Climate change caused by an increased greenhouse effect is expected to be associated with changes in the hydrological cycle and an increase in precipitation and more extreme rainfall . There are several physics-based explanations for the increased rainfall amounts. Higher surface temperature gives higher rates of evaporation and strengthens the moisture-holding capacity of air. The air moisture is part of the global hydrological cycle where it condenses to form clouds and returns to Earth's surface through precipitation. Furthermore, a recent analysis of the satellite-based Tropical Rainfall Measuring Mission (TRMM) suggests that there has been a change in the global rainfall area over the recent decades that may also imply changes in the mean rainfall intensity and there are similar indications in recent state-of-the-art reanalyses . There is also a possibility that some rain-producing phenomena become more prevalent under warmer conditions, such as convective systems. These theoretical aspects are underscored by trend analyses of the probability of heavy rainfall pointing to more extreme rainfall amounts . Extreme rainfall is a disruptive and damaging hazard but can to some extent be managed through a proper risk analysis. For example, precipitation intensity–duration–frequency (IDF) curves are commonly used tools in water resource management and planning .

One problem associated with IDF curves is that rain gauge data have limited geographical coverage, making it difficult to analyse the risk of extreme rainfall everywhere. The Intergovernmental Panel on Climate Change's Sixth Assessment Report also noted that the exact levels of regional IDF characteristics may depend on the method as well as the resolution of downscaling when derived from climate model simulations . Furthermore, short time series imply limited knowledge of extremes and challenge the extrapolation to long return periods. These caveats are particularly problematic for IDF statistics for short-duration rainfall (3 h or less), which are useful for the design of urban infrastructure and urban flood prevention. Short-duration rainfall is often a result of local convective activity and hence highly variable, but sub-hourly precipitation observations are relatively sparse. One way to circumvent this issue is by using systematic dependencies between IDF curves and climatological and geographical factors to regionalise the IDF curves from gauged to ungauged stations. A similar principle may be used to expand IDF curves from the current period (gauged or ungauged) to the future . While each precipitation event typically has low predictability, the statistical nature of such events, such as their probability, frequency or typical intensity, may still be highly predictable. developed a method for estimating return levels for sub-daily rainfall intensity over Norway based on high-resolution gridded climate observations and Bayesian hierarchical modelling. Remote sensing (radar and satellite) products of high spatial resolution can also be a complementary source of precipitation measurements in regionalisation to ungauged locations . While radar products have been shown to often underestimate precipitation and in particular high-intensity rainfall , the inherent bias can be locally adjusted with rain gauge measurements . One caveat with gridded data is that they are not optimal for analysing extreme values due to spatial inhomogeneities on sub-grid-box scales (; Richard Chandler, personal communication, 2016). While useful for regionalisation purposes in conjunction with rain gauge measurements, one should be careful estimating the frequency of heavy-precipitation events directly from gridded products.

Another problem is that IDF curves may be inconsistent across durations. proposed two post-processing approaches to deal with such inconsistencies. Another approach may involve a sort of “temporal downscaling” of sub-daily rainfall intensity based on 24 h precipitation statistics . In a similar vein, estimated future hourly extreme rainfall using temporal downscaling based on scaling properties of rainfall for a case study of Barcelona and the SRES A1B, A2, and B2 climate scenarios. used a different approach, proposing a flexible and consistent quantile estimation method for IDF curves that explored how to improve the estimation by adjusting parameters representing curvature, multi-scaling and flatness. Their approach involved using a duration-dependent formulation of the generalised extreme value (GEV) distribution to fit IDF models with a range of durations simultaneously. The degree of uncertainty associated with estimating IDF curves is substantial, and attempted to quantify uncertainties connected to both insufficient quantity and quality of data, leading to parameter uncertainty due to the distribution fitted to the data, and uncertainty as a result of using multiple global climate models (GCMs) from the CMIP5 ensemble (RCP2.6, RCP4.5, RCP6.0, and RCP8.5 scenarios). They used a Bayesian approach and a case study of the city of Bangalore in India and found that the uncertainty is larger for shorter than for longer durations for the rainfall return levels and also that parameter uncertainty was greater than the model uncertainty.

In this study, we test an empirical statistical modelling approach to estimate the shape of the IDF curves, rather than the return values for each duration and return period. This approach is based on principal component analysis (PCA) and Bayesian inference. Using PCA reduced the return values of all stations and return periods to a set of principal components (PCs) in the form of spatial patterns, with accompanying IDF shapes and eigenvalues. The leading PC spatial patterns were subjected to Bayesian linear regression and subsequently expanded to new stations based on climatological and geographical information. The analysis is included in its entirety in the R Markdown document provided in the Supplement. Predicting the shape of the IDF curves of all return periods simultaneously through PCA is a novel strategy which to our knowledge has not been done before in this context. The motivation for this approach was the observation and expectation that the curves have simple and smooth shapes with regional similarities. In other words, the return values for rainfall intensities over different durations are related to each other, and the IDF curves are associated with a substantial degree of redundant information that can be utilised through the application of PCA. The estimated return values were compared with the simple formula for estimating approximate values of return levels for sub-daily rainfall based on 24 h rain gauge data presented in . A strategy that involved using a weighted set of polynomials to represent the IDF shapes was also pursued but abandoned as it provided a poor representation of the return values. The purpose of the statistical modelling was (i) to explore the influence of meteorological and geographical conditions on the IDF curves, (ii) to establish an empirical model to be used for regionalisation where sub-daily precipitation data are not available, and (iii) to compare different and independent strategies for estimating sub-daily rainfall intensity.

The regression results presented in Table  and Fig.  indicated that the mean precipitation intensity in the warm season is connected to increased intensity over all timescales examined, whereas a similar increase in the cold season suggested reduced short-term intensity but increased long-term intensity. A similar increase over all timescales in the warm season is consistent with Eq. () δxτ=αL/24ζln⁡(fwτ)δμ, assuming a constant value for ζ, but the deviation from this in winter may suggest that ζ is not necessarily constant. The difference in seasonal response is likely related to the dominance of different precipitation-generating processes: stratiform in winter and convective in summer. It could also possibly be connected to the warm and cold initiation of precipitation . The locations with intense winter precipitation, which tend to have high return values for long durations and relatively low return values for shorter durations, are found along the west coast (Figs. , , S3 and S22). The climate of this region is characterised by the North Atlantic storm tracks, bringing a steady stream of cyclones that often meet land here in autumn and winter, giving rise to heavy-precipitation events with a long duration, and a relatively cold climate in summer, which is not conducive to convection. Further inland, the heavy-precipitation events often occur in summer as a result of convection, which is in line with the relatively high return values for shorter durations. Furthermore, there is a degree of orographically forced precipitation along the mountain ranges. The influence of the distance to the ocean and temperature in summer (Fig. ) also supports this picture, with the curvature of the IDF curves decreasing with increasing t2mJJA and docean.

An advantage of the proposed method, applying PCA to the IDF data, is that all durations and return periods are considered together. This is not only computationally efficient but also reduces the influence of uncertain or erroneous individual return levels. There are also some potential pitfalls with this approach. First of all, only the first two PCs could be modelled, which could have too much of a smoothing effect on the IDF curves. There could also be nonlinear effects that were not captured by the linear models, which could result in underestimated variations in the estimated PCs. The quality of the estimated return levels was limited by the quality of the IDF data that they were based on. A different set of IDF statistics would likely result in statistical models with similar predictors and coefficients, but the PCA and ultimately the estimated IDF curves would be defined by the shape of the IDF data.

IDF statistics tend to have large uncertainties attached to them, as illustrated by the confidence intervals in Fig. . It can therefore be difficult to evaluate what constitutes a skilful estimation of return values. Although the IDF statistics used in this paper are referred to as the “original” return values, they too were estimated and not directly observed. Is it enough that the confidence intervals of the two estimates overlap? If so, the regionalisation approach presented here is “good enough”. On the other hand, the confidence intervals of the PCA-based estimates are for sure underestimated because they represent only the distribution of the Bayesian regression coefficients, without taking into account the uncertainties of the IDF statistics or climatological information that went into the modelling.

A more direct evaluation of model skill might be between the original return values (that were also obtained as median values of a distribution; see ) and the confidence intervals of the estimated return values. Using this metric, the statistical modelling was successful at most but not all stations. For the 24 h duration and 200-year return period, which is where the largest discrepancies occurred (Fig. b), there were five stations at which the original return values were outside of the confidence interval of the estimated return values (Fig. S15): four stations along the southern coast of Norway, at which the return values were underestimated compared to the original values, and one station on the west coast, where the return value is overestimated instead. A closer examination revealed that at two of the stations where the return values were underestimated (Grimstad – Hia and Time – Lye), the problem could be traced to the discarding of higher-order PCs, as PC4 had a notable influence on the shape of the IDF curves (not shown). At the other stations with large discrepancies, the explanation was not as easy to find but likely connected to the Bayesian regression.

Expanding IDF curves from gauged to ungauged locations or from current to future periods leans on the assumption that the transfer functions are stationary and the climate data that go into the analyses are representative of the location/period that they are associated with. This is not always true. If the reference period is shorter than the relevant cycles of natural climate variability or if there is a trend or large interannual variability, the outcome will likely be sensitive to the precise period of the input data. found significant changes in heavy precipitation in the Nordic–Baltic region and concluded that return level calculations were sensitive to the time period of available data. One strategy, suggested by , is to investigate and quantify the uncertainty related to this issue by repeating the analyses with input data from varying periods. To test the robustness of the proposed method, we applied the statistical models to climatological data from two different periods (1970–1995 and 1995–2020) at 36 stations with long, complete observational records. The results (Fig. S24) showed that for long durations, the estimated return values could differ as much as 10–20 mm at some stations depending on the reference period. This uncertainty should ideally have been included in the confidence intervals of the estimated IDF curves, but short observational records made it difficult to do this at most locations. In future studies, additional sources of information such as high-resolution gridded products may be helpful to fill in the gaps of the observational network in space and time.

Given that the IDF statistics in Norway are estimated for each duration separately and often based on short time series , the smoother shapes of the IDF curves estimated by PCA-based Bayesian regression may, at some stations, be more representative of the precipitation climate than the original return values which they are based on. Hence, the PCA-based approach with Bayesian inference can be regarded as another “tool” together with the traditional approach for estimating IDFs and the simple formula proposed by . They are based on different assumptions and have different strengths and weaknesses, and together they can capture salient information and smooth over errors from single sites, as the different approaches to predicting IDF curves are based on independent methods, for example, modelling the return values , estimating future IDF curves based on changes in the precipitation intensity , and downscaling rainfall intensity with respect to timescale .

As an alternative to PCA, we tried using a weighted set of polynomials to represent the shapes of the IDF curves, fitting return values against time intervals (Fig. S6). This method was not successful. First- and second-order polynomials were not a good fit, and higher-order polynomials, while fitting the data reasonably well for the time intervals for which return value data were available, had wiggly shapes with local minima and maxima that did not make sense as IDF curves.

One interesting question is how to use the IDF modelling approach presented here in the context of climate change projections. For the case of Norway, the key variables expected to change are μwarm, μcold, and t2mJJA. By downscaling them, statistically or dynamically, we may be able to infer changes in the IDF through predicting new values for the PCs, assuming that their shapes will be valid in a future climate and the calibrated dependency holds. One limitation is our ability to get reliable estimates for the wet-day mean precipitation in the future, which can be challenging in the case of empirical–statistical downscaling. We can also use this approach to provide hypothetical IDF curves for stress testing . Other caveats may be that these results only apply to Norway and that similar analyses for different regions may find that different factors are important for the shape of the IDF curves. We expect that the shapes of the IDF curves depend on physical aspects such as mesoscale convection, synoptic frontal systems and cyclones, orographic precipitation, and atmospheric rivers and that they will be sensitive to changes in their occurrence relative to each other.

We obtained predictions of the shape of IDF curves in Norway with Bayesian inference applied to a PCA representation of the IDF data and conclude that it provides a useful strategy that can be utilised for regionalisation and downscaling of future climate projections.