The present study uses a more extensive daily precipitation dataset than that considered by , from a network of 51 stations at locations shown in Fig. . These data are a subset of the MIDAS precipitation dataset . Records cover the period from 1923 to 2022, although few stations were operational during the first few decades of this period. Moreover, data on atmospheric covariates (see below) are only available from 1959 onwards: we therefore restrict attention to the period 1959–2022. The complete MIDAS precipitation dataset includes data from 171 stations that were operational at some time during this period; the 51 selected stations were identified based on extensive quality checks. Specifically, records were retained only if they were nominally operational for at least 10 years and had fewer than 10 % of values missing. Furthermore, since operational difficulties and data quality problems during a given period can often lead to anomalies in the recorded proportions of wet days at a site e.g., such anomalies were identified from the residuals of a two-way Analysis of Variance (ANOVA) fitted to the annual proportions of wet days at each location. Periods of dubious data quality identified from this exercise were also excluded from the subsequent analysis, as were several values that were (i) unusually large (ii) recorded on the last day of a calendar month and (iii) the only value recorded at a site during that month: these values were considered likely to be monthly totals that were included in the dataset erroneously.

The recording resolution of the data has changed over time, partly due to a change from imperial to metric measurement units in the early 1970s: most values prior to this are recorded to a resolution of around 0.3 mm, while the resolution is typically 0.1 mm for more recent data. Such inconsistencies can give rise to spurious trends in rates of precipitation occurrence in particular, which can also be sensitive to inter-site differences in observer practice when recording small non-zero precipitation amounts “trace values”;. To reduce the effects of such issues, one option is to treat all non-zero values below some threshold τ≥ 0.3 mm as being known only to lie in the range  (0,τ) so that the exact recorded values are not used: this can be considered as a problem of censored data, as discussed in more detail by who also provide reasons for not adopting such an approach. Instead, we follow in transforming the original values {ỹ} to y=max⁡(ỹ-τ,0) where the threshold τ is set to 0.5 mm (here and subsequently, lower- and upper-case letters, respectively denote observed values and the associated random variables). This choice of threshold is hydrologically insignificant, in the sense that evapotranspiration rates in this area are well in excess of 0.5 mmd-1 in almost all months of the year so that precipitation amounts below this can be considered as effectively zero: see, for example, Fig. 1 of , noting that the rates shown there are in units of mm per week.

After fitting models to the transformed values, synthetic precipitation sequences on the original scale can be produced by first simulating from the fitted model and then adding τ to any non-zero simulated value. The resulting sequences will contain no values between 0 and τ; in most applications, however, this is unproblematic providing τ is small enough to be hydrologically insignificant as discussed above.

In the present application, the thresholding of trace values is designed in part to mitigate the effect of changes in recording resolution on precipitation occurrence. Its use to account for inconsistent recording practices is widespread more generally, however. For example, resolution changes may also affect the analysis of precipitation intensity, albeit to a lesser extent. One approach for mitigating this effect is to round the data to a common resolution prior to analysis e.g.. As noted in Sect. , however, and demonstrated via simulation in Appendix , we find that such rounding can have a surprisingly large impact on fitted GLMs and other models for precipitation intensity, with particular sensitivity in the upper tails of the distribution. For most of the analyses reported below therefore, the change in recording resolution is handled not by rounding the data, but rather by including appropriate adjustments within the fitted models.

In many applications, the effects of potential changes in climate must be incorporated into synthetic precipitation sequences. This is typically achieved by conditioning on indices of large-scale atmospheric structure that are known (i) to be related to local-scale precipitation; (ii) to capture the relevant climate change signal; and (iii) to be well represented by climate models such as atmosphere-ocean global circulation models (GCMs) e.g.Sect. 11.5. Generators can therefore be calibrated using historical information on large-scale covariates, after which “future” simulations can be produced by replacing the historical covariate information with the corresponding outputs from climate model simulations. Precipitation generators that incorporate such conditioning are among the more sophisticated tools available for statistical downscaling of climate model outputs .

In the work reported below, the required conditioning is obtained by including atmospheric indices as covariates in the statistical models. The covariates are all derived from the ERA5 Reanalysis dataset , averaged daily and over all grid cells with centres contained within the study area. The indices considered are 2 m temperature, 2 m dewpoint temperature, standardized mean sea level pressure as well as 10 m wind speed, derived from the 10 m u- and v-wind components. These indices are similar to those in previous studies , and consistent with guidance on covariate selection for precipitation downscaling which emphasises the importance of indices representing moisture availability and airflow e.g..

As noted in Sect. , the standard GLM-based approach treats occurrence and intensity separately. Specifically, if the random variable Yi represents the (possibly thresholded – see above) precipitation intensity for the ith case in a dataset, then the probability of a “wet” day is modelled using logistic regression: 1P(Yi>0)=πi, say,with log⁡πi1-πi=xi(π)β(π)=ηi(π),say.

Here, xi(π) is a row vector of covariates and β(π) is a corresponding column vector of coefficients: given these values, the probability of precipitation can be calculated as πi=[1+exp⁡(-ηi(π))]-1.

If Yi>0 (corresponding to a “wet” day), then the precipitation intensity is modelled as being drawn from a gamma distribution with mean μi (say) and shape parameter ψ. The mean is linked to another covariate vector xi(μ), as 2log⁡μi=xi(μ)β(μ)=ηi(μ), say, so that μi=exp⁡[ηi(μ)]. The shape parameter ψ is not linked to covariates, however: it is common to all cases, which implies that the standard deviations ({σi}, say) of the distributions are proportional to their means with σi=μi/ψ. This is analogous to the assumption of a constant residual variance in linear regression models.

The covariate vectors xi(π) and xi(μ) in Eqs. () and () need not be the same, although both of them typically include a “1” representing a constant term (so that the corresponding elements of β(π) and β(μ) are analogous to the intercepts in linear regression models), as well as quantities representing seasonal and regional variation, transformations of previous days' precipitation (accounting for temporal dependence in precipitation sequences) and indices of large-scale atmospheric structure. Interaction terms can also be included in situations where some covariates may modulate the effects of others – for example, in the case study of Sect.  one might reasonably anticipate temporal dependence to be weaker in summer than in winter, due to the seasonally-varying proportion of convective versus frontal precipitation events.

The coefficient vectors β(π) and β(μ) can be estimated from a dataset y:=(y1…yn)T, containing daily precipitation observations together with the corresponding covariate vectors {xi(π)} and {xi(μ)}. This estimation is typically done using maximum likelihood, under the assumption that the associated random variables {Yi} are conditionally independent of each other given the covariates. In this case, standard theory (and software output) also provides estimated standard errors for the coefficient estimates: these provide a way to determine whether the effect of each covariate is estimated sufficiently precisely to justify its inclusion in a model. By placing priors on the model parameters, Bayesian estimation might also be possible, as for example in . This potentially allows parameter uncertainty to be propagated into the synthetic sequences produced using the fitted models. However, given the large datasets used for estimation, the effect of parameter uncertainty is generally fairly small, particularly compared to the daily precipitation variability of interest.

To choose covariates, an alternative to the inspection of standard errors is to add additional terms to an existing model and determine whether they lead to a sufficiently large increase in log-likelihood. This can be done either using a likelihood ratio test or by choosing the model with the smallest value of a quantity such as the Akaike Information Criterion (AIC). For a model with generic coefficient vector θ:=(θ1…θk)T, AIC is defined as 3AIC=-2log⁡L(θ^y;y)+2k, where log⁡L(θ^y;y) is the log-likelihood constructed using the data y and evaluated at the maximum likelihood estimate, θ^y say, of θ (the reason for the notation will become clear below). By this criterion, a “good” model is one with a high log-likelihood (and hence a good fit to the data) despite having a small number k of parameters: for further justification of the precise definition, see the Supplement.

The use of likelihood ratio tests or AIC-type comparisons allows a structured approach to model-building in which simple models are gradually extended by adding related groups of covariates, at each stage evaluating the expanded model using a combination of formal comparisons and less formal model diagnostics, such as residual plots, that are designed to assess whether the modelling assumptions are broadly satisfied: see . However, the associated calculations are valid only under the conditional independence assumption underpinning the calculation of the (log-)likelihoods themselves. For the analysis of data from a single spatial location this assumption can be justified providing the covariate vectors contain appropriate information on previous days' precipitation values , but it is usually unrealistic in a multisite context: in the case study of Sect.  for example, large-scale frontal weather systems are likely to affect most or all sites simultaneously. To account for the resulting dependence, in a multisite context the standard errors and likelihood ratio tests can be adjusted as described in . The AIC can be adjusted similarly (see the Supplement), as 4AICadj=2-log⁡L(θ^y;y)+ trace (GH-1), where H is the negative Hessian matrix of the log-likelihood at its maximum and G is the covariance matrix of the gradient vector there: both of these quantities also feature in the adjustments to standard errors and likelihood ratio tests, and can be estimated straightforwardly as by-products of the usual algorithm for maximising the log-likelihood for a GLM. The quantity Eq. () is sometimes called the network information criterion or NIC e.g.Sect. 4.7. In passing, we note that these adjustments to standard errors, likelihood ratio tests, and AIC can also be applied in the absence of inter-site dependence and, in this case, all reduce to the “usual” definitions that apply when the log-likelihood is specified correctly because G=H in this case.

Having selected appropriate sets of covariates, estimated the coefficient vectors β(π) and β(μ) and carried out diagnostic checks on the fitted model, synthetic precipitation sequences can be generated one day at a time, first by sampling the wet/dry status of each site according to the probabilities defined by the occurrence model Eq. () and then, for wet sites, sampling their intensities from the corresponding gamma distributions. In a multisite setting, the generation of realistic sequences requires that the sampling is done in such a way as to respect the inter-site dependence structure: for example, on days when one location experiences precipitation, it is likely that neighbouring locations will also be wet. Inter-site dependence in occurrence and intensity is usually handled separately. For occurrence, modelled the distribution of the number of wet sites (on the grounds that in a small region the sites are usually either all wet or all dry) in such a way as to be consistent with the wet-day probabilities determined by the GLM; while, for application to larger regions, used inter-site dependence models based on thresholded latent Gaussian fields, which allow for a decay in the strength of dependence with intersite distance but can be challenging to estimate when the dependence is very strong. Other approaches also exist: for example, use a non-homogeneous hidden Markov model, accounting for spatial dependence directly in the model. Inter-site dependence in intensities is more straightforward and, for GLM-based approaches, is usually achieved via a Gaussian geostatistical model for the Anscombe residuals which are defined in such a way as to have a distribution that is very nearly Gaussian: this approach can also be regarded as corresponding to the use of (approximate) Gaussian copulas . Approaches based on latent multivariate Gaussians or latent Gaussian fields have also been used, for example, in .

There are two main limitations to GLM-based models for daily precipitation. These are: first, the restriction to linear combinations of covariates in equations Eqs. () and (); and second, the assumption of the constant shape parameter ψ in the gamma distributions for precipitation intensity. The GAMLSS framework of can potentially address both of these issues. A summary is provided here: and give further details.

In GAMLSS, the linearity assumption is relaxed by allowing smooth functions of covariates as well as linear combinations. Thus the respective equivalents of Eqs. () and () are, in their simplest forms, 5log⁡πi1-πi=xi(π,β)β(π)+s1(π)xi(π,1)+…+sMπ(π)xi(π,Mπ) and 6log⁡μi=xi(μ,β)β(μ)+s1(μ)xi(μ,1)+…+sMμ(μ)xiμ,Mμ.

In each of these expressions, xi(∗,β) is a row vector of covariates with effects that are represented linearly as before, with coefficient vector β(∗) (the asterisk “^∗” denoting either “π” or “μ” as appropriate). The terms s1(∗)(xi(∗,1)),…,sM∗(∗)(xi(∗,M∗)) are smooth functions of the covariates xi(∗,1),…,xi(∗,M∗), which are typically represented semiparametrically using flexible collections of spline basis functions that allow for the data-driven representation of nonlinear effects. Various families of spline basis are available, although our experience is consistent with the received wisdom that the precise choice is relatively unimportant. With one exception, the results reported below use cubic splines Chapter 5 as implemented in the gamlss and bamlss packages in the R programming environment . The exception is the representation of seasonal effects, for which cyclic splines are used to represent the repeating seasonal cycle – again, as implemented in the cited software.

As written, Eqs. () and () do not allow explicitly for smooth functions of more than one covariate, or for situations in which the linear coefficient of one covariate (i.e. the corresponding element of β(∗)) varies smoothly with the value of another so that (e.g.) βj(∗):=βj(∗)(xk) in an obvious notation. The representation of systematic regional variation is a canonical situation in which one might wish to consider two covariates (i.e. latitude and longitude) simultaneously: in this case, a bivariate spline basis of the two covariates can be specified, either as a tensor product or as a thin-plate spline Chapter 5, the latter of which is used in this work. These constructions are all implemented in the gamlss and bamlss software packages, as are the mixed linear-spline terms required for situations where βj(∗) varies smoothly with xk.

To relax the assumption of a constant shape parameter, GAMLSS allow both the mean and standard deviation (as well as, in principle, the skewness and kurtosis) of a distribution to depend explicitly on covariates. Thus, instead of the simple relationship σi=μi/ψ in a gamma GLM for daily precipitation intensity, GAMLSS represent the standard deviations as 7log⁡σi=xi(σ,β)β(σ)+s1(σ)xi(σ,1)+…+sMσ(σ)xi(σ,Mσ).

In the work reported here, the gamma distributional assumption for wet-day rainfall intensities is retained so that the implied shape parameter for the ith case in the dataset is ψi=μi2/σi2. The complete structure of the daily precipitation sequence at a single location is therefore determined by the three equations Eqs. ()–().

The increased flexibility of GAMLSS compared with GLMs comes at a price: given sufficiently rich collections of splines, almost any model can be made to fit a dataset almost perfectly, simply by specifying very “wiggly” functions {sj(∗)(⋅)} that (almost) interpolate the observations. Techniques such as maximum likelihood estimation therefore tend to yield model fits that are over-optimised to the data at hand and, consequently, are unsuitable for exploring potential variation beyond what has been observed (which, fundamentally, is the purpose of a precipitation generator). To overcome this, GAMLSS are usually fitted by maximising a penalised log-likelihood of the form 8log⁡L(θ^y;y)-P:=log⁡LPEN(θ^y;y), say, in which the “penalty” P is typically constructed using the integrated squared second derivatives of the functions {sj(∗)(⋅)}. The rationale for this is that the second derivatives measure curvature: wiggly functions have large integrated squared second derivatives, so that the penalty discourages wiggly model fits.

Equation () represents a compromise in model fitting: the log⁡L(θ^y;y) term encourages fidelity to the available observations, while the penalty P rewards models in which the functions {sj(∗)(⋅)} are smooth. The tradeoff between these two criteria is determined via smoothing parameters for each of the {sj(∗)(⋅)} which, in principle, must be chosen by the analyst. In practice, however, most modern software packages (including the gamlss and bamlss packages used here) incorporate automated, data-driven smoothing parameter selection, essentially by treating the smoothing parameters as auxiliary quantities to be estimated as described in .

GAMLSS do not have to include semiparametric terms: parametric GAMLSS for precipitation intensity would retain just the linear components of Eqs. () and (). In this case, as for GLMs, model selection can be done using likelihood ratio tests and AIC-type comparisons, with AICadj (defined in Eq. ) used to account for inter-site dependence where necessary.

For semiparametric models, however, the situation is more complicated. This is partly because AIC requires that models are fitted by maximising a log-likelihood rather than a penalised log-likelihood – although, in principle (see Supplement), a criterion such as AICadj is applicable more broadly. In practice, however, the matrices G and H in Eq. () must be estimated from the data and, if they are large, sampling errors in their estimation can compromise the accuracy of any associated inference: give an example of this effect. Semiparametric GAMLSS usually employ large numbers of spline basis functions for each smooth term. For these models therefore, the corresponding matrices G and H are themselves large and Eq. () cannot be estimated accurately.

For the semi-parametric GAMLSS we therefore use an alternative model selection criterion, which avoids this difficulty: the WIC . The underlying motivation is the same as that for AIC and AICadj, but WIC uses resampling and does not depend on poorly-estimated quantities. Specifically, let {y∗} be a collection of resampled datasets that can each be regarded as drawn from the same joint distribution as the observations y (see below). Then, in the current context, WIC is defined as 9WIC=-2log⁡LPEN(θ^y;y)+2log⁡L‾PENθ^y∗;y∗-log⁡L‾PENθ^y∗;y, where log⁡L‾PEN(⋅;⋅) denotes an average of penalised log-likelihoods computed over the resampled datasets.

In Eq. (), the “adjustment” in square brackets is a difference of two terms. The first is an average of maximised penalised log-likelihoods for each of the resampled datasets; the second, however, is an average of penalised log-likelihoods for the observations y, but evaluated at each of the resample-based estimates θ^y∗. The WIC thus aims to compensate for the extent to which the penalised log-likelihood log⁡LPEN(θ^y;y) is likely to be overoptimistic as a performance measure in alternative datasets generated from the same distribution as the observations y. The Supplement gives further details.

The WIC requires that the resampled datasets {y∗} can be regarded as drawn from the same distribution as the observations y. Standard bootstrap resampling – sampling individual observations with replacement – is not appropriate for multisite precipitation data, because it destroys the inter-site and temporal dependence structure present in each day's observations. Thus alternative resampling approaches are required: Appendix  describes the approach used here. Broadly the idea is to resample individual days instead of individual observations, whilst using additional stratification and adjustments in order to account for covariate dependence, handle missing observations and to ensure that resampled datasets have the same structure as the original one: specifically ensuring that both the full datasets and their dry and wet subsets contain the same number of observations and days with recorded values as the original dataset.

Finally, other resampling-based model selection criteria have been proposed, for example, by whose criterion is asymptotically equivalent to the WIC . The work reported below uses WIC; we have also examined the Cavanaugh-Shumway criterion and found that it yields very similar conclusions.

GAMLSS can be used to produce synthetic multisite precipitation sequences in the same way as was described for GLMs in Sect. , using models Eqs. ()–() to determine the daily occurrence probabilities and intensity distributions and with similar adjustments for capturing inter-site dependence. However, as outlined in Sect. , there is potential to improve on these options by considering dependence in occurrence and intensities simultaneously when simulating multisite precipitation.

To address this issue, in a similar spirit to we use transformed Gaussian fields or, equivalently, Gaussian copulas . Denoting by Yt:=(Y1t…YSt)T the random vector representing precipitation at S sites on day t of a simulation, the elements of Yt are constructed from a corresponding vector Qt:=(Q1t…QSt) of standard normal random variables with covariance matrix Σ, as 10Yst=0if Φ(Qst)<1-πstFst-1Φ(Qst)-(1-πst)πstotherwise.

Here, πst is the probability of non-zero precipitation as obtained from the occurrence model Eq. (); Fst-1(⋅) is the inverse cumulative distribution function (CDF) of the gamma distribution for wet-day amounts defined by Eqs. () and (); and Φ(⋅) is the CDF of the standard normal distribution. It is easily verified that under Eq. (), P(Yst>0)=πst and P(Yst≤y|Yst>0)=Fst(y) so that the procedure yields precipitation with the correct marginal (i.e. site-specific) distributions at each site.

The first line of Eq. () – yielding a simulated dry day – corresponds to a situation in which Qst is censored, meaning that its precise value is unknown: it is only known not to exceed Φ-1(1-πst). The {Qst} can be regarded as normalised quantile residuals , censored on dry days. There is, moreover, a connection with existing representations of inter-site dependence in precipitation intensities: if πst=1 so that the first condition in Eq. () is never met, then the transformation in the second line is similar in spirit to the use of transformed Anscombe residuals for this purpose by . The current proposal can therefore be seen as a natural extension of, and a way of linking, apparently distinct approaches from the literature.

Under Eq. (), inter-site dependence in the precipitation Yt is induced by the covariance matrix Σ of the Gaussian random vector Qt. In particular, if Σ is the identity matrix, then Qt contains independent standard normal random variables, and Eq. () produces independent precipitation values from the combined occurrence/intensity model at each site. To generate realistic multi-site precipitation sequences, however, non-zero off-diagonal elements of the matrix Σ must usually be estimated from the data y. Following standard practice for GLM-based precipitation generators, this is done after fitting the marginal occurrence and intensity models Eqs. ()–(): thus, when estimating Σ, the quantities {πst}, {μst} and {σst} are considered known.

In most situations involving more than a few sites, estimation of Σ is complicated by the fact that there are typically few, if any, periods during which data from all sites are simultaneously available. This can be handled e.g. by estimating correlations separately for each pair of sites using all dates for which both are operational. The resulting matrix of inter-site correlations is not guaranteed to be positive definite however: to address this, we subsequently fit a spatial correlation model to the pairwise estimates. This ensures a valid correlation matrix, and also provides correlations for pairs of locations with non-overlapping records or that are ungauged. We use a Matérn correlation function for this purpose, due to its flexibility in capturing a wide range of correlation behaviours . According to this model, the residual correlation between two locations separated by a distance d>0 is 11Cν(d)=α21-νΓ(ν)2νdρ2Kν2νdρ, where Γ(⋅) is the Gamma function and Kν(⋅) the modified Bessel function of the second kind. The form of the correlations is determined by the two parameters ν, ρ, and α, respectively controlling the smoothness of the residual fields, the distance-dependent rate of correlation decay, and the “nugget” effect representing the limiting correlation at small non-zero distances . In the work reported below, the parameters are estimated from the pairwise inter-site correlations using weighted least squares, weighting the correlation between each pair of sites according to the number of days' data from which it was calculated. This weighting ensures that the correlation model fit is not influenced unduly by pairwise correlations that are estimated imprecisely.

For schemes that treat dependence in occurrence and intensities separately, separate correlation models are needed for each component. For intensities, the pairwise correlations to which Eq. () is fitted can be calculated directly from the relevant (transformed) residuals e.g.; while those for occurrence can be inferred from the proportion of days for which both sites are simultaneously wet e.g.. For the combined occurrence/intensity scheme Eq. (), however, a single correlation model is needed, and neither of these approaches is feasible, because there are invariably days for which one site in a pair is wet while the other is dry. We thus estimate the pairwise correlations using maximum likelihood as described in Appendix : although this requires numerical optimisation for each pair of sites, it is a principled approach that is generally applicable and can in theory extend beyond bivariate estimation.

Having estimated the correlation matrix Σ, synthetic multisite precipitation sequences can be generated one day at a time: on day t of simulation a vector Qt of standard normal variates with correlation matrix Σ is simulated, and then converted to a corresponding precipitation vector Yt via the transformation Eq. ().

Each model is constructed using a structured approach . The parametric GLM and semiparametric GAM are constructed independently of each other, in each case starting with a simple baseline model and successively adding terms in perceived order of importance; see Appendix . The final GLM and GAM for precipitation intensity are then taken as starting points for the modelling of intensity standard deviations in the respective GAMLSS. Throughout, multiple effect forms (i.e. parametric forms and/or spline specifications) for potential covariates as well as interactions (see Sect. ) are tested; and alternative formulations are compared using the adjusted AIC () and WIC (), respectively for the parametric and semiparametric models. For the semi-parametric models always both parametric and spline-based effects are tested. To represent seasonal variation, sine and cosine functions of the day of the year are used in the parametric models; and cyclic splines in the semiparametric ones. Systematic regional variation is captured using functions of latitude, longitude, and altitude. For the parametric models, latitude–longitude effects are represented by Legendre polynomials; for the semiparametric models, a thin-plate bivariate spline in latitude and longitude is used instead. Following , the change in measurement resolution (see Sect. ) is accounted for via a binary covariate taking the value 0 for all observations before 1970 and 1 for all subsequent observations: the effect is to provide an adjustment within the models for any systematic changes arising from the coarser resolution of the data in the early part of the record, thus avoiding the need for data preprocessing that may have unintended consequences (see Appendix ).

After model selection, standard checks on residuals Chapter 12 are carried out as an initial assessment that the models appear to capture most of the systematic structure in the data, before carrying out more extensive tests via simulation (see next section). Finally, for each of the four options (GLM, GAM, parametric and semiparametric GAMLSS) a residual correlation matrix Σ is estimated as described in Sect. , for use in the inter-site dependence model Eq. ().

The performance of each precipitation generator is assessed by comparing statistics of the observed data over the period 1959–2022 with those for each of 19 simulated sequences over the same period. The number of simulations is limited by the computation time for the GAMLSS as implemented in the bamlss package: this is discussed further in Sect.  below. Each statistic is calculated separately for every simulated sequence, thus yielding a distribution of 19 values: if the precipitation generator is realistic, then the corresponding statistic computed from observations should lie within the range of this distribution 90 % of the time on average, since it has a 1/20 chance of being less (resp. greater) than the minimum (resp. maximum) from the simulations.

Each simulation is initialised with the first three observed days at each station. Simulations are carried out only on days and at stations where observations are available. When several consecutive observations are missing for a station, the simulation is reinitialised once new observed values become available, using these as the new lagged inputs. This guarantees a like-for-like comparison of simulations and observations. Much of the subsequent analysis will focus on the subset of 16 stations that report through most of the study period. The observations used for initialisation are not used in the comparison exercise, as the simulations contain no corresponding values.

For each month of the year, Figs.  and  compare the observed values of several key summary statistics with those derived from simulations of the parametric GLM and semiparametric GAMLSS (similar figures for the parametric GAMLSS and semi-parametric GAM can be found in the Supplement). These statistics all relate to the daily precipitation series averaged over the 16 stations reporting through most of the 1959–2022 period.

In both figures, most of the observed statistics fall within the 90 % range of the simulations: this indicates good performance of both the occurrence and intensity models in the parametric and semi-parametric case. Results for the parametric GAMLSS and semi-parametric GAM are similar (see Supplement). The overall mean is well captured across all models, however, there is some indication that the conditional (i.e. wet-day) mean might be underestimated. The proportion of wet days and the standard deviation statistics (both overall, as well as conditional) seem better captured by the semiparametric GAMLSS, compared to the parametric models. For the proportion of wet days, this is most likely due to the more flexible representation of covariate effects in the GAMLSS framework, since the GAMLSS-based modelling of standard deviations is not used in the occurrence model. By contrast, the improvement in representing the standard deviations is also seen for the parametric GAMLSS (see Supplement), whence this improvement probably comes from the ability to model variation in the shape parameter of the gamma distributions. In the Supplement (Sect. S4 in the Supplement) we further analyse the persistence of dry and wet spells which we find is well reproduced by all four precipitation generators.

Next, we focus on the generators' ability to reproduce the distribution of daily precipitation values, both overall and by season. This is done via quantile-quantile plots, in which the ordered values from each simulation (from smallest to largest) are plotted against those from the observations as in Fig. . Here, the daily values from all sites throughout the study period are pooled, and each panel shows the results from a location-only model (i.e. a GLM or a GAM) and a location–scale GAMLSS: covariate effects are represented parametrically in Fig. a but semiparametrically in Fig. b, and each vertical line indicates the range of values across the 19 simulations.

Figure  shows that overall, all four generators capture the rainfall distribution well: the ranges of ordered values from the simulations encompass the line of equality, and the medians of the 19 simulated values at each point fall close to this line. The ranges of simulated values are broadly similar for precipitation amounts up to around 50 mm (this is around the 99.96th percentile of the observed distribution); but there are larger differences in the upper tail of the distribution, where the GAMLSS yield less between-run variability, whilst still encompassing the line of equality. These results are also consistent when aggregating precipitation on a weekly scale (not shown).

The good reproduction of the overall precipitation distribution in Fig.  is consistent with the GLM-based results of , who nonetheless found systematic differences between observed and simulated distributions in individual seasons. To investigate this further, therefore, Fig.  compares the observed and simulated precipitation distributions from the two parametric precipitation generators (GLM and parametric scale-location GAMLSS) by season. Here, both sets of simulations tend systematically to overestimate the upper tail of the precipitation distribution in winter and, to a lesser extent, to underestimate in summer. The GLM-based result here is similar to that reported by . Some improvement can be seen for the GAMLSS, where in particular in DJF the amount of overestimation is reduced in some simulations. Nonetheless, the improvement appears relatively small since the medians of the simulated quantiles are similar in the GLM and the GAMLSS.

Figure  shows the corresponding plots for the semi-parametric models. The results are broadly comparable with those in Fig. . The semiparametric GAMLSS simulations show again some improvement in Summer and Winter, however, the overall biases are still present. This overall pattern also persists when focusing on individual months rather than entire season: see for example Fig. S3 in the Supplement showing results for January in which, as with the winter comparisons above, the simulations tend to overestimate the upper tail of the distribution; this is slightly less pronounced for the semiparametric GAMLSS than for the other three generators, however.

Taken together, these results suggest that modest improvements in the representation of seasonal precipitation tails can be gained by relaxing the assumption of a constant shape parameter although this does not fully resolve the issue. The more flexible representation of potentially nonlinear covariate effects compared with a standard GLM does not seem to improve tail behaviour much. We note, however, that whilst seasonal tail deviations are apparent in the pooled distribution of single-site rainfall studied above, no such deviation is seen in the distribution of catchment-averaged rainfall for any of the models (see Figs. S4 and S5 in the Supplement). This indicates that for most hydrological applications all weather generator models studied here will have satisfactory performance in terms of extremes, and stands in contrast to the result in who found seasonal tail deviations in the time series of average rainfall for GLM-based models. This is probably due either to different treatment of recording resolution (see Appendix ) or to the use of different atmospheric covariates: this work uses ERA5-derived variables, whilst used an index of the North Atlantic Oscillation (NAO).

An alternative way to characterise the upper tail behaviour is via extreme value theory , which is often used to study the expected frequency of rare events for purposes such as flood risk assessment. In particular, a common summary of potential extremal behaviour is the shape parameter, often denoted as ξ, of a generalized extreme value (GEV) distribution fitted to a collection of annual maxima. If ξ>0, as often reported for daily precipitation e.g., then the distribution of extremes is heavy-tailed which implies a potential for as-yet-unobserved extreme events to greatly exceed historical values, whereas if ξ=0 then the distribution is light-tailed, and if ξ<0 then the distribution has a finite upper bound.

To investigate the extremal behaviour of each precipitation generator over the period 1959–2022, the extRemes package is used to fit GEV distributions to the annual maxima of selected time series derived from each simulation: for each series, the mean and standard deviation of the 19 resulting estimates of ξ are compared with the estimate and standard error derived from the corresponding observations. This standard error is based on large-sample theory and aims to approximate the standard deviation of estimates obtained under repeated sampling: thus it should be comparable with the standard deviation of the simulation-based estimates. We follow the advice of and standardize the annual maxima prior to fitting: this does not affect the shape of the distribution.

Table  reports the results of this exercise for a representative set of time series: for four individual stations, and for the 16-site average. The observation- and simulation-derived estimates are all in good agreement: in particular, the point estimates of ξ are (slightly) positive for most of the considered series. This confirms that the extremal behaviours of the simulations and observations are qualitatively similar, despite the generators not being calibrated directly to reproduce such behaviour.

In their original study of precipitation in this catchment, found that inter-site dependence was extremely strong due to the relatively small size of the area. It is challenging to develop simulation strategies that can capture this, particularly when separate dependence models are used for precipitation occurrence and intensity: correlation-based dependence structures for occurrence are hard to identify in this case, because sites are typically either all wet or all dry on most days. To address this, introduced an indirect approach for capturing dependence in precipitation occurrence, based on the distribution of the number of wet sites on each day. It is possible, however, that correlation-based dependence structures can be used for small catchments by modelling dependence in occurrence and intensity simultaneously (e.g. via Eq. ) because, in this case, correlations learned predominantly from precipitation intensity are implicitly shared with the occurrence model.

To investigate this, we here evaluate the inter-site dependence structure in our simulations. The results presented are produced using the semiparametric GAMLSS; results obtained from the other models are not shown but are similar.