1 Introduction

Intergovernmental Panel on Climate Change (IPCC) estimated that the mean global temperature would exceed the higher limit of increase in temperature adopted in the Paris agreement by the 2040s (IPCC 2018). It was also reported in 2020 that the temperature in Australia has increased by 1.44 ± 0.24 °C since 1910 (CSIRO and Australian Government (Bureau of Meteorology) 2020). Generally, changes in maximum and minimum temperatures impact the environment more than mean temperature (Sein et al. 2018) and adversely influence the events related to extreme temperature (Chowdary et al. 2014). Extreme temperature triggers more natural hazards, i.e. droughts, bushfires, heatwaves, cyclones, and negatively affects human health, agriculture, ecosystems and infrastructure (Omer et al. 2020; Suman and Maity 2020). Therefore, predicting extreme temperatures for a particular duration and return period is important in many aspects of our lives. Some are relevant to health, energy and agriculture, urban planning, and ecology management. In this context, the design temperatures for the selected durations and return periods can be estimated following the same principle of rainfall Intensity–Duration–Frequency (IDF) curves, which have been utilised for many years for the design and management of hydrological infrastructure as well as flood risk management (Yan et al. 2019). Extreme value theory has been accepted globally to construct IDF curves by fitting theoretical probability distribution functions to annual maximum rainfall time series (Cheng and Aghakouchak 2014; Yan et al. 2020).

Most of the works found in the literature on IDF curves are based on the temporal stationarity concept (Singh and Zhang 2007; Jakob 2013), implying that the occurrence probability of precipitation events will not change significantly over time. However, in reality, the frequency, magnitude and duration of hydroclimatic extremes, i.e. extreme rainfall (Galiatsatou and Iliadis 2022), floods (Berghuijs et al. 2019), droughts (Spinoni et al. 2017), and heatwaves (Lorenz et al. 2019) fluctuates beyond the stationary envelope of variability. Furthermore, it is also observed that the IDF curve considering stationarity, often underestimates the return level of rainfall (Sugahara et al. 2009; Cheng and Aghakouchak 2014). Therefore, to increase the reliability of IDF curves, it is suggested to incorporate the non-stationarity of the distribution parameters in hydrological models (Sarhadi and Soulis 2017), especially at their extreme levels (Ganguli and Coulibaly 2017).

In the construction of IDF curves, covariates are generally introduced to apprehend the non-stationarities in time series. Any drivers that are correlated to the selected events can be considered as a candidate for such covariates (Katz et al. 2002; Hundecha et al. 2008), which can be low-frequency climatological signals (Coles 2001) as well as time (Sugahara et al. 2009; Yilmaz and Perera 2014). In this approach, the parameters of the Generalised Extreme Value (GEV) distribution are expressed as a linear or non-linear function of covariates (Kwon and Lall 2016; Sarhadi and Soulis 2017). This approach has been effectively employed in extreme rainfall events around the world (Ouarda et al. 2019), including Australia (Yilmaz and Perera 2014).

Although there have been extensive works carried out on the development of IDF curves all over the globe, very limited works are found in the literature on the construction of Temperature-Duration-Frequency (TDF) curves, which relate temperature intensities corresponding to varying durations and return periods (Wang et al. 2013; Ouarda and Charron 2018). Moreover, no TDF curves have been constructed for Australia till date.

In this study, time series data of the daily maximum temperature of Australia at selected stations are analysed, and the best climate-informed covariates are selected from the most influential covariates for the annual maximum temperature (AMT) to characterise the physical process related to the dynamics of the extreme temperature. For this purpose, three climate drivers (CDs), namely—El Niño Southern Oscillation (ENSO), Southern Annular Mode (SAM) and Indian Ocean Dipole (IOD), are selected along with the Time as potential covariates. After that, stationary TDF curves are constructed following the frequency analysis method, already adopted in the construction of IDF curves, for different durations and return periods. The present work also extends these stationary TDF curves to non-stationary conditions according to the framework outlined by Ouarda and Charron (2018). Then, based on the Akaike Information Criterion (AIC) value, the best model for each selected station is identified, and the TDF curves are constructed under stationary and non-stationary conditions. Finally, the influence of non-stationarity on the TDF curves is assessed.

2 Study area and data

The whole Australia is considered as the study area except Tasmania (Fig. 1). Australia experiences a variety of climates across the country because of a wide range of geographical extent consisting of tropical-influenced climate in the northern and north-eastern zones, Mediterranean-like climate in the southern coastal zones and desert climate in most of the central interior zones (Turney et al. 2007).

Fig. 1
figure 1

Study area and selected stations (stations marked with black circles are described in detail in this paper)

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2.1 Temperature time series

All the recorded daily maximum temperatures across Australian weather stations were examined for inconsistency, outliers and gaps. The stations, which had less than 5% missing data over the last five decades covering the time period from 1969 to 2021, were selected for analysis. According to the above mentioned criterion, 82 weather stations in Australia (out of 1415 weather stations) were initially selected as candidate stations. The gaps in the daily maximum temperature time series in the initially selected stations were spatially interpolated with the same time series data from surrounding weather stations located within a 30 km radius. The locations of the initially selected temperature stations are shown in Fig. 1. For the subsequent analysis, a total of 12 stations were selected, two from each of the six Australian states (New South Wales, Queensland, The Northern Territory, Western Australia, South Australia and Victoria) as shown in Fig. 1.

The annual maximum daily temperatures were calculated for each selected weather station from the recorded daily maximum temperature data. Particular attention was given to the summer season of Australia, which is considered to be December to February, and the extended summer is considered to be from November to March. Therefore, the annual maximum daily temperature calculated from each calendar year would have been divided into summer or extended summer seasons into two consecutive years. Therefore, the hydrological year in Australia, starting from April to March, instead of the calendar year, was considered in this study to calculate the annual maximum daily temperature.

2.2 Covariates

Australia is extremely susceptible to changes in the ocean–atmosphere system. The inter-annual dynamics of Australia's seasonal weather are regulated by the climatic variability of three neighbouring oceans—the Pacific, Indian, and Southern Oceans (Cai et al. 2014; Maher and Sherwood 2014; Oliveira and Ambrizzi 2017), known as CDs. Three primary CDs strongly influence Hydroclimate in Australia—The El Niño Southern Oscillation (ENSO) (Power et al. 1999; Cai and Van Rensch 2012), Southern Annular Mode (SAM) (Thompson et al. 2000; Risbey et al. 2009) and Indian Ocean Dipole (IOD) (Ummenhofer et al. 2009; Ummenhofer et al. 2011).

Therefore, in this study, three physical processes, in which two of them from the sea surface temperature—ENSO, IOD and one from air pressure—SAM, and also Time, were considered to be one of the potential covariates to identify the best covariate for developing non-stationary TDF curves.

2.2.1 El Niño-Southern Oscillation (ENSO) cycle

The El Nino-Southern Oscillation (ENSO) is considered to be a major CD over many regions of the globe (Ropelewski and Halpert 1988; Halpert and Ropelewski 1992), including Australia (Nicholls 1985). The SST index is considered the monthly sea surface temperature anomaly over NINO3.4 (17°E–120°W, 5°S–5°N) region (Bellenger et al. 2014) and was used as a covariate representing ENSO. Nino3.4 was obtained from https://psl.noaa.gov/gcos_wgsp/Timeseries/Nino34.

ENSO is considered to be a strong CD in Australia (Power et al. 2006; Risbey et al. 2009; Arblaster and Alexander 2012), especially in southeastern Australia (Nicholls and Lucas 2007) and northeast Australia (Min et al. 2013; Cowan et al. 2014; Perkins et al. 2015), but weak in the far southeast of Australia (Parker et al. 2013; Boschat et al. 2015). In Australia, it has been observed that the strength of ENSO decreases along a north–south gradient.

2.2.2 Indian Ocean Dipole (IOD)

The Indian Ocean Dipole (IOD) is represented with Dipole Mode Index (DMI) (Saji et al. 1999; Meyers et al. 2007; Liu et al. 2014) and calculated as the difference of SST anomalies in the western equatorial Indian Ocean (50 to 70°E, 10°S to 10°N) minus those in the east (90 to 110°E, 10°S to 0° N). The monthly DMI was derived from https://psl.noaa.gov/gcos_wgsp/Timeseries/DMI/ and considered as a covariate that represents IOD.

IOD has been identified as the major driver for Eastern Australia (Meyers et al. 2007; Risbey et al. 2009; Cai et al. 2011) and has a positive correlation with maximum temperatures during winter and spring (May and November) (Min et al. 2013; White et al. 2013).

2.2.3 SAM

The Southern Annular Mode (SAM) describes the north–south movement of the westerly wind belt that circles Antarctica and is a monthly mean sea level pressure (MSLP) anomaly time dataset. The data for this index were obtained from http://www.nerc-bas.ac.uk/icd/gjma/sam.html.

SAM considerably impacts the rainfall in Australia (Hendon et al. 2007), particularly in southern Australia. SAM is even associated with 10–15% of the rainfall in southwest Australia. SAM also affects the temperature in Australia and is identified as the main CD for the southwestern land division in western Australia (Guthrie 2021).

2.2.4 Time

Time has been utilised as a covariate to investigate the non-stationarity in the IDF curves all over the world (Sugahara et al. 2009; Yilmaz and Perera 2014). Therefore, in this study, Time was also considered as one of the potential covariates to construct the non-stationary TDF curves. The hydrological years of the AMT series were transformed into a series of integers from 1 to the number of years of the selected time duration and considered as a covariate representing Time.

3 Methodology

In this study, firstly, the non-stationarity of the temperature data series was assessed to justify the requirements of the incorporation of the non-stationarity in the construction of TDF curves. Then, the correlations between the selected three physical processes—ENSO, SAM and IOD were computed to identify the best covariate. In this approach, Time was also used as one of the covariates. Different non-stationary models were constructed considering different covariates stated earlier as well as their combination. Finally, Time or any of the selected physical processes (ENSO, IOD and SAM) was considered a covariate in models with one covariate.

Similarly, Time and one climate index (Time—ENSO, Time—IOD and Time—SAM) were adopted in models with two covariates. First, the best model for the AMT series was chosen based on the Akaike information criterion (AIC). After identifying the best non-stationary model, the non-stationary design temperature was computed using the parameters of the best non-stationary model.

GEV distribution was selected to develop both stationary and non-stationary TDF curves as it is widely used to model climatic extremes (Ouarda and Charron 2018; Ouarda et al. 2020; Haddad 2021; Devi et al. 2022). Under the condition of stationary, three parameters of the GEV distributions, μ, σ, ξ, which are the location, scale and shape parameters, respectively, were assumed to be constant, whereas, under non-stationary conditions, the parameters were expressed as a linear or quadratic function of the covariate(s). However, the shape parameter was assumed to be constant for all the cases since the reliability of modelling the shape parameter is low (Cheng et al. 2014; Ganguli and Coulibaly 2017). The distribution parameters were calculated using the maximum composite likelihood method utilising the optimisation function fmincon in MATLAB. Finally, the stationary and non-stationary TDF curves were compared to assess the influence of non-stationarity in the construction of TDF curves.

3.1 Detection of non-stationarity in temperature time series

All the historical records of AMT at the selected stations over the selected time period were tested for non-stationary signals. An augmented Dickey-Fuller (ADF) test was applied to the AMT data series for each of the selected stations. In this test, the null hypothesis assumes that the data series is non-stationary and thereby considered stationary when the null hypothesis is rejected at the selected significance level.

3.2 Correlation between climate drivers (CDs) and temperature time series

The relations between the AMT for all the selected stations and CDs—ENSO, SAM and IOD were tested using correlation analyses considering both concurrent and time-lagged relationships.

The covariates representing CDs in this study were computed as the moving averages of corresponding CDs over three consecutive months starting from April and ending in March of the next year, covering the full hydrological year and denoted in this study by the first letter of the three months (AMJ, MJJ, JJA, JAS, ASO, SON, OND, NDJ, DJF, JFM). To determine the appropriate season of the CD acting as the predictor of the AMT, correlations between seasonal CDs and AMTs were explored. Best covariates and the suitable season were selected based on the higher correlation values between the AMT series and CDs averaged over the selected season. This selection process was also validated graphically by plotting all the correlations between the AMT time series and all the CDs for all seasons.

3.3 Construction of TDF curves

  1. (i)

    General TDF relationship

The general TDF relationship can be expressed by following the formulation for IDF curves proposed by Rossi and Villani (1994) and Koutsoyiannis et al. (1998):

$$t_{R} \left( d \right) = \frac{a\left( R \right)}{{b\left( d \right)}}$$
(1)

where d is the duration and R is the return period. In this formulation, the dependency of return level, tR(d) on R and d can be modelled separately. The distribution of maximum average temperature T(d) governs the function a(R) that defines the TDF curves, which remain parallel for various return periods. The function b(d) controls the shape of the TDF curves and is given as:

$$b\left( d \right) = \left( {d + \theta } \right)^{\eta }$$
(2)

where \(\theta\) and \(\eta\) are the shape parameters subjected to the boundary conditions of \(\theta >0\) and \(0<\eta <1\).

If the probability distribution of \(T\left(d\right)\) is denoted by \({F}_{T(d)}\left(t;d\right)\), where t indicates the maximum average temperature and d represents the durations, \(Y=T\left(d\right)b(d)\), which is the scaled maximum average temperature distributed as \({F}_{T(d)}\left(t;d\right)\) (i.e., \({F}_{T(d)}\left(t;d\right)={F}_{Y}\left({y}_{R}\right)=1-\frac{1}{R}\)). Consequently, the expression \(a\left(R\right)\) is given by:

$$a\left( R \right) = F_{y}^{ - 1} \left( {1 - \frac{1}{R}} \right)$$
(3)
  1. (ii)

    Stationary TDF curves

The GEV distribution is the most widely used probability distribution that is used to model climate extremes, and hence, the GEV distribution is used here to model T(d). The cumulative distribution function of the GEV is given by:

$$F\left( x \right) = \exp \left\{ { - \left[ {1 + \kappa \left( {\frac{{x - {\varvec{\mu}}}}{\varvec{\sigma} }} \right)} \right]^{{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 k}}\right.\kern-0pt} \!\lower0.7ex\hbox{$k$}}}} } \right\},$$
(4)

where \(\varvec{\mu} , \varvec{\sigma} \, and \, \kappa\) are the location, scale, and shape parameters, respectively. \(F\left(x\right)\) is defined for \(1+\frac{\kappa \left(x-{\varvec{\mu}}\right)}{\varvec{\sigma} }>0\), where \({\varvec{\sigma}} >0\).

The general stationary TDF relationship based on the GEV distribution can be expressed as (Ouarda and Charron 2018):

$$t_{R} \left( d \right) = \frac{a \left( R \right)}{{b\left( d \right)}} = \frac{{\varvec{\mu} - \frac{\varvec{\sigma }}{\kappa }\left\{ {1 - \left[ {\log \left( {1 - \frac{1}{R}} \right)^{ - \kappa } } \right]} \right\}}}{{\left( {d + \theta } \right)^{\eta } }}$$
(5)
  1. (iii)

    Non-stationary TDF curves

To incorporate non-stationarity in Eq. (5), the three parameters of the GEV distribution described in Eq. (4) are considered to be dependent on time to incorporate the non-stationarity in TDF curve development and Eq. (4) can be expressed as:

$$ F\left( x \right) = \exp \left\{ { - \left[ {1 + \kappa \left( t \right)\left( {\frac{x - \varvec{\mu} \left( t \right)}{{\varvec{\sigma} \left( t \right)}}} \right)} \right]^{{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 k}}\right.\kern-0pt} \!\lower0.7ex\hbox{$k$}}}} } \right\},\;{\text{for}}\;1 + \frac{{\kappa \left( t \right)\left( {x - \varvec{\mu} \left( t \right)} \right)}}{\varvec{\sigma} \left( t \right)} > 0 $$
(6)

In this exploratory study, the location and scale parameters are considered to vary with covariates linearly or quadratically, while all other shape parameters, θ, η and κ for the GEV distribution are assumed to be constant (Katz et al. 2002; Adlouni and Ouarda 2009).

The vectors of the distribution parameters ψ = (μ, σ, θ, η) and ψ = (μ0, μ1, …, σ0, σ1, …, κ, θ, η) are estimated using the maximum composite likelihood for stationary and non-stationary TDF curves, respectively.

4 Results and discussion

The results for 12 stations (two stations from each state in Australia and shown as black dots in Fig. 1) are presented in greater detail (out of the 82 selected stations) in the following sections.

4.1 Detection of non-stationarity

The augmented Dickey-Fuller (ADF) test was applied to each AMT data series (covering 1969–2021) separately for each selected station. Based on the ADF tests, the test statistics and p-value for 12 stations are summarised in Table 1. Four stations showed ADF test statistics below the critical value (− 3.507), and the p-values were less than 0.05 at the 95% confidence level, as shown in boldface in Table 1. For these stations, the null hypothesis was rejected, and the data series was considered stationary. Therefore, the null hypothesis could not be rejected for the rest of the stations, and the non-stationarities in the data series were detected. Although all the available stationarity tests have some drawbacks and are not decisive alone (Cai et al. 2009a, b), the results of ADF tests here highlighted the presence of the non-stationarity in the AMT time series and consequently emphasised the importance of incorporating the non-stationarity in the construction of TDF curves.

Table 1 Summary of ADF test statistics and p-values for selected 12 stations in Australia
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4.2 Correlation analysis and covariate selection

The correlations between the AMT and the ENSO, IOD and SAM for all the seasons are presented in Supplementary Section (Figures S1, S2 and S3).

The correlations between ENSO and the AMT were the strongest and statistically significant during the spring and summer seasons, particularly in the eastern regions of Australia. The positive influence of the ENSO on the temperature was weaker at the beginning of the hydrological year and increased gradually over the year. At the end of the hydrological year, this influence became weak, even negative for some stations.

Similar to the ENSO, positive influences of IOD on the temperatures were also identified all over Australia. This influence was enhanced during the spring season and was statistically significant in the southeastern region of Australia. However, at the end of the hydrological year, the impact of IOD also became weaker and negative at some stations.

Considering the correlation values between the CDs averaged over different seasons and the temperature, one CD was selected for each station along with the occurring season, as shown in Fig. 2.

Fig. 2
figure 2

Selected climate drivers (CDs) with season based on the correlations between AMT and the CDs (ENSO, IOD and SAM) during 1969–2021

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One of the main findings of this study was that ENSO in spring acted as the dominant CD for the stations located in the inland NSW, Queensland and Western Australia, which is similar to the findings by a few other researchers (Min et al. 2013). On the other hand, IOD had the strongest relations with temperature along the east coast and almost all stations in the coastal and inland regions of Victoria and South Australia. It should be noted that there is a significant decrease in rainfall across southern Australia as the frequency of positive IOD events has increased, which contributed to significant bushfires over southeastern regions of the continent (Cai, Cowan and Raupach, 2009; Cai et al. 2009a, b).

Although SAM in Autumn had strong relations with stations in NSW and Victoria, ENSO and IOD were the dominant CDs. Furthermore, one of the Department of Primary Industries and Regional Development studies reported a decline in rainfall due to SAM in southern Australia, particularly in the southwest region (Guthrie 2021). However, SAM appeared as the selected CD at the stations located in the coastal regions of Western Australia. In the cases of stations located in the Northern Territory, the temperature recorded at the station on the coastal side was associated with ENSO, while at the inland, it was correlated with the IOD.

The Pearson correlation coefficient values for the AMT series from April–May–June (AMJ) to January–February–March (JFM) of the same hydrological year recorded at 12 stations are presented in Fig. 3. These illustrations validated the selection of the best CDs and seasons presented in Fig. 2. Selected CDs with the seasons for the selected 12 stations are summarised in Table 2.

Fig. 3
figure 3

Correlations between AMT and CDs. The blue, black and green lines represent correlations between AMT and ENSO, IOD and SAM, respectively

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Table 2 Pearson correlation coefficients between the AMT series and the selected seasonal climate index
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4.3 Stationary TDF

Figure 4 illustrates the stationary TDF curves based on GEV distribution as described in the methodology section. Estimated temperatures were plotted against the selected durations (1, 2, 3, 4, 5, 6, 7 and 10 days), with each curve indicating a different return period such as 2, 5, 10, 25, 50 and 100 years at stationary scale. These TDF curves demonstrated a significant rise in temperature with higher return periods and decreased with the increase in duration for all the selected stations.

Fig. 4
figure 4

Stationary TDF curves for 2, 5, 10, 25, 50 and 100 years return periods

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4.4 Non-stationary TDF surfaces

For each TDF model and station, the maximal independence log-likelihood, the CL-AIC statistic, and the model parameters are summarised in Table 3. According to the values obtained by the AIC criterion from each model, the table only shows the optimal parameter relationship among the constant, linear and quadratic relationships to the covariate or the combinations of the covariates.

Table 3 Summary of parameters for the selected models and their error estimation
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The CL-AIC statistic and log-likelihood suggested that the Time and covariate individually or combinedly increased the goodness-of-fit compared to the stationary model for all the stations. It's worth noting that employing a non-stationary model or the GEV enhances the log-likelihood in every case. However, the performances of different stationary and non-stationary TDF models were compared by the CL-AIC values, which were penalised due to the inclusion of more variables and thereby provided more reliable results.

Most stations (9 out of 12) with a combination of Time and CD as covariates showed the best goodness-of-fit. This suggested that the combination of the two covariates considerably impacted severe temperatures. Time was more prominent than CDs and qualified as the best covariate in cases of Sydney Airport AMO and Laverton RAAF stations. On the other hand, for Barcaldine Post Office, the influence of covariate alone was stronger than Time or the combination of Time and CD as the covariate.

4.4.1 Non-stationary TDF surfaces—One covariate (Time or CD)

Figures 5 and 6 present the non-stationary TDF graphs with the model considering Time and CD as a covariate for the typical stations. For all the illustrated stations in this study, non-stationary TDF models with Time as a covariate, either scale or location parameters were found to be varied linearly with time, except for Darwin, Alice Springs Airport, Adelaide Airport, Walpeup Research, where either of the parameters varied quadratically with time. TDF curves with Time as a covariate for Kalgoorlie-Boulder Airport stations linearly varied with time but were not parallel to each other for different return periods. It needs further investigation, which is left for future research. In the case of Perth stations, the estimated temperature increased with return periods and decreased with duration but did not vary significantly with time. However, in the case of non-stationary TDF models with CDs as covariates, the scale or location parameters varied quadratically with selected CDs for most stations.

Fig. 5
figure 5

Non-stationary TDF surfaces with Time covariates for 2, 5, 10, 25, 50 and 100 years return periods

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Fig. 6
figure 6

Non-stationary TDF surfaces with CD covariates for 2, 5, 10, 25, 50 and 100 years return periods

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Consequently, the TDF curves varied parabolically with CDs. Exceptions were observed for Sydney Airport AMO, Alice Spring Airport, Amberly AMO, Adelaide Airport and Walpeup Research stations, where either scale or the location parameter changed linearly with CDs and exhibited linear TDF curves for different return periods and durations. These characteristics of non-stationary TDF models can be seen distinctively in Figures S4—S7 (in supplementary sections).

4.4.2 Non-stationary TDF surfaces—Two covariates (Time and CDs as covariates)

For non-stationary TDF models incorporating two covariates, five variables are required to be present in the graph. Therefore, in this paper, non-stationary TDF surfaces are presented in two ways—either the duration is kept fixed and TDF surfaces for different return periods are presented, or return period is kept fixed and TDF surfaces for different durations are illustrated. Figure 7 shows the TDF surfaces for 5-day duration and all the return periods considered in this study, whereas Fig. 8 presents the TDF surfaces for 50-year return period and all the considered durations.

Fig. 7
figure 7

Non-stationary TDF surfaces with Time and CD as covariates of 5-days duration for 2, 5, 10, 25, 50 and 100 years return periods

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Fig. 8
figure 8

Non-stationary TDF surfaces with Time and CD as covariates of 50-year return period for 1, 2, 3, 4, 5, 6, 7 and 10 days durations

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4.5 Impacts of non-stationarity on TDF curves

The temperature quantiles calculated at a given time are significantly affected by incorporating one or more covariates. Figure 9 shows the graphs of the 50-year quantiles versus the duration for the stationary TDF model and the non-stationary TDF models considering different Time and covariates representing different scenarios for each station. The non-stationary quantiles for the first and last years of the study period were computed with Time covariate and denoted as Time (1970) and Time (2021) to display the quantiles' temporal movement. The years with the highest and lowest values of the selected seasonal CDs throughout the period 1970–2021 were selected for "CD" and "Time + CD" non-stationary models to highlight the influence of the extreme conditions on the estimated quantiles. All the graphs illustrated in Fig. 9 use the model that provides the greatest overall fit.

Fig. 9
figure 9

Comparison between stationary and non-stationary TDF curves (50-year quantiles)

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In all stations, it was observed that the influence of the duration on the difference in quantile estimation between the stationary and non-stationary models was negligible. For all the 12 stations, the stationary model always overestimated and underestimated the return levels in the case of the non-stationary models with the earliest (1970) and latest (2021) years of the study period as Time covariate, respectively. The stationary model overestimated upto 4.2 °C compared to the former cases, while underestimated upto 2.3 °C for the latter cases at all the12 stations. The only exception was for the Perth Airport station, where Time covariate had no significant influence. This can be explained by the fact that the Perth Airport station showed no non-stationarity (Table 1) and no temporal trend in the AMT during the study period (Figure S8 in the supplementary section).

In general, for all the stations, the difference between the stationary and non-stationary models incorporating the CDs as covariates decreased with the return period, as shown in the supplementary section (Figures S9–S13). Also, the stations where IOD was selected as the CD showed a significant difference between stationary and non-stationary models compared to the other stations where ENSO and SAM were selected as the CDs.

The average difference between the stationary model and the non-stationary models considering the CD alone as covariate ranged from − 3.0 to 1.5 °C (+ CD) and − 0.6 to 3.4 °C (– CD). In contrast, this difference became − 1.9 to 2.2 °C ("Time + CD") and − 1.6 to 4.7 °C ("Time–CD") in the case of the combination of two covariates—Time and selected CD.

Although Time had the overall dominance as a covariate on the computed quantiles for all the return periods compared to the selected climate indices, different covariates or combinations of them yielded the maximum quantiles for different return periods and stations. It should be noted that non-stationary TDF curves should be adopted in practice irrespective of its difference with the stationary TDF curves.

5 Conclusion

In this study, the formulation of stationary and non-stationary TDF curves was based on temperature time series from 82 weather stations located in Australia using the GEV distribution. Initially, the augmented Dickey-Fuller (ADF) test was conducted to identify non-stationarity in temperature time series and non-stationarity was found to be present in the data series.

The long-range relationships between the seasonal CDs (ENSO, IOD and SAM) and temperature data from 1969 to 2021 were investigated using the Pearson correlation coefficient to find out the best covariates in non-stationary TDF models. The magnitude of correlation coefficients of ENSO increases towards the east of Australia, and these coefficients are significant during SON and DJF seasons. The AMT observed at stations in the southeastern region of Australia, especially in the inland and coastal region of Victoria and South Australia, showed a significant correlation with IOD during SON. SAM showed a strong correlation with the annual temperature at the stations located in the coastal regions of Western Australia.

Stationary TDF curves showed an increase and decrease in design temperatures with higher return periods and an increase in duration, respectively. In the case of non-stationary TDF models, the location and scale parameters were modelled as being dependent on time and climate indicators for the selected stations. Inclusion of the selected covariates in non-stationary TDF models enhanced goodness-of-fit compared to the stationary TDF model for the corresponding station. Similar results were found by Ouarda and Charron (2018) where the influence of the climate oscillation pattern was found to be more prominent than the temporal trend. Furthermore, the best goodness-of-fit of the TDF model based on the AIC values was obtained with a combination of both covariates Time and selected CD for most of the stations. These results highlighted the importance of considering the combined effect of the temporal trend caused by global warming and CDs in statistical models used to predict design temperature.

The non-stationary quantiles computed with the first and last years of the study period and with the years of highest and lowest values of the selected seasonal CDs were compared with the stationary model to display the quantiles' temporal movement and the influence of the extreme conditions on the quantiles. In most cases, the stationary TDF model underestimated the design temperature compared to the non-stationary model, including Time as a covariate. This conveys a crucial message that the non-stationary framework for designing temperature facilities in Australia could be considered a stronger option than the traditional stationary approach. In addition, TDF curves developed in this study can be applied to a range of sectors such as agriculture, health care and energy production and can be a useful tool for policymakers and planners.