## 1 Introduction

[3] Seasonality is an important source of variation in many processes especially those in the natural world such as rainfall and has usually been incorporated into rainfall models in a deterministic way. Either the year is simply divided into a number of fixed parts (4 for seasonal resolution, 12 for monthly resolution) with each part being modelled individually so that the parameters are independent from part to part. Or, the parameters are allowed to vary continuously over the year as Fourier components. Both of these can be termed *deterministic* as the model parameters on any particular day of the year are the same for all years. Alternatively, the common experience of what can be described as, for example, “an early winter” or “a long summer” implies that seasons can start and/or finish at times of the year that vary from year-to-year. Such seasonality was proposed by [*Sansom and Thomson*, 2007] and can be termed *stochastic*.

[4] Although season onsets vary from year-to-year, in any particular year the onset of a season is likely to be contemporary across neighbouring rainfall stations within a region. A recommendation of the single-site high temporal resolution modelling in [*Sansom and Thomson*, 2010] was that stochastic season models should be extended to encompass regional multi-site networks over longer time periods using the more conventional, and abundant, daily rainfall accumulations. Carey-Smith et al. (manuscript in preparation, 2012) have adopted this strategy for including seasonality in a new stochastic model of daily rainfall which allows seasons to be earlier or later, and longer or shorter, than usual. This leads to increased rainfall variability over and above that explained by the standard fixed (deterministic) seasons.

[5] New Zealand rainfall shows relatively weak seasonality with significant rainfall occurring throughout the year and no markedly dry or wet seasons. This is illustrated in Figure 1 where monthly rainfall accumulations for Invercargill in the far south of New Zealand at 46°25'S, 168°20'E are presented. So transitions between homogeneous rainfall seasons will generally be more subtle, diffuse and difficult to determine than the marked changes experienced in, for example, monsoons. The weak rainfall seasonality reinforces the need to consider multi-site daily rainfall data over longer time periods as recommended by [*Sansom and Thomson*, 2010]. Furthermore despite its weakness in New Zealand, rainfall seasonality does exist and needs to be allowed for in models if they are to truly represent the behaviour of rainfall and provide valid simulations as required, for example, in risk analysis.

[6] The modelling and prediction of the onset of more definite seasons such as monsoons has been considered by a number of other authors. [*Stern*, 1982] using a simple Markov chain model for rainfall occurrence, considered wet season onset where this was defined as the first day when the daily rainfall, or two day total, exceeded a given threshold. More recently, [*Lima and Lall*, 2009] modelled multi-site rainfall occurrence using a Bayesian seasonal model with conventional Fourier components and season onset defined as the time when the posterior probability of rainfall occurrence exceeds 0.5. [*Slocum et al*., 2010] used cumulative rainfall anomalies to estimate the onsets of dry and wet seasons.

[7] Both the [*Sansom and Thomson*, 2010] and (Carey-Smith et al., manuscript in preparation, 2012) models are such that within each season, rainfall is modelled as a homogeneous hidden Markov model (HMM) with each season having its own dynamics, but common precipitation mechanisms. Thus, for example, a heavy rain episode in summer is stochastically equivalent to a heavy rain episode in winter (non-seasonal mechanisms), but the frequency and clustering of rainfall and dry episodes varies from season to season (seasonal dynamics). Furthermore, both depend on the specification of one day of the year for each season being in the particular season. So, if the seasonality is to be represented by four seasons then it is necessary to provide four dates on each of which it can be said that, every year, the season is of the first, second, third or fourth type respectively on that day of the year. The model fitting can only proceed once these dates, the mid-seasons, have been provided to a precision of days rather than weeks or months.

[8] The study presented in this paper is the estimation of these mid-seasons from the 55-year long rainfall records of daily data from 141 stations spread across New Zealand. Section 2 presents the data and general approach of analysing running accumulations on a regional basis. Section 3 is a more detailed explanation of the two stage method that was used to produce the results presented in Section 4. Any method applied to data will produce an answer, so Section 5 explores the physical significance of the results and Section 6 is a summary with some conclusions.