wgen

Asked to gemini:
In order to simulate daily rainfall in a Mediterranian climate I wish to have references to models which use some given data for previous times and produce plausible numbers.

Amongst the given data might be
  mean rainfall on a given day of the year  rm(n)  day n
  standard deviation of this                rsd(n)
  correlation between rainfalls on successive days at day n
I would be happy with the first few terms of a Fourier series in 2*pi*n/365 for rm(n) and rsd(n)

I want the model to capture that
Pr(rain on day n+1 | rain on day n) > Pr(rain on day n+1 | dry on day n)

I am not interested in the extreme variablity in the dry season as the model isfor a task related to winter crops.

The model that best fits your requirements for winter crop simulation in a Mediterranean climate is the 

Richardson-type Stochastic Weather Generator, exemplified by WGEN and its derivatives like LARS-WG. 

1. Model Structure (Two-Stage Process) 

This framework separates rainfall into occurrence and amount: 

• Occurrence: Modeled by a first-order, two-state Markov Chain. It explicitly satisfies your condition 

• Pr(rainn+1|rainn)>Pr(rainn+1|dryn)

• 𝑃𝑟(𝑟𝑎𝑖𝑛𝑛+1|𝑟𝑎𝑖𝑛𝑛)>𝑃𝑟(𝑟𝑎𝑖𝑛𝑛+1|𝑑𝑟𝑦𝑛) by using two transition probabilities:

  • P(W|W)

  • 𝑃(𝑊|𝑊): Probability of a wet day following a wet day (capturing persistence).

    1. P(W|D)

    2. 𝑃(𝑊|𝐷): Probability of a wet day following a dry day.

  • Amount: Modeled using a skewed distribution, typically the Gamma distribution or Mixed Exponential distribution. 

2. Parameterization with Fourier Series 

The seasonal variation of the model parameters is captured by fitting Fourier series (usually the first 1–3 harmonics) to represent the annual cycle. This allows you to derive daily values for: 

• Mean Rainfall  rm(n)

• 𝑟𝑚(𝑛)): The expected value on day n

• Standard Deviation (rsd(n)

• 𝑟𝑠𝑑(𝑛)): The variability on day n

• Transition Probabilities: P(W|W) and P(W|D) are also smoothed using Fourier coefficients to ensure continuous daily transitions. 

3. Key References for Implementation

• WGEN (Richardson and Wright, 1984): The foundational model for agricultural applications. It uses Fourier series to interpolate daily means and standard deviations.

• LARS-WG (Semenov et al., 1998): An advanced version that handles the specific persistence patterns (wet/dry spells) critical for Mediterranean winter crops more effectively than standard Markov chains.

• Woolhiser and Pegram (1979): A seminal paper specifically describing the use of Fourier series to fit the seasonal variation of parameters in daily precipitation models.

• RGENERATEPREC (R Package): A modern 2025/2026 toolset for R that implements these stochastic generation methods using Fourier series for seasonal parameters.