A C-Vine structure was selected for Vine Copula construction, with testing (not shown; utilizing the algorithm of Dissmann et al., 2013) revealing the conditional/root variable precipitation as the optimal variable. Figure S1 (top right) shows the variable order represented by precipitation-soil moisture-temperature, which in real terms results in precipitation having a direct impact on soil moisture, and an indirect impact on temperature (as temperature can vary depending on soil moisture and the associated energy balance partitioning, Seneviratne et al., 2010). Bivariate copulas are therefore required to be established between precipitation and soil moisture (edge 1,3), precipitation and temperature (edge 1,2) and temperature-soil moisture, conditional on precipitation (edge 2,3;1). As noted by Aas et al. (2009), vine copulas can be used to model multivariate data acting on two variables at a time, offering a way in which to construct higher-dimension copulas. For a three variable example, the vine copula density c is given as: 3cu1,u2,u3;Θ=c1,2u1,u2;Θ1,2c1,3u1,u3;Θ1,3c2,3;1h2|1u2,u1;Θ1,2h3|1u3,u1;Θ1,2;Θ2,3;1 where c1,2, c1,3 and c2,3;1 are the pair copula densities corresponding to the copulas C1,2, C1,3 and C2,3;1. The h-functions involved are defined as hb|a(ub,ua;Θ):=Cb|a(ub,ua;Θ), where Cb|a denotes the conditional distribution function of Ub given Ua (Erhardt and Czado, 2018). Copula selection was from a family made up of Clayton (lower tail), Frank (symmetrical) and Joe (upper tail), with optimal copula selection across New Zealand identified in Fig. 1, which in turn captures the regional hydro-climatic variation across the country. Additional methodological detail is contained in Sects. S1 and S2 in the Supplement and Table S1.

After fitting a copula for each grid cell and calendar day, the resultant copula data (encompassing precipitation, maximum temperature and soil moisture; Table 1) were transformed into independent data in the [0, 1] space using a probability integral transformation (the Rosenblatt transformation, Rosenblatt, 1952). The Rosenblatt transforms v:=(v1,…,vd) on the basis of the selected vine copula C for the data u= (u1,…,ud) is defined as: 4vd:=Cd|1,…,d-1ud|u1,…,ud-1 where Cj|1,…j-1 is the conditional cumulative distribution function for variable j given the variables 1,…,j-1 for all j=2,…,d. In the Canonical Vine (C-Vine) context the order of the variables is determined by the selected order of root variables (Erhardt and Czado, 2018). Following the transformation into the [0, 1] space, the data is further transformed into a standard normal distribution using the inverse of the cumulative distribution function of a standard normal distribution: 5pj=Φ-1(vj) before being aggregated and standardised to the final Standardised Multivariate Index (SMI): 6SMI=1d∑j=1dpj

Limits of -3/3 were also imposed to ensure reasonableness with the extrapolation of return periods (Stagge et al., 2015). Each calendar day was then placed back into a sequential time series within each grid cell, resulting in the construction of the Standardised Multivariate Index (SMI). The SMI was then employed to characterise compound events. The 30 d accumulation was chosen due to the desired focus on monthly accumulations, with existing research on compound (Feng et al., 2021) and seesaw events (He and Sheffield, 2020) employing a similar one-month accumulation. Figure S7 illustrates a clustering performed on a 1 d accumulation of the SMI, highlighting known climatic regions of New Zealand when clustered outside temporal auto-correlation, while additional methodological detail is contained in Sects. S1 and S2 in the Supplement and Table S1.

An additional bivariate copula index was also constructed, the Standardised Bivariate Index (SBI), following the same procedure listed above (with the exclusion of the vine copula components; Fig. 2). A bivariate copula index was constructed between soil moisture and precipitation (Table 1), to enable a comparison of these variables outside of the influence of temperature (i.e. the SBI is used to characterise seesaw event), with the bivariate copula construction between precipitation and soil moisture being a common copula pairing in multivariate hydroclimatic studies (AghaKouchak, 2015; Hao and AghaKouchak, 2013, 2014).

To investigate the differences in event detection between the coincident (consecutive) and SMI (SBI) approaches, binary event occurrences were developed at each grid cell. These were then evaluated by comparing the event detection considering the SMI (SBI) approach as the true detection and developing a confusion matrix (e.g. error matrix). To illustrate, any day where both the SMI (SBI) and coincident (consecutive) approaches detect the same compound (seesaw) event is classed as a true positive, while any day where both approaches do not detect an event is classed as a true negative. If the SMI (SBI) approach detected a compound (seesaw) event day, which was not detected by the coincident (consecutive) approach, then this was classed as a false negative, while the detection of a compound (seesaw) event day under a coincident (consecutive) approach which is not present in the SMI (SBI) approach is classed as a false positive. The results at each grid cell were summarised by taking the mean across all days. For all compound event days at each grid cell, as defined by the SMI threshold, the mean of the STI and either the SPI or SSMI were extracted, as well as the STI value in isolation. These were then visualised as density plots of all grid cells. For seesaw events, density plots were constructed using the mean value for each grid cell as defined by the SBI approach i.e. the value of the SPI, SSMI and their mean for the same days that SBI identifies a seesaw event.

Run theory, commonly applied to drought analysis to calculate severity, intensity, duration and frequency (Panu and Sharma, 2002; Yevjevich, 1972), was used here to further investigate the differences in the representation of both compound and seesaw events under each classification. Frequency was defined as the total number of events. Duration is defined as the number of days below the exceedance thresholds (compound) or the average length of all seesaw events. Severity is defined as the cumulative sum of all index values. For compound events, the severity and intensity metrics for the coincident classification criteria were treated as the mean of SSMI/STI or SPI/STI. To minimise the effects of minor compound events, only those events which exceeded 14 d were investigated. Intensity is defined as the average of the index value for compound event days, while for seesaw events the intensity metric was replaced with a new metric: phase domination. Phase domination was defined as the combined total of both the lowest (dry) and highest (wet) value during the event, with negative values thereby indicating relatively stronger dry phases compared to the wet phase of the seesaw event (and vice versa).

The onset rates (maximum compound event value over number of days to reach said value from the start of the event) and termination rates (maximum compound event value over number of days to cessation of event) were compared between each classification criteria for compound events, with the multivariate (SMI) approach used as the baseline i.e. SMI compared to coincident SSMI/STI, SMI compared to coincident SPI/STI (Fig. S2a–b). For seesaw events, the average transition time for each event was established for each grid cell, by taking the average number of days between the peak dry period and peak wet period, for all events at each grid cell, following the methods of Rashid and Wahl (2022). Differences between dry termination rates (minimum dry phase value over number of days to reach zero) and wet onset rates (maximum wet phase value over number of days to reach said value) were also calculated on a grid cell level (termed slope rates). Statistical significance was calculated using a t-test (Student, 1908), with 2000 iterations of a bootstrapping procedure performed (Wilkes, 2019), and adjusting p-values for spatial autocorrelation using the False Discovery Rate (FDR) approach (Wilkes, 2016). This method of comparison enables statistical confidence in the identification of events, but it should be noted that the suitability of the method at detecting historical events remains to be tested. In turn, this work will enable this future alignment by providing the necessary event identification dataset to ensure robust statistical testing.

For the east coast of both islands (examples of transitional regimes, being regions where soil moisture constrains evapotranspiration variability, Seneviratne et al., 2010), SPI/STI events have the strongest agreement in severity and duration to the SMI (Figs. 3a–b, 4a,d). This is in contrast to the longer duration and stronger severity observed within the coincident SSMI/STI detection method, which also drives their faster onset and termination rates (Figs. 3a–b, 4 b, e, 5a–b, 6a–b). Thus, a more common (and more extreme) temperature anomaly is visible within coincident soil moisture/high temperature, resulting in longer duration (and subsequently stronger severity) events, reflective of the underlying surface energy balance exchanges (e.g. increased sensible heat) taking place during compound event occurrence (Seneviratne et al., 2010).

The strong relationship between low soil moisture and high temperature (given precipitation – i.e. lower tail dependence) in the vine copula results in a more common occurrence of compound events within the SMI than coincident approaches (i.e. not accounting for the dependence between soil moisture and temperature). A higher frequency of events across the east coast of both islands is also shown in Bennet et al. (2023), demonstrating the importance of compound events for these regions of the country, and in agreement with the findings that extreme temperatures impact these regions (Harrington, 2021; Harrington and Frame, 2022). This is expressed across the east coast regions as high event occurrence across all metrics (SMI: 72 events; SSMI/STI: 51 events; SPI/STI: 42 events). Put another way, a 1 in 100 year event defined by the SMI across these east coast regions becomes a 1 in 171 (SPI/STI) or 1 in 141 (SSMI/STI) year event. Such variation has vital implications for hazard management and planning, and illustrates the importance of understanding uncertainty in compound hot and dry event detection which is driven by the choice of dry indicator (Hosseinzadehtalaei et al., 2024).

The identification of agricultural dry conditions in wet, energy limited regimes (i.e. displaying lower tail dependence; Cammalleri et al., 2024) is impacted by the vine copula approach to a greater extent than transitional regimes. The representation of agricultural drought via the SMI captures a higher occurrence of dry phases, with the lower tail dependence resulting in the largest difference in event frequency compared to the coincident approaches (Fig. 3a–c). Similar to transitional regimes, precipitation metrics for agricultural drought identify the least number of compound events, reflecting the delay in moisture deficit propagation into soil moisture (Zhu et al., 2021). The lower number of agricultural droughts identified using precipitation metrics is further supported by Bachmair et al. (2018) who identified that meteorological indices (e.g. SPI) were less informative of agricultural drought across colder/wetter regions in Europe.

Building on the findings of Bachmair et al. (2018) that precipitation metrics are less informative of agricultural drought in wet regions, the present research generates the greatest disparity between detection methods across wet regions. For example, the west coast of both islands indicate large variation in events across all three metrics (SMI: 62; SSMI/STI: 44: SPI/STI: 33), which simultaneously indicate a weaker intensity (SMI: average intensity of -1.66; SSMI/STI: average intensity of -1.77: SPI/STI: average intensity of -1.83) i.e. more frequent, less intense under the SMI approach. For the west coast of the South Island, the high occurrence of events captured by the SMI results in events which are shorter, less severe and less intense than the coincident SSMI/STI approach or the high intensity coincident SPI/STI approach. For example, low severity events (defined as the 25th percentile of SMI events) with a 1 in 100 year return period equate to a 1 in 181 (SSMI/STI) or 1 in 219 (SPI/STI) year event. Meanwhile, high severity events (75th percentile of SMI) with a 1 in 100 year event (SMI) equate to a 1 in 160 (SSMI/STI) or a 1 in 210 (SPI/STI) year event. While dry indicator uncertainty for compound hot and dry events is significant in general (Hosseinzadehtalaei et al., 2024), the regional variation shown here indicates that the greatest sensitivity to dry indicator selection is found across wet energy limited regions.

The spatial expression in event occurrence for New Zealand shown here (0.64 to 1.21 events per year based on the SMI) was also shown in Bennet et al. (2023), where a range of 0.21 to 1.14 events per year was identified using coincident SSMI/STI. The coincident metrics currently employed, being SPI/STI (0.19 to 0.92 events per year) and SSMI/STI (0.21 to 0.81 events per year), both reveal an overall lower frequency compared to that reported in Bennet et al. (2023). An overall higher frequency of events is present across the entire country for the SMI (median of 65 events) compared to the coincident approaches (SSMI and STI (47 events); SPI and STI (34 events)) (Figs. 3c, 4j–l). This is expressed as a 63 % difference under the SMI compared to the coincident SPI/STI approach, and a 32 % difference compared to the coincident SSMI/STI approach. Such differences suggest careful consideration should be made to the choice of dry indicator for any study involving compound hot and dry events (e.g. establishing trends or projected changes) (Hosseinzadehtalaei et al., 2024).

Nationally, the soil moisture/high temperature anomaly is more common than precipitation/high temperature (Fig. 3d), indicating the difficulty of detecting co-occurring low precipitation and high temperature anomalies with the coincident approach (Tabari and Willems, 2023; Zscheischler and Seneviratne, 2017). The use of precipitation as a proxy for agricultural drought (coincident SPI/STI) identifies the least number of compound events, an outcome of the delay in moisture deficit propagation from atmosphere to soil moisture (He and Sheffield, 2020). The usage of any precipitation metric as a proxy for soil moisture and agricultural dry conditions should only be made after careful consideration of the appropriate accumulation period, necessitating a regional specific prior analysis and acknowledgement of possible spatial variation in appropriate accumulation periods (Wang et al., 2022). Without this prior analysis, soil moisture-based approaches should be the default for investigating compound hot and dry events.

Meanwhile, representing agricultural drought as the dependence between soil moisture and precipitation (Hao and AghaKouchak, 2013) results in the highest detection of compound events (SMI) nationally. The use of the vine copula approach for agricultural drought representation in compound event detection enables a greater understanding of the impact regional differences in variable anomalies have on compounding conditions (e.g. stronger termination rates for lower tail dependency regions (east coast of New Zealand) or more frequent events in wet, energy limited regions). The vine copula approach simultaneously reveals a greater frequency of this agricultural dry phase than solely precipitation or soil moisture (Hao and AghaKouchak, 2013). Thus, while uni variate soil moisture-based approaches are still recommended for investigating compound hot and dry events, multivariate approaches (SMI) identify events which would go undetected when solely using soil moisture. Multivariate approaches may therefore provide a more accurate description for both risk and hazard management as well aid in accurate identification of wider atmospheric controls and teleconnections. This accurate description and identification by multivariate approaches is achieved by revealing the true probability for hot and dry co-occurrence and therefore should be utilised whenever possible

Similarities are strongest between approaches across wet, energy limited regions. The west coast of the North Island shows a similar pattern of high frequency occurrence (SBI: 64 events; SSMI: 62 events; SPI: 80 events) and phase domination (SBI: mean value of 0.33; SSMI: mean value of -0.20; SPI: mean value of -0.33). Meanwhile, the average transition time is lowest across the west coast of the South Island under all measurement techniques (Fig. 9a–c). Differences are most noteworthy between approaches in the representation of slope rate differences between drought termination and wet onset (Fig. 10a–c). Consecutive SPI reveals significant parts of the country (upper-mid North Island, north and south of South Island) as having rapid drought cessation relative to wet onset (Fig. 10a), driven by the return to relatively wet conditions in these wet regimes (Bennet et al., 2023). For the west coast of the South Island, the SSMI representation of seesaw events reveals a stronger dry termination rate, evidence of the comparatively wet environment witnessing a return to the wet, normal conditions. Contrasting this, the copula approach reveals much of the country has quicker wet phase onsets – it is noted that this is the generally expected response given the required time to recover from dry or drought conditions (Rashid and Wahl, 2022).

The emergence of stronger wet phases in the SBI across wet energy-limited regions becomes apparent due to the lower tail dominance (Fig. 1), whereby agricultural drought detection is greater (Cammalleri et al., 2024) in the west coast (Fig. 8l) (an outcome of the more common agricultural drought phase driven by the slow responding soil moisture i.e. weaker but more frequent agricultural drought). In turn, this more common drought phase and lower tail dominance drives weaker termination rates relative to the onset rate of the wet phase in the SBI (slope difference of -0.02). Contrasting this, consecutive metrics reveal stronger dry termination rates (Fig. 10c), a result of the stronger dry phases relative to wet phase (i.e. stronger, less frequent agricultural drought; Fig. 8g–i).

The west coast of the South Island is characterised by its exposure to the westerly passage of air movement which drives New Zealand weather systems (Macara, 2018), as well as large atmospheric river events which bring substantial precipitation (Prince et al., 2021). Therefore, the high occurrence of seesaw events (Fig. 8c) is somewhat expected. Further, De Luca et al. (2020) note a domination of wet over dry extremes in wet climates across the globe. Combined with a greater identification of agricultural drought across these wet regions (i.e. lower tail dependence; Cammalleri et al., 2024) (Fig. 1), a relatively high number of both dry and wet events results, with a subsequent increased likelihood of these occurring consecutively. However, of note is the severity of such events being comparatively minor, likely driven by their short duration (Fig. 8a–c).

The consistent finding that seesaw events on the South Island west coast are typically high frequency, short duration and low severity for each of the SBI, SSMI and SPI (Figs. 6, 7, 8) is consistent with the findings for this region in the global-scale study of Rashid and Wahl (2022). Similarly, return periods are similar between the three indices – overall, indicating little difference in detection metrics across wet energy-limited regions such as this, and in stark contrast to these regions having the strongest differences between metrics for compound hot and dry events (Figs. 3, 4, 7, 8).

The strong differences in seesaw event characteristics across the transitional regime of the North Island east coast (long duration/high severity/low frequency for SSMI and SBI, vs. opposing characteristics for the SPI) are associated with notable slope rate differences between drought termination and wet onset (Fig. 9). The slower responding soil moisture indicates much of east coast as having a significantly quicker onset of wet phases (relative to drought cessation) (Fig. 10b). Consecutive SPI also reveals parts of the east coast as having quicker wet onset phases, although such differences are not significant. Contrasting this, the copula approach reveals weaker wet phase dominance across the east coast of both islands (Fig. 10c).

The weaker wet phase dominance in the SBI approach is present due to the upper tail dominance (Fig. 1) resulting in a more frequent wet or pluvial phase under the joint probability framework (Fig. 8). With evapotranspiration (including seasonal variation) becoming a controlling factor in moisture loss in transitional regimes (Seneviratne et al., 2010), no distinguishable lower tail relationship is present across these east coast regions under the SBI approach (Fig. 1), resulting in more rapid shifts out of the dry phase into the more common wet phase (e.g. the upper tail dependence) (Fig. 10c).

East coast regions reveal the largest variation between event detection (SBI: 53 events; SSMI: 42 events: SPI: 82 events), as well as the longest transition time in the SBI (88 d) and SSMI (132 d) driven by the slow responding soil moisture and upper tail dominance in the SBI. The high occurrence across east coast regions described by the SPI is similarly reflected in Bennet et al. (2023), where up to 18 % of droughts were terminated by a pluvial. Translated into return periods, a 1 in 100 year event as defined by the SBI results in a 1 in 126 (SSMI) and 1 in 65 (SPI) year return periods. Dry indicator selection therefore has the most uncertainty across the east coast of both islands (transitional regimes), with key implications for water management practices illustrated in the range in return periods.

Collectively, the SBI representation of seesaw events exists in the middle between the SSMI (57 events) and SPI (79 events) detection approaches, with a spatial average of 62 events across the country. As a spatial average, SBI events are dominated on the wet side (mean value of 0.31), characterising SBI-defined transitions as having longer or more intense wet phases during seesaw events in comparison to the dry phase dominance shown in SPI (mean value of -0.26) and SSMI (mean value of -0.06) (e.g. more intense or longer dry phases). Nationally this is expressed as a range in return periods of 30 years using SBI as the 1 in 100 year baseline (SPI: 1 in 79 years, SSMI: 1 in 109 years).

Wet, energy limited regions show the greatest agreement between approaches, with precipitation and soil moisture seesaw progressing at a comparatively similar speed. In contrast, seesaw events across transitional areas have the largest deviation between precipitation and soil moisture seesaw, indicative of evapotranspiration becoming a controlling factor (including seasonal variation) in dry phase development during soil moisture seesaw. Consistent with the existing research on drought quantification uncertainty (Mishra and Singh, 2010) (Stagge et al., 2017) (Vicente-Serrano et al., 2012), the present work illustrates variation in seesaw event detection and characteristics dependent on the chosen representative variable (Figs. 7a–c; 9a–c, 10a–c).

Similar to compound hot and dry events (e.g. Hosseinzadehtalaei et al., 2024), sensitivity to selection of dry indicator for seesaw events is expected. However, unlike the recommended use of soil moisture metrics for compound hot and dry detection (and preference for multivariate representation of agricultural drought), no one method should be ultimately termed “superior” for seesaw detection, reflecting the more general comments by Lloyd-Hughes (2014) on drought detection. Accordingly, care is needed to fully differentiate studies according to the hydrological domain(s) implied by the dry phase/drought type targeted and subsequent index selection (Hoffmann et al., 2020) (e.g. precipitation seesaw, soil moisture seesaw). The differences in frequency and return periods of water deficits in different hydrological cycle domains can have vital implications for water management practices as well as the understanding of driving mechanisms and wider teleconnections. For New Zealand, east coast regions show the largest difference in detected events, and with these regions being key agricultural centres and sources of significant hydropower generation, not framing the study around the desired hydrological cycle component could have significant implications.