# Sub-Hourly to Daily Rainfall Intensity-Duration-Frequency Estimation Using Stochastic Storm Transposition and Discontinuous Radar Data

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## Abstract

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## 1. Introduction

## 2. Data and Study Area

#### 2.1. Study Area

#### 2.2. Rain Gauges

#### 2.3. Regional Model

#### 2.4. Radar Rainfall

## 3. Methods

#### 3.1. Stochastic Storm Transposition

- We specify the island of Zealand (7842 km${}^{2}$) as our transposition domain from which rainfall will be resampled, see Figure 1. A prerequisite/assumption for random transposition to the area of interest (151 km${}^{2}$) is that the selected domain’s extreme rainfall climatology is homogeneous. This assumption is assessed and discussed in Section 4.1.
- The 500 largest storms at durations: 10, 30, 60, 180, 360, 720, and 1440 min are identified by ranking storms with the same shape and size as the area of interest (Figure 1). This collection of storms is henceforth known as the storm catalog (e.g., 60-minute storm catalog or simply 60-minute catalog). The creation of the storm catalogs is further described in Section 4.2.
- The number of yearly storm occurrences for each duration is assumed to follow a Poisson distribution with a rate parameter: $\lambda =88$ (average 88 storms per year). This derivation will be further explained in Section 4.2.
- From the storm catalog in question (e.g., 60-min), we randomly select a storm. The assumption of homogeneous climatology, within the transposition domain, allows us to stochastically transpose the series of radar images in the x, y directions, since the storm is assumed to have equal likelihood to have appeared anywhere within the domain. Every time step of the radar image series is transposed with the same vector, so the original motion of a rain storm is retained, and only the spatial occurrence is altered.
- Step 4 is repeated k times where k is a random integer drawn from the Poisson distribution explained in step 3. The k storms represent one year of rainfall.
- For each transposed storm in step 4, we compute the t-minute catchment-average/point rainfall depth, for the area/point of interest.
- Steps 4–6 are repeated 1000 times to create 1000 years of synthetic storm events. The 1000 largest values from all of the synthetic years are retained (following a general “partial-duration-series” approach), and IDF values are estimated.
- Steps 4–7 are repeated, for this study, 100 times to quantify the uncertainty of the estimated IDF values.

#### 3.2. Rainfall Statistics

#### 3.3. Pixel-Scale Duration Bias

#### 3.4. Areal Reduction Factors

## 4. Results and Discussion

#### 4.1. Transposition Domain Selection and Assessment

#### 4.2. Rainfall Catalog Creation

#### 4.3. IDF Curves Based on Empirical Plotting Position, Extreme-Value Distribution Fit, and Comparison to SST

#### 4.4. Point-to-Pixel Comparison of IDF Curves

#### 4.5. Catchment-Scale Comparison of IDF Curves

- ARFs are applied to the regional model (WPC) using Equation (5).
- Radar pixels within the AOI are, likewise, used to create an area-mean rainfall time series, and a GPD model is fitted, albeit with significantly more data points than the collection of gauges.
- Instead of only sampling one point for the SST framework, all points within the AOI are used to calculate a catchment average for the randomly generated rainfall event.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Island of Zealand: the study area for the present study. The shaded area represents the transposition domain used for the SST method, as presented in Section 3.1. All rain gauges used in the study are shown and labeled with the length of the observation period. The red area encapsulates the city of Copenhagen, which serves as the area of interest for the study. The division in the red-shaded area is a Voronoi polygon which is used to create area-weighted rainfall series.

**Figure 2.**Ranked maximum point (dashed lines) and transposition domain average (solid lines) rainfall depths for short-term rain-gauges (orange) and generated radar-data storm catalogs (black). Blue lines show the short-term gauge dataset adjusted so that the dates coincide with the radar dataset.

**Figure 3.**Examples of the spatial variability of accumulated rainfall depths for the duration-specific storms added to the final rainfall catalogs. The unit for the colorbars are in mm rainfall depth. Lowest, median, and highest refer to the rank of the domain mean rainfall depth.

**Figure 4.**GPD-based frequency analysis for long-term gauge (orange) and the corresponding radar pixels (black). The lines indicate the fitted GPD-model and shaded area depicts model uncertainty. Ranked values of synthetic rainfall years generated by the SST procedure are also shown (blue). Due to the high density of points, only every 50th SST point is shown on the figures to the left.

**Figure 5.**IDF-curves for all of the short-term rain gauges (orange) and their corresponding radar pixels (black). The solid line indicates median values and the shaded area depicts the total ensemble spread between gauges. The curves represent return levels of 2, 5, and 10 years.

**Figure 6.**Biases for duration of 10, 30, and 1440 min. Biases are calculated by Equation (4).

**Figure 7.**IDF-curves (10 min to 1440 min) for the regional model (green, WPC); long-term rain gauge, derived from GPD fit (orange); single radar pixel, derived from GPD fit (black, Radar), and the SST framework (blue, SST). The shaded area depicts 95% confidence interval, and total ensemble spread for SST values. Each curve represents a return level of: 2, 5, 10, and 100 years.

**Figure 8.**IDF curves (10 min to 1440 min) for the regional model with ARFs applied (green, WPC); collection of medium-term gauges, derived from GPD fit (orange, Gauge); radar pixels within the AOI, derived from GPD fit (black, Radar) and the SST framework for AOI (blue, SST). Shaded area depicts 95% confidence interval and total ensemble spread for SST values. Each curve is representing a return level of: 2, 5, 10 and 100 years.

**Figure 9.**Generated 5-year storms for a duration 1440 min, shown on the left. The storms was arbitrarily selected. On the right, time series for each individual pixel is shown (black) and the largest pxiel based on accumulated rain depth (dashed red line).

**Table 1.**Bias, root mean square error (RMSE) and mean absolute error (MAE) for all investigated rainfall durations.

Rainfall Duration [min] | 10 | 30 | 60 | 180 | 360 | 720 | 1440 |
---|---|---|---|---|---|---|---|

Bias [-] | 1.56 | 1.27 | 1.19 | 1.11 | 1.06 | 1.03 | 1.00 |

RMSE [mm] | 8.71 | 4.43 | 3.09 | 1.42 | 0.83 | 0.47 | 0.25 |

MAE [mm] | 4.49 | 2.28 | 1.46 | 0.69 | 0.41 | 0.23 | 0.12 |

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**MDPI and ACS Style**

Andersen, C.B.; Wright, D.B.; Thorndahl, S.
Sub-Hourly to Daily Rainfall Intensity-Duration-Frequency Estimation Using Stochastic Storm Transposition and Discontinuous Radar Data. *Water* **2022**, *14*, 4013.
https://doi.org/10.3390/w14244013

**AMA Style**

Andersen CB, Wright DB, Thorndahl S.
Sub-Hourly to Daily Rainfall Intensity-Duration-Frequency Estimation Using Stochastic Storm Transposition and Discontinuous Radar Data. *Water*. 2022; 14(24):4013.
https://doi.org/10.3390/w14244013

**Chicago/Turabian Style**

Andersen, Christoffer B., Daniel B. Wright, and Søren Thorndahl.
2022. "Sub-Hourly to Daily Rainfall Intensity-Duration-Frequency Estimation Using Stochastic Storm Transposition and Discontinuous Radar Data" *Water* 14, no. 24: 4013.
https://doi.org/10.3390/w14244013