# Value of Spatially Distributed Rainfall Design Events—Creating Basin-Scale Stochastic Design Storm Ensembles

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}(Figure 1). The area is relatively flat, with the water level difference along the Kokemäenjoki river being 57.5 m. The study area was selected for the comprehensive rainfall radar coverage, as well as for the large basin size enabling assessments across varying spatial scales. It comprises 494 subbasins (3rd-delineation-level), with sizes varying from 2.7 to 535.9 km

^{2}and with a median size of 40.4 km

^{2}[26,27]. The numbers of higher-level subbasins are 81 and 9 for the 2nd- and 1st-delineation-levels, respectively. The median subbasin size is 261.3 km

^{2}for the 2nd-level and 3135.8 km

^{2}for the 1st-level.

#### 2.2. Radar Data and Rainfall Event Selection

^{2}spatial resolution. The product has been described in more detail by Schleiss et al. [28].

_{10}(aR

^{b}),

#### 2.3. Rainfall Simulation Model

_{0}(the correlation length) and H (the temporal scaling exponent), which control the autocorrelation function of the μ time series. Similarly, the broken line model was also used to generate time series for the magnitude and direction of rain field advection. The spatial structure for each rain field was created in pySTEPS by generating a field of white noise, transforming it into the Fourier domain, and filtering it with a power-law filter. Gaussian bandpass filtering from pySTEPS was applied to decompose the spatially correlated rain field into cascade levels (L

_{k}) representing k different spatial scales in rainfall structures [24]. Assuming that the lifespan of a rainfall structure is related to its size [36], the temporal evolution of rainfall structures was described as a second-order autoregressive process in pySTEPS with the assumption that temporal autocorrelations follow the power law relationship between the temporal correlation length and the spatial scale k. The fields were advected by moving the pixels according to the generated advection time series. The advection routine in pySTEPS was modified to allow the fields to “wrap around” the simulation domain to ensure spatially continuous fields [37]. To minimize the impact of wrap-around effects near the edges of the field domain, final fields of size N×N were clipped from the center of the 2N×2N simulation domain. Finally, the statistical characteristics of the fields, i.e., μ, the field standard deviation σ, and the wetted area ratio WAR describing the ratio of rainy cells to total cells in the field, were adjusted as in the work of Niemi et al. [25].

#### 2.4. Parameter Estimation

_{0}for μ time series, denoting the memory of the process, was not estimated as the decorrelation time, but rather determined by optimizing the fit between the autocorrelation function for the observed μ and the mean of autocorrelation functions for 100 simulated stochastic μ time series. For the advection time series, estimating a

_{0}as the decorrelation time as in the work of Seed et al. [35] was deemed to give satisfactory results. As in the work of Seed et al. [40] and Niemi et al. [25], the observed σ and WAR were found to be closely related to the observed μ, and they were therefore modeled as quadratic functions of μ (Table S2).

_{1}and β

_{2}) and one breaking point, or scale break, separating β

_{1}and β

_{2}were estimated for each field. The scale break was considered to have a constant value of 18 km. As in the work of Seed et al. [40], the time series of β

_{1}and β

_{2}were related to μ using quadratic functions (Table S3).

_{k}using bandpass filtering in pySTEPS [24]. The mean lag-1 and mean lag-2 autocorrelation coefficients were estimated for each L

_{k}over the event duration and related to k as in the work of Seed et al. [40] (Table S4).

#### 2.5. Ensemble Variation of Simulated Rainfall Events

^{2}), the entire Kokemäenjoki river basin (ca. 27,000 km

^{2}), and three selected subbasins from different delineation levels. A 3rd-level subbasin (66.5 km

^{2}) was first picked, and then the corresponding 2nd- and 1st-level subbasins (564.6 km

^{2}and 2189.9 km

^{2}) were selected with the condition that they contain the selected 3rd-level subbasin. Locations of the studied subbasins are presented in Figure 1. Cumulative areal rainfall was calculated across the five spatial scales for each ensemble member and for the two rainfall events.

#### 2.6. Comparison of Rainfall from Areal and Gauge Estimates

_{tot}) calculated using two different methods and the selected spatial scales. In the first method, referred to as “areal estimate”, R

_{tot}was calculated using rainfall values from all pixels included in the study region of a given scale. The second method exemplified the procedure for the estimation of subbasin rainfall in operational runoff modeling [41,42]. In this case, R

_{tot}in the 3rd-delineation-level subbasin scale was calculated using the inverse-distance-weighted average of those pixel values that accommodate the three closest point measurement gauge locations from the subbasin centroid. This method is referred to as “gauge estimate”. R

_{tot}for the 2nd- and the 1st-delineation-level subbasins were computed as area-weighted averages from the 3rd-level results. Locations of gauge stations measuring hourly rainfall [43] are presented in Figure 1.

## 3. Results

#### 3.1. Model Performance

^{2}field (Figure 3c and Figure 4c). For both studied events, the generated ensemble average temporal autocorrelation function closely followed the observed autocorrelation for lags up to ca. 150 min. The model was capable of reproducing the triangular shape of the event field mean intensity, and the peak intensities of the observed events were on par with the peaks of the simulated ensemble members (Figure 3c and Figure 4c). The negative autocorrelations for longer lags (Figure 3b and Figure 4b) indicate the typical pattern in storm events as increase in rainfall in the rising limb of the hyetograph is matched with a decreasing rainfall intensity in the residing limb when the lag time is long enough.

#### 3.2. Ensemble Variation of Cumulative Areal Rainfall across Spatial Scales

#### 3.3. Variability in Total Event Rainfalls between Areal and Gauge Estimates

_{tot}) calculated either as areal estimates over the area of interest using all pixel values or as gauge estimates calculated only using values from pixels residing at the three closest gauge locations to the area of interest (Section 2.6) were studied. From the hydrological point of view, rarely occurring high-intensity rainfalls are of particular interest. Hence, the ensemble maximum R

_{tot}was calculated as the largest R

_{tot}of all ensemble members for each subbasin, and the impact of the sampling method was evaluated (Figure 8 and Figure 9, Table 3).

_{tot}can result in clear deviations as compared to utilizing rainfall values from all pixels (areal estimate). This is explained by the relatively sparse gauge network that can completely miss small but intensive rainfall structures in small subbasins, or alternatively can include rainfall from a distant gauge at times of no rain within the subbasin. Similar to the results in cumulative areal rainfall estimates (Section 4.2), event 1 with a lower WAR value produces more pronounced discrepancies than event 2, when changing the sampling method from the areal estimate to the gauge estimate. The deviations in R

_{tot}between the larger basins are smaller due to spatial averaging.

_{tot}in different spatial scales is almost always lower for gauge estimates than for areal estimates. The difference increases towards smaller subbasins.

_{tot}in terms of CV remains relatively stable between the subbasin scales (Table 3). However, std increases significantly when moving into smaller spatial scales from the 1st-level subbasins. In event 1 (event 2), the std of ensemble maximum R

_{tot}calculated from areal estimates doubles from 9.19 (3.86) to 20.90 (7.11) mm between the two lowest spatial scales (the 1st- and 2nd-level subbasins). The increase in gauge estimates is similar but ca. 60% lower, from 10.93 (5.66) to 15.72 (7.01) mm. The increase in the std between the two highest spatial scales (the 2nd- and 3rd-level subbasins) is much lower for both calculation methods.

_{tot}values strongly increase when moving from lower (1st) to higher (3rd) spatial scales (Table 3). The increase is more pronounced for the areal estimates than for the gauge estimates. In the former case, the increase is from 61.98 (33.47) to 157.09 (88.94) mm in event 1 (event 2), and in the latter case, it is from 63.31 (38.24) to 119.46 (58.55) mm. A similar pattern can be seen in the areal mean of ensemble maximum R

_{tot}, which increases from 48.24 (27.10) to 72.97 (35.15) mm in event 1 (event 2) using the areal estimate and from 47.12 (25.24) to 55.46 (28.11) mm in event 1 (event 2) using the gauge estimate. The mean increases more when using the areal estimate than when using the gauge estimate. Patterns in median values are similar, but increases are not as large.

## 4. Discussion

#### 4.1. Benefits of Spatial Rainfall Data and Ensemble Simulations

^{2}.

#### 4.2. Skill of the Model

#### 4.3. Limitations of the Model

^{2}were clipped from the center of the 512 × 512 km

^{2}simulation domain. The size of the simulation domain was considered to be sufficient for countering the wrap-around effect. However, there were some rare cases where arbitrary rain appeared and accumulated on the edges of the study area. These rare cases can be eliminated by increasing the size of the simulation domain, even though this would increase the simulation time.

## 5. Conclusions

## Supplementary Materials

^{2}radar field for (

**a**) observed rain event and (

**b**–

**i**) eight randomly selected ensemble members; Figure S2: Event 2 total event rainfall (mm) of the entire 256 × 256 km

^{2}radar field for (

**a**) observed rain event and (

**b**–

**i**) eight randomly selected ensemble members; Table S1: Parameters for modeling mean areal rainfall (μ), advection magnitude (V

_{mag}), and advection direction (V

_{dir}) time series [35]; Table S2: Parameters for modeling field standard deviation (σ) [40] and wetted area ratio (WAR) [25] as quadratic functions of mean areal rainfall (μ); Table S3: Parameters for modeling spectral slopes (β

_{1}and β

_{2}) for power-law filter as quadratic functions of mean areal rainfall (μ) [40]; Table S4: Parameters for modeling mean lag-1 and mean lag-2 Lagrangian temporal autocorrelations (ρ

_{1,k}and ρ

_{2,k}) [40]. References [25,35,40] are cited in the Supplementary Materials.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Location of the Kokemäenjoki river basin (light blue area) and the radar network (green circles with a 120 km radius); (

**b**) 3rd-delineation-level subbasins within the study area (light blue), as well as selected 1st (yellow)-, 2nd (orange)-, and 3rd (red)-level subbasins. Rainfall gauge network (dark blue circles) is also shown.

**Figure 3.**Model performance for event 1. (

**a**) Ensemble average radially averaged 1D spatial power spectra; (

**b**) temporal autocorrelation functions of the mean areal rainfall; (

**c**) mean areal rainfall broken line time series; (

**d**) advection velocities to southern and eastern directions with marginal plots presenting ensemble distribution. Observed event is illustrated with blue, simulated ensemble members with gray, and ensemble average with red color.

**Figure 4.**Model performance for event 2. (

**a**) Ensemble average radially averaged 1D spatial power spectra; (

**b**) temporal autocorrelation functions of the mean areal rainfall; (

**c**) mean areal rainfall broken line time series; (

**d**) advection velocities to southern and eastern directions with marginal plots presenting ensemble distribution. Observed event is illustrated with blue, simulated ensemble members with gray, and ensemble average with red color.

**Figure 5.**Event 1 cumulative areal rainfall (mm) time series for (

**a**) the entire radar field, (

**b**) the entire Kokemäenjoki river basin, (

**c**) the selected 1st-delineation-level subbasin, (

**d**) the selected 2nd-delineation-level subbasin, and (

**e**) the selected 3rd-delineation-level subbasin. Observed event is illustrated with blue (

**a**,

**b**), simulated ensemble members with gray

**(a**–

**e**), realization with ensemble mean cumulative areal rainfall with red (

**c**–

**e**), realization with ensemble maximum cumulative areal rainfall with orange (

**c**–

**e**), and realization with ensemble median cumulative areal rainfall with yellow color (

**c**–

**e**).

**Figure 6.**Event 2 cumulative areal rainfall (mm) time series for (

**a**) the entire radar field, (

**b**) the entire Kokemäenjoki river basin, (

**c**) the selected 1st-delineation-level subbasin; (

**d**) the selected 2nd-delineation-level subbasin; (

**e**) the selected 3rd-delineation-level subbasin. Observed event is illustrated with blue (

**a**,

**b**), simulated ensemble members with gray (

**a**–

**e**), realization with ensemble mean cumulative areal rainfall with red (

**c**–

**e**), realization with ensemble maximum cumulative areal rainfall with orange (

**c**–

**e**), and realization with ensemble median cumulative areal rainfall with yellow color (

**c**–

**e**).

**Figure 7.**Cumulative areal rainfall (mm) time series of each 3rd-delineation-level subbasin for (

**a**) event 1 and (

**b**) event 2. 3rd-level subbasins are illustrated with gray, subbasin mean cumulative areal rainfall with red, subbasin with maximum cumulative areal rainfall with orange, and subbasin with median cumulative areal rainfall with yellow color.

**Figure 8.**Event 1 ensemble maximum of total event rainfall (mm) utilizing areal estimates for (

**a**) 1st-level subbasins, (

**b**) 2nd-level subbasins, and (

**c**) 3rd-level subbasin. Ensemble maximum of total event rainfall (mm) utilizing gauge estimates for (

**d**) 1st-level subbasins, (

**e**) 2nd-level subbasins, and (

**f**) 3rd-level subbasin. The difference (mm) between two calculation methods (areal–gauge) for (

**g**) 1st-level subbasins, (

**h**) 2nd-level subbasins, and (

**i**) 3rd-level subbasin. Gauge locations are denoted with black circles.

**Figure 9.**Event 2 ensemble maximum of total event rainfall (mm) utilizing areal estimates for (

**a**) 1st-level subbasins, (

**b**) 2nd-level subbasins, and (

**c**) 3rd-level subbasin. Ensemble maximum of total event rainfall (mm) utilizing gauge estimates for (

**d**) 1st-level subbasins, (

**e**) 2nd-level subbasins, and (

**f**) 3rd-level subbasin. The difference (mm) between two calculation methods (areal–gauge) for (

**g**) 1st-level subbasins, (

**h**) 2nd-level subbasins, and (

**i**) 3rd-level subbasin. Gauge locations are denoted with black circles.

Event | Start and End Time | Duration (h:mm) | Average (Peak) μ (mm/h) | Average (Maximum) WAR (%) | Cumulative Mean Areal Rainfall (mm) |
---|---|---|---|---|---|

Event 1 | 27 June 2013, 08:10– 27 June 2013, 19:55 | 11:45 | 1.11 (2.17) | 30 (49) | 12.28 |

Event 2 | 28 October 2013, 18:10– 29 October 2013, 03:45 | 9:35 | 0.68 (1.73) | 55 (91) | 6.16 |

Spatial Scale | Max (mm) | Min (mm) | Mean (mm) | Median (mm) | Std (mm) | CV (-) |
---|---|---|---|---|---|---|

Radar field | 13.05 (8.83) | 7.19 (6.00) | 9.22 (7.36) | 9.06 (7.34) | 0.99 (0.48) | 0.11 (0.07) |

Kokemäenjoki river basin | 20.53 (12.67) | 1.19 (2.73) | 7.72 (6.62) | 7.33 (6.42) | 3.58 (1.73) | 0.46 (0.26) |

1st-level | 62.00 (29.20) | 0.02 (0.68) | 9.50 (6.86) | 6.02 (5.50) | 10.20 (5.24) | 1.07 (0.76) |

2nd-level | 77.09 (42.76) | 0.01 (0.07) | 9.65 (6.90) | 5.03 (5.05) | 12.39 (6.12) | 1.28 (0.89) |

3rd-level | 87.53 (54.64) | 0.00 (0.04) | 10.01 (7.14) | 4.39 (5.04) | 13.39 (6.67) | 1.34 (0.93) |

**Table 3.**Statistics of ensemble maximum of total event rainfall for event 1 (event 2) utilizing areal and gauge estimates at different spatial scales.

Spatial Scale | Statistic | Areal Estimate | Gauge Estimate | Difference |
---|---|---|---|---|

1st-level | Max (mm) | 61.98 (33.47) | 63.31 (38.24) | −1.33 (−4.77) |

1st-level | Mean (mm) | 48.24 (27.10) | 47.12 (25.24) | 1.12 (1.86) |

1st-level | Median (mm) | 47.35 (26.74) | 47.74 (23.95) | −0.39 (2.80) |

1st-level | Std (mm) | 9.19 (3.86) | 10.93 (5.66) | −1.74 (−1.80) |

1st-level | CV (-) | 0.19 (0.14) | 0.23 (0.22) | −0.04 (−0.08) |

2nd-level | Max (mm) | 137.87 (62.89) | 100.76 (50.10) | 37.11 (12.79) |

2nd-level | Mean (mm) | 68.81 (32.99) | 54.88 (27.61) | 13.93 (5.38) |

2nd-level | Median (mm) | 62.73 (31.51) | 54.48 (26.61) | 8.25 (4.90) |

2nd-level | Std (mm) | 20.90 (7.11) | 15.72 (7.01) | 5.18 (0.10) |

2nd-level | CV (-) | 0.30 (0.22) | 0.29 (0.25) | 0.02 (−0.04) |

3rd-level | Max (mm) | 157.09 (88.94) | 119.46 (58.55) | 37.64 (30.39) |

3rd-level | Mean (mm) | 72.97 (35.15) | 55.46 (28.11) | 17.51 (7.05) |

3rd-level | Median (mm) | 68.23 (33.79) | 55.12 (27.07) | 13.11 (6.72) |

3rd-level | Std (mm) | 21.05 (7.75) | 16.70 (7.47) | 4.35 (0.28) |

3rd-level | CV (-) | 0.29 (0.22) | 0.30 (0.27) | −0.01 (−0.05) |

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## Share and Cite

**MDPI and ACS Style**

Lindgren, V.; Niemi, T.; Koivusalo, H.; Kokkonen, T.
Value of Spatially Distributed Rainfall Design Events—Creating Basin-Scale Stochastic Design Storm Ensembles. *Water* **2023**, *15*, 3066.
https://doi.org/10.3390/w15173066

**AMA Style**

Lindgren V, Niemi T, Koivusalo H, Kokkonen T.
Value of Spatially Distributed Rainfall Design Events—Creating Basin-Scale Stochastic Design Storm Ensembles. *Water*. 2023; 15(17):3066.
https://doi.org/10.3390/w15173066

**Chicago/Turabian Style**

Lindgren, Ville, Tero Niemi, Harri Koivusalo, and Teemu Kokkonen.
2023. "Value of Spatially Distributed Rainfall Design Events—Creating Basin-Scale Stochastic Design Storm Ensembles" *Water* 15, no. 17: 3066.
https://doi.org/10.3390/w15173066