# Estimation of Storm-Centred Areal Reduction Factors from Radar Rainfall for Design in Urban Hydrology

^{*}

## Abstract

**:**

^{2}, this leads to significant overestimation of the design rainfall intensities, and thus potentially oversizing of urban drainage systems. By extending methods from rural hydrology to urban hydrology, this paper proposes the introduction of areal reduction factors in urban drainage design focusing on temporal and spatial scales relevant for urban hydrological applications (1 min to 1 day and 0.1 to 100 km

^{2}). Storm-centred areal reduction factors are developed based on a 15-year radar rainfall dataset from Denmark. From the individual storms, a generic relationship of the areal reduction factor as a function of rainfall duration and area is derived. This relationship can be directly implemented in design with intensity–duration–frequency curves or design storms.

## 1. Introduction

^{2}[22,23,24]. In comparison with other studies, the focus at the small scales in space and time is novel.

## 2. Materials and Methods

#### 2.1. Data

^{2,}including the greater Copenhagen area and the island of Sealand in Denmark as well as parts of south-western Sweden (Figure 1). The spatial resolution of the cartesian rainfall product is 500 × 500 m

^{2}generated in a 1 km altitude pseudo-CAPPI layer. This product is selected due to consistency in data having a fixed elevation as a function of range and significantly fewer errors and clutter compared to lower altitude pseudo CAPPI products. The temporal resolution of the original dataset is 10 min, but the developed dataset has been regenerated by using a mixed forward and backward advection interpolation method by Nielsen et al. [30] to a 1 min resolution. Radar reflectivity is converted into rainfall intensities using a fixed Marshall–Palmer relationship [31] and bias-adjusted against with 67 rain stations/gauges using a daily mean field bias adjustment approach as described in Thorndahl et al. [30] and Smith and Krajewski [32]. As documented in Thorndahl et al. [29], the mean-field bias adjustment is applied within the 100 km range domain of the radar on a daily time scale. Regional variability of the bias is insignificant and can be neglected [29].

^{2}pixel. For this study, 534 individual rainy days with more than 1 mm of rainfall (in at least one of the rain stations/gauges) were selected. All days with significant noise (defect filters), missing images due to hardware or communication failures, poor bias adjustment (defined as a Nash–Sutcliffe Efficiency below 0, e.g., due to little gauge data, gauge data failures) were omitted. Consequently, the dataset consists of 534 discrete days and is thus discontinuous for the period (2002–2016). In traditional rainfall design statistics of rainfall, an incomplete dataset with periods of missing data is subject to uncertainty in the estimation of return periods. However, since this study disregards estimating return periods, the discontinuous data are not considered any further.

#### 2.2. Correction for Pixel Scale Error

^{2}as applied here, there is a need to acknowledge the scale error between rain gauge and radar pixel [25,33,34,35,36,37]. This representativeness error between rain gauge and radar can originate from several sources. It can be due to rainfall variability itself within a radar pixel meaning that the average rainfall over a 500 × 500 m

^{2}is not the same as recorded in a rain gauge with a surface area of less than 0.05 m

^{2}. It can also be due to artefacts measuring with radars, such as the difference between the atmosphere and ground, wind drift, timing errors. Rather than attempting to estimate the rainfall variability at subpixel scale, a data-driven approach for estimation of the mean error is suggested in this paper. This involves calculating a ratio between maximum intensities at different durations (aggregation levels) from rain gauge and radar, respectively. The estimation of the pixel scale error is thus represented through duration-dependent bias factor

_{G}is the rainfall intensity from rain stations/gauges (G) averaged over the duration d, and correspondingly i

_{R}is the rainfall intensity averaged over duration d which is calculated from daily mean-field bias adjusted radar (R) estimates based on Thorndahl et al. [29]. n is the total number of rain station/gauge radar pixel pairs within the radar range for a specific day t. T is the total number of selected rainy days. The max function implies that the maximum the daily maximum intensity over duration d is applied in rain gauge data and radar, respectively. The developed biases are presented in the result section.

#### 2.3. Method Development

_{R,max}), averaged over duration d, area A and the maximum rainfall intensity in a point (A→0). Since a storm-centred approach is applied, the maximum point rainfall, ${i}_{R,max}\left(d,A\to 0\right)$ is estimated in the point of the maximum rain intensity within the extent of the averaged area, A, (Figure 2). For each rainy day and selected discrete values of A and d, the maximum radar rainfall intensity within the domain i

_{R,max}is calculated. In this paper, the combination of rainy day, A and d defines a storm, s. The estimation of the ARF is conducted for a number of individual storms (s).

_{1}and c

_{2}are coefficients calibrated from record data. Based on the ARF as a function of area, λ is calibrated for each duration and storm using a non-linear least squares approximation. To limit the degrees of freedom, Equation (4) (i.e., Equation (5) with parameters c

_{1}= 0.5 and c

_{2}= 0.5) is initially applied to fit the ARF.

_{1}and c

_{2}(Equation (5)) are fitted by the non-linear least squares approximation method. This leads to a modification of the initial values of parameters from Equation (4). In this way, all four parameters are calibrated, and Equation (7) can be simplified to the following relationship:

_{1}, b

_{2}, and b

_{3}will be calibrated by a stepwise procedure of Equations (3), (4), (6–8). The calibration procedure is explained along with the data-processing and examples in the results section.

## 3. Application and Results

^{2}(one pixel) to 100 km

^{2}(20 × 20 pixels). This corresponds to 240 combinations of area and duration for each of the 534 storms leading to 6408 individual storm-centred areal reduction factor relationships.

- (1)
- Applying Equation (1), a correction of the pixel scale error is performed. Results are shown for selected durations in Figure 3 and Table 1. It is evident that the error between rain gauge intensities and radar intensities are significantly larger for short rainfall durations. This is a result of the daily mean-field bias adjustment and leads to a bias factor of 1 for the 1440 min durations (1 day). As shown in Figure 3, there is a considerable scatter between maximum rain gauge intensities and the corresponding radar intensities, which is also explained by the Nash–Sutcliffe Efficiency (NSE)-values in Table 1 and Figure 3. Furthermore, the scatter is larger for the shorter durations indicating high uncertainties. However, as the study aims for a mean pixel scale error, the dispersion of the pixel scale error is not considered any further.
- (2)
- (3)
- The correlation lengths, λ are fitted (Equation (6)) as a function of duration (Figure 6). From Figure 6, it is evident that there is a large variability from storm to storm, but that the mean fit well to the power function with r
^{2}of 0.98. It shows that the power-law function in Equation (6) can be further used to derive a relationship of the storm-centred ARF as a function of area and duration. In addition to the mean relationship, the uncertainty corresponding to mean plus/minus one standard deviation (assuming a Gaussian distribution) is investigated. This uncertainty will provide insight into the variability from storm to storm. - (4)
- Applying the obtained function of correlation length and duration, each storm is re-fitted by the relationship in Equation (7) to derive an ARF function. Examples of this fit are shown in Figure 4 and Figure 5 for durations of 60 and 360 min, respectively. Comparing with the mean ARF functions, the fitted relationships show a slight overestimation for the small areas and correspondingly an underestimation for large areas. For some durations, the opposite case occurs (not shown). This uncertainty is a trade-off of fitting a fixed parameter relationship to all durations.
- (5)

## 4. Discussion

#### 4.1. Comparison with Previous Studies

^{2}are examined in Figure 8. Few authors [11,16,17,25] have reported ARFs for sub-hourly durations and areas less than 100 km

^{2}. This comparison is not exhaustive, but it gives an indication of differences between ARF values using the area-fixed or storm-centred approach as well as differences using rain gauge and radar data.

^{2}, [15]) and for rainfall durations of 1 day or more. Calibrated relationships which cover these large rural scales might potentially be less precise on smaller scales relevant in urban hydrology. It is acknowledged that the ARFs derived in this paper are smaller than the ones reported in the majority of previous studies. Mean values and standard deviations are, however, comparable to the ones reported by Wright et al. [18], which implies that the storm-centred approach will provide larger areal reductions, thus smaller ARFs, compared to the fixed-area counterparts. From the comparison, it is not possible to determine whether ARFs obtained from rain gauge data is different from the ones based on radar data.

#### 4.2. Implementation in Urban Drainage Design

^{2}) are of importance in urban hydrology. An example relevant to urban scales is presented in the following, along with an evaluation of the impact of the implementation of an ARF.

^{2}with a time of concentration corresponding to 60 min to the outlet, the developed mean ARF is 0.78 with 0.58 and 0.87 for the lower and upper confidence limits (+/- the standard deviation), respectively. This corresponds to a reduction of the design rainfall on average of 25% ranging from 13% to 46%. Using the mean ARF, the design rainfall for Danish conditions with a return period of 10 years [39] can be adjusted as presented in Figure 9a for IDF curves and Figure 9b for design rainfall of the Chicago design storm (CDS) type [40]. Compared to the current design practice where no areal reduction is implemented; this will reduce the design intensities and thus lead to smaller designs.

## 5. Conclusions

^{2}with short concentration times (thus short rainfall design durations), to prevent unnecessary oversizing of designs.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Domain of radar (EKXS) rainfall dataset and rainfall stations/gauges (blue dots) used of bias adjustment of radar data.

**Figure 2.**Principle of averaging radar images spatially over an area (A) and temporally over a duration (d).

**Figure 3.**Example of estimation of bias between radar and rain gauge intensities at rainfall durations of 10 (

**a**), 60 (

**b**) and 1440 (

**c**) min for the 500 × 500 m

^{2}resolution.

**Figure 4.**Areal reduction factors (ARF) for durations of 60 min. Dashed lines are the confidence limits (mean +/− std. dev.) of the fitted ARF.

**Figure 5.**ARF for durations of 360 min. Dashed lines are the confidence limits (mean +/− std. dev.) of the fitted ARF.

**Figure 6.**Fitted power-function (Equation (6)) between correlation length and rainfall duration in solid black with confidence limits (mean +/− std. dev.) in dashed lines. Grey dots indicate values for individual storms, and black circles indicate mean values for each duration.

**Figure 8.**Derived ARF relationship for 60 min durations compared to previous studies for areas of 10, 50 and 100 km

^{2}estimated using fixed-area (FA) and storm-centred (SC) approaches applying both rain gauge (RG) and radar data (RA).

**Figure 9.**Example of intensity–duration–frequency curves (

**a**) and Chicago design storms (

**b**) with a return period of 10 years [2] adjusted with the developed mean areal reduction factors.

**Table 1.**Bias between radar and rain gauge intensities at different rainfall durations for the pixel size of 500 × 500 m

^{2}.

Duration, d (min) | 1 | 10 | 30 | 60 | 180 | 360 | 720 | 1440 |

Bias, B (-) | 1.63 | 1.36 | 1.21 | 1.15 | 1.07 | 1.04 | 1.03 | 1.00 |

Nash–Sutcliffe Efficiency, NSE (-) | 0.21 | 0.40 | 0.52 | 0.60 | 0.63 | 0.62 | 0.62 | 0.61 |

Root mean square error, RMSE (mm/h) | 22.47 | 9.82 | 4.61 | 2.65 | 1.12 | 0.66 | 0.38 | 0.21 |

b_{1} | b_{2} | b_{3} | |
---|---|---|---|

mean | 0.31 | 0.38 | 0.26 |

mean – 1 × std. dev. | 0.21 | 0.45 | 0.36 |

mean + 1 × std. dev. | 0.47 | 0.37 | 0.17 |

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**MDPI and ACS Style**

Thorndahl, S.; Nielsen, J.E.; Rasmussen, M.R. Estimation of Storm-Centred Areal Reduction Factors from Radar Rainfall for Design in Urban Hydrology. *Water* **2019**, *11*, 1120.
https://doi.org/10.3390/w11061120

**AMA Style**

Thorndahl S, Nielsen JE, Rasmussen MR. Estimation of Storm-Centred Areal Reduction Factors from Radar Rainfall for Design in Urban Hydrology. *Water*. 2019; 11(6):1120.
https://doi.org/10.3390/w11061120

**Chicago/Turabian Style**

Thorndahl, Søren, Jesper Ellerbæk Nielsen, and Michael R. Rasmussen. 2019. "Estimation of Storm-Centred Areal Reduction Factors from Radar Rainfall for Design in Urban Hydrology" *Water* 11, no. 6: 1120.
https://doi.org/10.3390/w11061120