1. Rainfall generator Whollstoptherain2
Gerrit de Rooij, February 2018

2. Stochastic rainfall generation: principles and parameters
The models presented here are four variations of the modified Bartlett-Lewis model introduced by Rodriguez-Iturbe et al. (1988). The key features of this type of models are
Storms are periods during which rainfall can occur. Their arrival times and durations are random. Storms may overlap.
The rainfall in a storm arrives in rectangular cells of constant rainfall rate. Rate, starting time, and duration are random. Cells too may overlap.
The rainfall rate at a given time is the sum of the rainfall rates of all active cells at that time.
The last cell in a storm is the last cell that starts within the duration of the storm. It is allowed to end after the end time of the storm to which it belongs and can overlap with cells in later storms.

3. Stochastic rainfall generation: principles and parameters
The parameters of the various distributions vary over the year. The last storm generated for a given period will start in a later period. Its parameters are drawn from distributions valid for the period in which it occurs.
The differences between the models address weaknesses in previous versions. The original Bartlett-Lewis model was too static and is not included in the code. The modified BL model (MBL) and its three refinements all generate rain storms by a Poisson process. This leads to exponentially distributed intervals between storm starting times. The distribution has a single parameter: λ [T-1].(Input parameters are red.)
Generating random variates requires a simple transformation of the standard uniform variate U:

4. Modified Bartlett-Lewis model (MBL)
The original BL model used the same pdf to generate cell durations for all storms. This led to incorrect probabilities of zero rainfall. The MBL therefore has a Gamma-distributed variate η [T-1] that governs the duration of rain cells, the duration of the storms and the time intervals between cells within a storm:
η has a shape parameter α and a rate parameter ν [T] (incorrectly labeled scale parameter in the literature).
where
Generating the gamma distributed variate is elaborate and beyond the scope of this presentation.
η is the parameter of the exponential distribution governing the cell duration

5. Modified Bartlett-Lewis model (MBL)
The duration of a storm and the intervals between arrival times of cells in a storm are exponentially distributed with parameter γ and  [T1], respectively.
Parameters γ and  are related to η through dimensionless parameters  and :
This relationship between storm duration, cell distribution, and cell duration creates a smooth consistent spectrum between frontal systems with long rain periods (long storms with cells) and convective systems with short storms and short showers.
The rainfall rate of a cell is exponentially distributed. The average rainfall rate is given by μx [LT-1]. The pdf thus becomes:

6. MBL with Gamma-distributed rainfall rates: MBLG
MBL does not perform so well in reproducing extreme rainfall: the exponential distribution of μx apparently is too limited.
The Gamma distribution can have a much wider range of shapes than the exponential distribution. In fact, the exponential distribution is a special case of the Gamma distribution. Allowing the rainfall rate to be Gamma distributed thus adds flexibility at the expense of one extra parameter. Instead of μx, we have a shape parameter p and a rate parameter δ [L-1] (again incorrectly called a scale parameter). The MBL with Gamma-distributed rainfall rates is labeled MBLG.

7. MBL and MBLG with truncated Gamma distributions: TBL and TBLG
The moments of the Gamma distribution can become infinite for values of the shape parameter α that are quite commonly found when MBL and MBLG are fitted to existing rainfall records. This leads to rare occurrences of unrealistically long cells, which is a problem if rainfall records of several decades are generated.
This can be avoided if small values of η are not allowed. Thus, a small truncation value ε for η is specified. Any value of η that is generated that does not exceed ε is rejected and a new value needs to be produced. The versions of MBL and MBLG that use these truncated Gamma distributions are labeled TBL and TBLG, respectively.
If non-truncated model versions are preferred (i.e., MBL or MBLG), ε simply needs to be set to zero in the input file.

8. Input parameters required to generate rainfall records
In the above, we identified the following parameters that are needed to generate storms and cells:
λ [T-1] α
ν [T] κ φ
μx [LT-1] or p
δ [TL-1]
ε [T-1] > zero for truncated Gamma distributions
A label given on input informs the code whether the rainfall rate is exponentially or gamma distributed. Depending on that, the input records with parameters have an entry for μx or for p and δ.

9. Physical meaning of the input parameters
λ-1 is the average time interval between the arrival of subsequent storms.
η-1 is the mean duration of a rain cell. For MBL and MBLG, η-1 = να-1.
γ-1 = (φη)-1 is the mean duration of a storm.
β-1 = (κη)-1 is the mean cell inter-arrival time.
μx is the mean rainfall rate of the cells. For MBLG and TBLG, μx= pδ-1.
ε is the smallest permitted value of η. If fcrit is the exceedance probability of a critical duration tcrit of a rain cell, ε = − ln(fcrit)/ tcrit.

10. Physical meaning of the input parameters
For climate change scenarios this gives a convenient set of possibilities to tune the generated rainfall. The parameters of the two gamma distributions and ε are a bit tricky.
For TBL and TBLG, an approximate relationship between α, ν and ε on one hand and η on the other exists, but it is too complicated for practical use. For ε small enough, η-1 = να-1 might work as a (first) approximation.

11. Input requirements of Whollstoptherain
Typically, the parameters are given for each month, requiring twelve records of input parameters if the input label informing the code that the input is for months equals ‚Yes‘. Alternatively, the code also permits to specify the parameters for arbitrary periods within the year. In that case, the number of periods is required, and the data record for each period includes its starting date (for a non-leap year).
The random generator in Fortran needs a large arbitrary integer number (seed) to initialize it. This too needs to be specified.
The number of years for which a rainfall record needs to be generated is required. Because the program takes into account leap years, this number is rounded up internally to the nearest multiple of four. Every fourth year of the simulation period is a leap year. The code determines internally which parameters are valid for February 29th.

12. Input requirements of Whollstoptherain
N.B. The rainfall community normally uses the hour as the preferred unit of time. Soil hydrologists often prefer the day as the unit of time. The code requires the unit of time to be the day, and accepts arbitrary units of length. Some variables need to be converted if they are indeed based on the hour as the time unit according to the following table:
Parameter Dimension Value in terms of
______________________________________________________________________
original unit unit with time unit with time expressed in days
expressed in days in case original time unit is 1 hour
_________________________________________________________________________________________________
lambda(I) /T x x * day/unit x
mux(I) L/T x x * day/unit x (ModelType /= Gamma; mux = μx)
mux(I) x x x (ModelType = Gamma; mux = p)
muscale(I) T/L x x * unit/day x/24
alpha(I) x x x
nu(I) T x x * unit/day x/24
kappa(I) x x x
phi(I) x x x
epsilon(I) /T x x * day/unit x
StartTimes(I) T x x * unit/day x/24

13. Input file of Whollstoptherain
All input is presented to the code in a single file: Rainpar.In. The parameters can be specified in free format, but the lines on which they occur and their required order must be strictly adhered to.
The next two slides give two example input files, the first for monthly data with gamma-distributed rainfall rates for Uccle in Belgium, the second for a hypothetical two-season year in a semi-arid climate with exponentially distributed rainfall rates.

14. Input file of Whollstoptherain
Test 1 Monthly rainfall parameters with gamma distribution. Rainfall rate in mm per day
Are time periods given as months?
Yes
Is detailed output about cells and storms desired?
Are hourly rainfall sums required?
Does the rainfall rate have a gamma or an exponential distribution?
Gamma
Seed Duration of the generated rainfall record (years)
lambda(I) mu-shape par(I) mu-scale par(I) alpha(I) nu(I) kappa(I) phi(I) epsilon(I)
E-14
E-14
E-14
E-14
E-12
E-13
E-13
E-14
E-13
E-14

15. Input file of Whollstoptherain
Test 1 Biseasonal rainfall parameters with exponential distribution. Rainfall rate in mm per day
Are time periods given as months? No
Is detailed output about cells and storms desired?
Are hourly rainfall sums required?
Does the rainfall rate have a gamma or an exponential distribution?
Exp
Seed Number of periods in the year Duration of the generated rainfall record (years)
lambda(I) mu-x (I) alpha(I) nu(I) kappa(I) phi(I) epsilon(I) Start time of each period (I)

16. Output of Whollstoptherain
File Diagnostics.Out echos part of the input and contains information about the mass convergence and other details of the numerical process. It also gives the number of lines in file Rainfall_Rate.OUT, as this is required on input by my groundwater modeling code DupuitFlow.
File Rainfall_Rate.OUT contains the continuous rainfall record of the simulation period. The first column contains the time until which the rainfall rate in column 2 is valid: the rainfall rate in row i of column 2 is valid for the time period between the times in rows i-1 and i in column 1. The rainfall rate in row 1 is valid from time zero through the time in row 1.

17. Output of Whollstoptherain
Files Hourly_Rainfall.Out, Daily_Rainfall.Out, and Annual_Rainfall.OUT contain hourly, daily, and annual sums of rainfall, respectively. The first column contains the time (in days) at which each hour or day of the simulated period ends, and the second column contains the amount of rainfall (in equivalent water layer) that fell during that hour or day.
File Rain_Statistics.OUT gives the averages of the annual and monthly rainfall and their standard deviations.
N.B. Time is expressed relative to the start of the simulation period.

18. Literature
Kim, D., F. Olivera, H. Cho, and S.A. Socolofsky Regionalization of the modified Bartlett-Lewis rectangular pulse stochastic rainfall model. Terr. Atmos. Ocean. Sci. 24: Doi: /TAO (Hy). Parameter maps for continental U.S.A. except Alaska. Also has a good explanation of the nature of MBL.
Onof, C., R.E. Chandler, A. Kakou, P. Northrop, H.S. Wheater, and V. Isham Rainfall modelling using Poisson-cluster processes: a review of developments. Stochastic Environmental Research and Risk Assessment 14: Review paper that also looks at spatio-temporal modeling.
Onof, C., W.J. Vanhaute, T. Meca-Figueras, J. Kaczmarska, R. Chandler, L. Hege, S. Vanderberghe, P. Willems, and N.E.C. Verhoest A truncated random parameter version of the Bartlett-Lewis model. Unpublished communication. An internal document with the mathematical details of the truncated MBL (TBL).
Pham, M.T., W.J. Vanhaute, S. Vanderberghe, B. De Baets, and N.E.C. Verhoest An assessment of the ability of Bartlett-Lewis type of rainfall models to reproduce drought statistics. Hydrol. Earth Syst. Sci. 17: Doi: /hess This paper compares the four models I coded for a Belgian dataset. I adopted their notation and abbreviations.
Rodriguez-Iturbe, I., D.R. Cox, and V. Isham A point process model for rainfall: further developments. Proc. R. Soc. Lond. A 417: This paper introduces MBL.
Xi, B., K.M. Tan, and C. Liu Logarithmic transformation-based gamma random number generators. J. Statistical Software 55 (4): This paper presents the algorithms I used to generate Gamma-distributed variates and gives the mathematical background.