{01}= 0. More generally, it is impossible for W to be large or small enough for R

_{L}

_{+1}to exceed PMP

_{L}

_{+1}. If R

_{L}> PMP

_{L}

_{+1}, then:

_{x}is assumed to be a uniform distribution, for example [1], for which the only parameters are the upper and lower bounds, deriving P

_{x}from (3) is straightforward. However, in general, alternative distributions are preferable for P

_{x}, in which case the parameters cannot be directly derived from (3).

_{L}= 2 × PMP

_{L}

_{+1}because R

_{L}

_{+1}cannot exceed PMP

_{L}

_{+1}:

_{x}

_{0}is the Dirac delta. Although hypothetical, (4) provides an asymptote that may be useful for the extrapolation of the observed volume dependency, as illustrated by the case study results.

## 3. Methods

#### 3.1. Case Study

_{01}and β, were estimated per level. One of the conclusions was that, while the model performance was considered good, the model tended to over-estimate the high extreme rainfall volumes.

#### 3.2. MDRC Models

_{x}because it is considered to adequately represent the histograms of observed W values over a broad range of volumes [6].

_{01}using logistic regression:

_{01}values to be estimated for any value of R, including those higher than observed during the model fitting period. A comparable log-normal function was used by [4] to model the dependence between R and 1 − P

_{01}. For P

_{x}, the values of β used in the baseline model, β

_{BAS}, were maintained because a relationship between R and β was not easily identifiable for all levels from the observed W in the fitting period.

_{01}is employed but adjusted so that, from (3), P

_{01}is zero when R equals the PMP of the sub-interval (PMP

_{L}

_{+1}).

_{L}

_{+1}, P

_{01}= 0.

_{L}

_{+1}.

#### 3.3. Parameter Estimation

_{BAS}are estimated by maximising the log-likelihood of the model given the fitting period observations of W. An obvious fitting period would be 1987 to 2015 in order to be consistent with the evaluation of the baseline model in [6]. However, that period includes some of the highest extremes in the historical record, and hence evaluating the model on the remaining years would require a limited degree of extrapolation. Instead, therefore, the 12 years in the historical record with the fewest extreme events were identified and used for model fitting. For this purpose, the frequency of extreme events in year Y was defined as N

_{99Y}/N

_{99}, where N

_{99}is the number of R values that lie above the 99th percentile of non-trace values, R

_{99}, over the entire record and N

_{99Y}is the same number but only counting those values in year Y, scaled according to how many of the missing data lie in that year. N

_{99Y}/N

_{99}was calculated for each of the seven levels of disaggregation and the average taken. The 12 years with the smallest averages were used as the fitting period. In order of an increasing number of extremes these 12 years are 1993, 1908, 1962, 1989, 1913, 1978, 1936, 1911, 1918, 1952, 1944, and 2007.

_{x}being widely inconsistent with (3). Instead, (3) is used to synthesise a theoretically-based value of β at R = PMP, here called β

_{PMP}, and (10) is required to pass through this value. To estimate β

_{PMP}, a lower bound on W corresponding to R = PMP, here called W

_{PMP}, is identified from (3), and β

_{PMP}is optimised so that the probability of W < W

_{PMP}is equal to an arbitrarily low probability of 0.001. Due to the assumption that P

_{x}is symmetrical, this leads to a 0.002 probability of a W value being sampled from outside the bounds defined by (3) at R = PMP. The baseline model value of β, β

_{BAS}, is maintained up to the maximum observed value of R, M. For R > M, (10) is applied and both c and d are fixed by requiring β to pass through the point [PMP, β

_{PMP}] and also through [M, β

_{BAS}]. The latter means that there is no discontinuity in the volume dependency when moving from R < M to R > M. This approach to estimating P

_{x}therefore has two questionable aspects: the use of β

_{PMP}allows β to be extrapolated to extreme values in a way that is guided by the theoretical bounds defined in (3) but does not strictly honour them; and the value of M is arbitrary rather being estimated from the observations. Why this is a reasonable approach and why alternative models are not useful for the case study will be discussed later in the paper.

#### 3.4. Model Evaluation

_{99}are considered in the evaluation. As well as a visual comparison of the histograms of the observed and modelled rainfall, the observed and modelled count of all values above R

_{99}(a metric of frequency of extremes) and the observed and modelled means of all R values above R

_{99}(a metric of magnitude of extremes) are also compared. Achieving a realistic time-structure of rainfall events would require a more complex disaggregation approach, so only the frequencies and magnitudes are evaluated.

_{01}, and any non-zero values of W are then sampled from P

_{x}. The value of R in that sub-interval is then W.R, and the value in the second sub-interval is (1 − W)R.

#### 3.5. Sensitivity Analysis

## 4. Results

_{99}values, which are derived from the entire record) indicates the high degree of extrapolation required of the model in its evaluation.

_{01}estimated by fitting to the observed values of W within each second percentile range of R, plotted against the mean values of R in these ranges, using data from the entire historical record. Although there is considerable noise in these estimates, they show that the logistic regression models are reasonably consistent with the observed volume dependency and that the model identified using only the fitting period translates reasonably to the entire historical record. Also included in Figure 2 are the theoretical point, R

_{L}= PMP

_{L+}

_{1}, at and above which P

_{01}= 0 in the bounded model and the results of a generalised model (see Discussion). The results for the bounded model are barely distinguishable from those of the unbounded model so are not included in Figure 2.

_{PMP}, illustrating the degree to which the theoretical bounds on W are compromised. Figure 4 shows the baseline model estimate of β that is used in all three versions of the model and also the volume dependency of β obtained from (10). Superimposed on these curves are the ‘observed’ values of β estimated using the entire historical record, as previously explained for P

_{01}. These ‘observed’ values indicate the errors that arise from using only the fitting period for estimation, most notably at L = 1.

_{99}, and the frequencies are relative to the total number of included observed values. Therefore the differences in the shape and magnitude of the observed and simulated histograms can be used to visually assess the magnitude and frequency performance of the model. For the same set of observed and simulated extremes, Table 4 compares the magnitude and frequency metrics. Table 4 also shows the means and standard deviations of the results over the 100 realisations, which, for practical purposes, were the same between models so are only given once.

## 5. Discussions

_{01}values at log

_{10}R > ~1.4, resulting in weaker disaggregation than observed. This error persists even if fitting the logistic regression to the entire record. This may be explained by diurnal convective processes that dominate wet season rainfall in Brisbane, whereby extreme values of daily rainfall are often concentrated in the latter half of the day. In other words, while in general higher volumes mean weaker disaggregation, this is not necessarily the case due to strong convective processes that dominate extremes in this case. As the time–interval decreases, this effect seems to become less important.

_{01}and β deteriorated the performance for all models. This illustrates that restricting the data used for model fitting to a more relevant range of volumes is not useful here due to the lower number of observations and hence higher variance in the baseline estimates of P

_{01}and β. Reducing the PMP estimates to the HNPR values caused the bounded model to substantially underestimate the number of extremes above the threshold R

_{99}. This sensitivity confirms that the asymptote defined by (4) is an active constraint and hence that reasonable approximations of PMPs are valuable. Since in practice there is considerable uncertainty in PMP estimates [9,11], applications of the bounded approach to the PMP should ideally be described by a probability distribution function; for example, the approach described by [14] could be adapted. Changing the arbitrary values of β

_{PMP}and M as described in Table 2 also affected the simulated magnitudes and frequencies as expected, although to a relatively small extent.

_{L+}

_{1}< R

_{L}< PMP

_{L}. An alternative translation of (3), which would avoid this compromise, would be scaling the Beta distribution so that it’s upper and lower bounds are equal to the theoretical bounds calculated from (3). The β parameter could then, in the absence of a better estimate, be fixed at zero (a uniform distribution) or at the baseline model value. An alternative adjustment would be to maintain the standard Beta distribution but to curtail it so that the probability density of W is zero outside the theoretical bounds defined by (3). However, forcing the model to honour (3) is not useful for the case study because a very small number of the observed R are high enough (R

_{L}> PMP

_{L+}

_{1}) to be directly affected by (3), and for each of these the probability of an unrealistic sample of W is small. Rather, the value of (3) is supporting the extrapolation of β in the range M

_{L}< R

_{L}< PMP

_{L+}

_{1}. Another subjectivity in developing the bounded model of β is the arbitrary value of M, at which the model transitions from the constant baseline value of β to volume dependence. M can be eliminated from the model by applying (10) over the full range of R, in which case c can be fitted to the observations using maximum likelihood and d can be fixed by still requiring (10) to pass through the point [PMP, β

_{PMP}]. This produces a reasonable volume dependency over the higher ranges of R, similar to the results in Figure 4; however it only matches the performance of the original bounded model in the less relevant case that the observed values of R are used as inputs to every disaggregation level.

_{x}must transition, as the PMP value is approached, from a distribution function bounded by the values W = 0 and W = 1 to a distribution function with narrower bounds and (2) sufficient observations of extremes do not exist to examine the transition. Therefore a reasonable judgment of how the theoretical bounds on W should be used to bound P

_{x}as R tends to PMP was necessary in the case study and seems likely to be necessary in future applications. It is possible that pooling extremes from a large number of sites, which adhere to the same volume dependency model, will increase the number of extreme value observations and hence permit a reduced level of subjectivity in developing the model.

_{01}and β can be generalised over the seven disaggregation levels using linear regression. This is potentially advantageous for generating rainfall at unobserved time–resolutions; for example, extrapolating the regression would provide eighth and ninth levels of disaggregation to generate rainfall at approximately 5.6 and 2.8-min intervals. Furthermore, reducing the number of parameters by generalising over levels may assist with the regionalisation of the model. To transfer these potential benefits to the improved, volume–dependent models would require the parameters a and b to be generalised over levels as well as the baseline value of β (recalling that parameters c and d can then be fixed given β

_{PMP}and M). Figure 6 plots the estimates of log

_{10}β for the baseline model against L and the estimates of a and b for the bounded model against L. More or less the same results for a and b are obtained for the unbounded model.

^{2}= 0.98). There is much less of a relationship between level and log

_{10}β than previously found by [6], which must be due to the different fitting periods used (here, the 12 years with fewest extremes; there the 12 available years between 1987 and 2015). It seems that either using a lower range of volumes for fitting the model or relying more on the older pluviograph record rather than more modern tipping bucket data produces less well–behaved β values. To generalise the volume dependency of β, a generalised estimate of the point of transition, M, is also required. For the purpose of illustrating the potential for generalising the model, an arbitrary value, M = PMP/10, is adopted. The generalised volume dependencies are included in Figure 3 and Figure 5. As should be expected, overall, the generalised models are less consistent with the observations than the originals. However, for some levels, the generalisation is actually better, presumably because in these cases the regression smooths out random errors in the original estimates; for example, errors associated with the relatively short fitting period used. The model output statistics for the generalised bounded models, included in Table 4, show a slightly reduced performance compared to the original bounded model but still a considerably better performance than the baseline model. In summary, there is evidence that scaling relationships exist in the bounded and unbounded models, and it is speculated that these are useful for extending the application to unobserved time intervals and regionalisation.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 5.**Comparison of observed and simulated extreme rainfall using the baseline, unbounded and bounded models.

**Table 1.**Method of parameter estimation for different ranges of R for the unbounded baseline and bounded models.

Model | R_{L} ≤ M_{L} | M_{L} < R_{L} ≤ PMP_{L+}_{1} | R_{L} > PMP_{L+}_{1} | |
---|---|---|---|---|

Baseline model | P_{01} | Maximum likelihood estimate from the observed ^{[1]} W values. | ||

β | Maximum likelihood estimate from the observed ^{[1]} W values (0 < W < 1). | |||

Unbounded model | P_{01} | Logistic regression to describe volume dependency. Estimate parameters using maximum likelihood using all the observed W values. | ||

β | Maximum likelihood estimate from the observed ^{[1]} W values (0 < W < 1). | |||

Bounded model | P_{01} | Logistic regression to describe volume dependency bounded by R = PMP_{L+}_{1}. Estimate parameters using maximum likelihood using all the observed W values. | P_{01} = 0 | |

β | Maximum likelihood estimate from the observed ^{[1]} W values (0 < W < 1). | Equation (10) to described volume dependency. Parameters of (10) are fixed as described in text. |

^{[1]}Using only the values of W corresponding to the upper quartile of non-trace R

Subject of Sensitivity Analysis | Original Setting | Perturbed Setting | Rationale | Mean of All Values > R_{L99} for 11.25 Mins (mm) | Count of All Values > R_{L99} for 11.25 Mins |
---|---|---|---|---|---|

The range of
R volumes used to estimate P_{01} and P_{x} (baseline model, and all models for P_{x}) | Upper quartile of non-trace observed R volumes in fitting period | Upper 2% of non-trace observed R volumes in fitting period | The perturbation may improve the estimate of β applicable to modelling high extreme rainfall | 12.9 | 806 |

The translation of theoretical bounds on
W to values of β_{PMP} (bounded model) | β_{PMP} is set so there is a 0.002 probability of W values being sampled from outside the theoretical bounds | 0.02 probability is used instead of 0.002 | This will reduce β estimates for R > M, leading to stronger disaggregation and higher extremes | 12.7 | 686 |

The identification of the point of transition, M, from a constant value of β to volume-dependency (bounded model) | M is the highest value of R observed in the fitting period | M is the highest value of R observed in the entire record | M is unknown and arbitrarily fixed. A higher value of M will reduce β estimates at extremes, leading to stronger disaggregation and higher extremes | 13.1 | 690 |

The probable maximum precipitation (PMP) estimates (bounded model) | PMP estimates are derived using the Australian standard approaches [9,10] | The Historical Notable Point Rainfall (HNPR) values values are used instead | Although there is no ambiguity about the parameters used to calculate the PMP values, they are generalised across Australia and are high compared to the HNPR values. The latter may be considered a valid empirical rather than theoretical bound | 12.8 | 610 |

Observed values | 11.9 | 688 | |||

Original results of bounded model | 12.7 | 688 |

Level | Time Interval (mins) | PMP (mm) | HNPR (mm) | R_{99} (mm) | M (mm) | Baseline Model | Unbounded Model | Bounded Model | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Full Period | Fitting Period | P_{01} | log_{10}(β) | a | b | a | b | c | d | |||||

1 | 1440 | 1440 | 960 | 91 | 311 | 70 | 0.40 | −0.06 | −1.78 | 2.04 | −1.87 | 2.08 | −1.19 | 0.60 |

2 | 720 | 1242 | 717 | 69 | 228 | 66 | 0.29 | −0.12 | −0.95 | 1.72 | −1.02 | 1.73 | −1.15 | 0.56 |

3 | 360 | 1043 | 589 | 49 | 167 | 42 | 0.29 | −0.06 | −0.54 | 1.69 | −0.61 | 1.69 | −1.09 | 0.55 |

4 | 180 | 790 | 356 | 36 | 111 | 41 | 0.24 | 0.00 | 0.01 | 1.67 | −0.04 | 1.66 | −1.05 | 0.55 |

5 | 90 | 550 | 279 | 26 | 103 | 35 | 0.18 | −0.04 | 0.76 | 1.65 | 0.72 | 1.63 | −1.07 | 0.52 |

6 | 45 | 371 | 176 | 18 | 82 | 35 | 0.12 | 0.04 | 1.48 | 2.12 | 1.46 | 2.12 | −0.99 | 0.48 |

7 | 22.5 | 250 | 95 | 12 | 82 | 27 | 0.05 | 0.19 | 2.71 | 2.34 | 2.71 | 2.31 | −1.06 | 0.65 |

− | 11.25 | 184 | 75 | 8 | 75 | 18 | − | − | − | − | − | − | − | − |

**Table 4.**Observed and simulated mean R above R

_{99}and the number of R above R

_{99}over 1908–2015.

Mean R above R_{99} (mm) | Number of Peaks above R_{99} | |||||||
---|---|---|---|---|---|---|---|---|

12 h | 3 h | 45 min | 11.25 min | 12 h | 3 h | 45 min | 11.25 min | |

Observed R | 93.9 | 50.1 | 26.6 | 11.9 | 118 | 215 | 398 | 688 |

Baseline model | 103.3 ^{[1]} | 57.7 | 31.4 | 14.8 | 121 [^{[2]} | 283 | 552 | 1101 |

Unbounded model | 100.5 | 54.1 | 28.4 | 13.2 | 103 | 210 | 370 | 710 |

Bounded model | 98.7 | 51.9 | 27.1 | 12.6 | 100 | 203 | 365 | 688 |

Bounded model (generalised across levels) | 100.6 | 53.4 | 27.8 | 12.9 | 107 | 206 | 342 | 678 |

Standard deviation of model results | 2.0 ^{[3]} | 1.1 | 0.5 | 0.2 | 5 | 10 | 15 | 21 |

^{[1]}This and all subsequent mean values in the table are calculated by: generate 100 realisations of the 1908 to 2015 time-series; for each realization, calculate the mean volume, R

_{mean}, of all modelled R above the observed R

_{99}during 1908 to 2015; calculate the mean of R

_{mean}over the 100 realisations.

^{[2]}This and all subsequent numbers of peaks in the table are calculated by: generate 100 realisations of the 1908 to 2015 time-series; for each realisation count the number of modelled R values, R

_{num}, above the observed R

_{99}during 1908 to 2015; calculate the mean of R

_{num}over the 100 realisations.

^{[3]}This and all subsequent standard deviation values in the table are the standard deviations over the 100 samples of R

_{mean}or R

_{num}for the baseline model (the standard deviation values for the other models are within ±5% of this value).

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**MDPI and ACS Style**

McIntyre, N.; Bárdossy, A.
Using Probable Maximum Precipitation to Bound the Disaggregation of Rainfall. *Water* **2017**, *9*, 496.
https://doi.org/10.3390/w9070496

**AMA Style**

McIntyre N, Bárdossy A.
Using Probable Maximum Precipitation to Bound the Disaggregation of Rainfall. *Water*. 2017; 9(7):496.
https://doi.org/10.3390/w9070496

**Chicago/Turabian Style**

McIntyre, Neil, and András Bárdossy.
2017. "Using Probable Maximum Precipitation to Bound the Disaggregation of Rainfall" *Water* 9, no. 7: 496.
https://doi.org/10.3390/w9070496