# An Area-Orientated Analysis of the Temporal Variation of Extreme Daily Rainfall in Great Britain and Australia

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

- Generate ROIs with varying locations, sizes and shapes and extract the annual maximum daily rainfall (AMDR) time series with the assistance of high-performance computing (HPC).
- Fit the time series obtained at every ROI with stationary and nonstationary GEV models with different parameter estimation methods.
- Evaluate the performance of all models and analyse the changes of time-varying parameters with regard to the geographical locations, sizes, and shapes as well as the level of extremity.

#### 2.1. Data, ROIs and Extreme Rainfall in Two Countries

^{2}for GB case and 125–9900 km

^{2}for AU case) and shapes (controlled by the spatial index $sp$ varying from elongated to rounded with east–west and north–south orientations). For each ROI, the areal daily rainfall is calculated by taking the arithmetic average before the ROI-orientated AMDR is generated. This procedure is carried out using the HPC-Wales resources due to the huge amount of data.

#### 2.2. Stationary and Nonstationary Generalised Extreme Value Models and Return Period

## 3. Results

#### 3.1. Selection of Stationary and Nonstationary Models and Spatial Nonstationary Patterns

^{2}and a relatively rounded shape is used. Geographically, those ROIs in GB that prefer nonstationary models are located along or near the coastal regions, especially in eastern and northern GB and the Scotland Highland. In AU, nonstationary models dominate the inland area and the majority of the south-western coastline while the north coastline of AU and the majority inland of the Northern Territory favour stationary model.

#### 3.2. Spatial Variation in Nonstationary Patterns over ROI Size

#### 3.3. Spatial Variation in Nonstationary Patterns over ROI Shape

#### 3.4. Implication of Return Period

## 4. Conclusions

- (1)
- In general, the majority of the ROIs in both countries favour the nonstationary GEV model (NS-GEV) and most of them prefer the condition that only $\mu $ assumed to be linearly changing with time; most NS-GEV applications show the ML method performs better than the B-MCMC method (60% and 90% in GB and AU). AMDR of over 80% ROIs in both countries follows Fréchet distribution.
- (2)
- Geographic location is the most significant factor that affects not only the average status of the baseline climate (with respect to ${\mu}_{0}$ and ${\sigma}_{0}$) but also the time-varying changes due to climate change (with respect to ${\mu}_{1}$ and ${\sigma}_{1}$). During the last century in GB, the changes in the level of the most frequent AMDR (with respect to $\mu $) are in the range of $\pm 10\%$ and the majority of areas show a non-decreasing trend. However, in AU, the south-middle zone and the eastern coasts are dominated by an increasing $\mu $ up to the rate of +20% while the north coast of Northern Territory and west-south coast of Western Australia are controlled by a decreasing $\mu $ with the rate of $-5\%$. The majority of regions of GB and AU are observed to a still $\sigma $ while some specific regions with a decreasing $\sigma $ scattering near the coasts of England indicate a decreasing occurrence probability of extremes.
- (3)
- Region size is shown to be a secondary factor. Generally, the two countries show a decreased average status of climate with an increase in size because of the statistical average. However, near the coastal regions of GB and the boundary of the south-middle dry zone of AU, some ROIs have an increasing status. Although the effect of region size on time-varying changes is insignificant, the climate change impact is not always decreased with the increase in region size but is influenced by geographical locations.
- (4)
- Region shape does not significantly affect either the average climate status or time-varying changes; however, a symmetric pattern of average climate status is found for regions with reciprocal spatial indexes and the average climate status in ROIs with a relatively rounded shape is usually higher than the elongated ones
- (5)
- The stationary GEV models underestimate the risk in several specific regions such as the coastal regions in both countries where the nonstationary model is preferred. It may inspire a re-consideration of the current design storm determination procedure.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**ROIs and their locations spatially distributed in GB (

**a**) and AU (

**b**) for this study: each subfigure can be divided into three panels where the left one demonstrates an ROI with the same size of 500 km

^{2}and a relatively rounded shape for the analysis in Section 3.1; the middle demonstrates a group of 10 ROIs which have a fixed shape but an incrementally increased size by defining an 20% increase in the main axis (marked as red dash line) in each iteration for the analysis in Section 3.2; the right panel depicts a group of 7 ROIs which have fixed size but varying shape described by spatial index $sp$. “$\times $” marks the central location of these single or group of ROI(s) which are distributed uniformly over two countries. More details for generating ROIs can be checked in [20].

**Figure 4.**Spatial distribution of ROIs with the size of 500 km

^{2}and relative rounded shape in terms of (1) the best-selected model type in GB (

**a**) and AU (

**e**); (2) the best-fitted GEV type in GB (

**b**) and AU (

**d**); (3) the changes of location (

**c**,

**g**) and scale parameters (

**f**,

**h**) in percentage within the record periods (113 years for GB and 129 years for AU) and please noted that white colour (“w” specified in the colour bar) indicates the ROI with no change in the parameters.

**Figure 5.**Spatial distribution of ROI groups whose parameters (${\mu}_{0},{\mu}_{1},{\sigma}_{0},{\sigma}_{1}$) change with the increase in ROI size in GB (

**a**–

**d**) and AU (

**e**–

**h**). Please noted that the white colour indicates the ROI with an insignificant change in the parameters. Noted that “w” shown in the colour bar indicates the colour white which means the changes of the parameter is zero.

**Figure 6.**Both baseline and time-varying parameters (${\mu}_{0},{\mu}_{1},{\sigma}_{0},{\sigma}_{1}$) change over the ROI shape indicated by the shape index of sp in GB (

**a**,

**b**) and AU (

**c**,

**d**). The vertical axis indicates the location of the groups of ROIs (e.g., in Figure 6a, 10 groups (those at 240 km, 280 km, 320 km, 360 km, 400 km, 440 km, 480 km, 520 km, 560 km and 600 km) are presented from west to east while in each group, ROIs are presented from south to north (e.g., 5 ROIs presented between 240 km and 280 km are all geographically located at 240 km Easting from 580 km Northing to 900 km Northing)) and the colour bar shows the values of parameters. Due to the limited length of figures, the completed AU case with 2646 ROIs is shown in Supplementary Figure S1.

**Figure 7.**Nonstationary return periods corresponding to the return levels estimated by the stationary model and the spatial distribution referring to 1-in-5 years, 1-in-25 years and 1-in-50 years in GB (

**a**) and AU (

**b**). An overall comparison between the nonstationary and stationary return levels corresponding to the same return year is presented as a boxplot in (

**c**) where the upper panel shows the GB case and the lower shows the AU case. Noted that “w” in the colour bar indicates the colour white which means there is no difference between stationary and nonstationary return periods.

Description | Parameters | Estimation Method(s) |
---|---|---|

Stationary model: $F\left(x;{\sigma}_{0},{\mu}_{0},\xi \right)$ | ${\sigma}_{0},{\mu}_{0},\xi $ are constant | ML ^{1} |

Nonstationary model 1: ${F}_{t}\left({x}_{t};{\sigma}_{0},{\mu}_{t},\xi \right)$ | $\begin{array}{c}{\mu}_{t}={\mu}_{0}+{\mu}_{1}\times t\\ {\sigma}_{0},\xi \mathrm{are}\mathrm{constant}\end{array}$ | ML and B-MCMC ^{2} |

Nonstationary model 2: ${F}_{t}\left({x}_{t};{\sigma}_{t},{\mu}_{t},\xi \right)$ | $\begin{array}{c}{\sigma}_{t}={\sigma}_{0}+{\sigma}_{1}\times t\\ {\mu}_{t}={\mu}_{0}+{\mu}_{1}\times t\\ \xi \mathrm{is}\mathrm{constant}\end{array}$ | ML and B-MCMC |

Nonstationary model 3: ${F}_{t}\left({x}_{t};{\sigma}_{t},{\mu}_{t},\xi \right)$ | $\begin{array}{c}{\sigma}_{t}=\mathrm{exp}({\sigma}_{0}+{\sigma}_{1}\times t)\\ {\mu}_{t}={\mu}_{0}+{\mu}_{1}\times t\\ \xi \mathrm{is}\mathrm{constant}\end{array}$ | ML and B-MCMC |

^{1}short for “Maximum Likelihood” method.

^{2}short for “Bayesian Markov-Chain Monte-Carlo” method.

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**MDPI and ACS Style**

Wang, H.; Xuan, Y. An Area-Orientated Analysis of the Temporal Variation of Extreme Daily Rainfall in Great Britain and Australia. *Water* **2023**, *15*, 128.
https://doi.org/10.3390/w15010128

**AMA Style**

Wang H, Xuan Y. An Area-Orientated Analysis of the Temporal Variation of Extreme Daily Rainfall in Great Britain and Australia. *Water*. 2023; 15(1):128.
https://doi.org/10.3390/w15010128

**Chicago/Turabian Style**

Wang, Han, and Yunqing Xuan. 2023. "An Area-Orientated Analysis of the Temporal Variation of Extreme Daily Rainfall in Great Britain and Australia" *Water* 15, no. 1: 128.
https://doi.org/10.3390/w15010128