Conditioning Synthetic Weather Generation on Climate Change Projections
This is the fourth blog post in a five part series on synthetic weather generators. You can read about common single-site parametric and non-parametric weather generators in Parts I and II, respectively, as well as multi-site generators of both types in Part III. Here I discuss how parametric and non-parametric weather generators can be modified to simulate weather that is consistent with climate change projections.
As you are all well aware, water managers are interested in finding robust water management plans that will consistently meet performance criteria in the face of hydrologic variability and change. One method for such analyses is to re-evaluate different water management plans across a range of potential future climate scenarios. Weather data that is consistent with these scenarios can be synthetically generated from a stochastic weather generator whose parameters or resampled data values have been conditioned on the climate scenarios of interest, and methods for doing so are discussed below.
Parametric Weather Generators
Most climate projections from GCMs are given in terms of monthly values, such as the monthly mean precipitation and temperature, as well as the variances of these monthly means. Wilks (1992) describes a method for adjusting parametric weather generators of the Richardson type in order to reproduce these projected means and variances. This method, described here, has been used for agricultural (Mearns et al., 1996; Riha et al., 1996) and hydrologic (Woodbury and Shoemaker, 2012) climate change assessments.
Recall from Part I that the first step of the Richardson generator is to simulate the daily precipitation states with a first order Markov chain, and then the precipitation amounts on wet days by independently drawing from a Gamma distribution. Thus, the total monthly precipitation is a function of the transition probabilities describing the Markov chain and the parameters of the Gamma distribution describing precipitation amounts. The transition probabilities of the Markov chain, p01 and p11, represent the probabilities of transitioning from a dry day to a wet day, or a wet day to another wet day, respectively. An alternative representation of this model shown by Katz (1983) is with two different parameters, π and d, where π = p01/(1 + p01 – p11) and d = p11 – p01. Here π represents the unconditional probability of a wet day, and d represents the first-order autocorrelation of the Markov chain.
Letting SN be the sum of N daily precipitation amounts in a month (or equivalently, the monthly mean precipitation), Katz (1983) shows that if the precipitation amounts come from a Gamma distribution with shape parameter α and scale parameter β, the mean, μ, and variance, σ2, of SN are described by the following equations:
(1) μ(SN) = Nπαβ
(2) σ2(SN) = Nπαβ2[1 + α(1-π)(1+d)/(1-d)].
For climate change impact studies, one must find a set of parameters θ=(π,d,α,β) that satisfies the two equations above for the projected monthly mean precipitation amounts and variances of those means. Since there are only 2 equations but 4 unknowns, 2 additional constraints are required to fully specify the parameters (Wilks, 1992). For example, one might assume that the frequency and persistence of precipitation do not change, but that the mean and variance of the amounts distribution do. In that case, π and d would be unchanged from their estimates derived from the historical record, while α and β would be re-estimated to satisfy Equations (1) and (2). Other constraints can be chosen based on the intentions of the impacts assessment, or varied as part of a sensitivity analysis.
To modify the temperature-specific parameters, recall that in the Richardson generator, the daily mean and standard deviation of the non-precipitation variables are modeled separately on wet and dry days by annual Fourier harmonics. Standardized residuals of daily minimum and maximum temperature are calculated for each day by subtracting the daily mean and dividing by the daily standard deviation given by these harmonics. The standardized residuals are then modeled using a first-order vector auto-regression, or VAR(1) model.
For generating weather conditional on climate change projections, Wilks (1992) assumes that the daily temperature auto and cross correlation structure remains the same under the future climate so that the VAR(1) model parameters are unchanged. However, the harmonics describing the mean and standard deviation of daily minimum and maximum temperature must be modified to capture projected temperature changes. GCM projections of temperature changes do not usually distinguish between wet and dry days, but it is reasonable to assume the changes are the same on both days (Wilks, 1992). However, it is not reasonable to assume that changes in minimum and maximum temperatures are the same, as observations indicate that minimum temperatures are increasing by more than maximum temperatures (Easterling et al., 1997; Vose et al., 2005).
Approximating the mean temperature, T, on any day t as the average of that day’s mean maximum temperature, µmax(t), and mean minimum temperature, µmin(t), the projected change in that day’s mean temperature, ΔT(t), can be modeled by Equation 3:
where CX0 and CN0 represent the annual average changes in maximum and minimum temperatures, respectively, and CX1 and CN1 the corresponding amplitudes of the annual harmonics. The phase angle, φ, represents the day of the year with the greatest temperature change between the current and projected climate, which is generally assumed to be the same for the maximum and minimum temperature. Since GCMs predict that warming will be greater in the winter than the summer, a reasonable value of φ is 21 for January 21st, the middle of winter (Wilks, 1992).
In order to use Equation 3 to estimate harmonics of mean minimum and maximum temperature under the projected climate, one must estimate the values of CX0, CN0, CX1 and CN1. Wilks (1992) suggests a system of four equations that can be used to estimate these parameters:
(4) ΔT = 0.5*(CX0 + CN0)
(5) Δ[T(JJA) – T(DJF)] = -0.895(CX1 + CN1)
(6) ΔDR(DJF) = CX0 − CN0 + 0.895(CX1 − CN1)
(7) ΔDR(JJA) = CX0 − CN0 − 0.895(CX1 − CN1)
where the left hand sides of Equations (4)-(7) represent the annual average temperature change, the change in temperature range between summer (JJF) and winter (DJF), the change in average diurnal temperature range (DR = µmax – µmin) in winter, and the change in average diurnal temperature range in summer, respectively. The constant ±0.895 is simply the average value of the cosine term in equation (3) evaluated at φ = 21 for the winter (+) and summer (−) seasons. The values for the left hand side of these equations can be taken from GCM projections, either as transient functions of time or as step changes.
Equations (4)-(7) can be used to estimate the mean minimum and maximum temperature harmonics for the modified weather generator, but the variance in these means may also change. Unlike changes in mean daily minimum and maximum temperature, it is fair to assume that changes in the standard deviation of these means are the same as each other and the GCM projections for changes in the standard deviation of daily mean temperature for both wet and dry days. Thus, harmonics modeling the standard deviation of daily minimum and maximum temperature on wet and dry days can simply be scaled by some factor σd’/ σd, where σd is the standard deviation of the daily mean temperature under the current climate, and σd’ is the standard deviation of the daily mean temperature under the climate change projection (Wilks, 1992). Like the daily mean temperature changes, this ratio can be specified as a transient function of time or a step change.
It should be noted that several unanticipated changes can occur from the modifications described above. For instance, if one modifies the probability of daily precipitation occurrence, this will change both the mean daily temperature (since temperature is a function of whether or not it rains) and its variance and autocorrelation (Katz, 1996). See Katz (1996) for additional examples and suggested modifications to overcome these potential problems.
Non-parametric Weather Generators
As described in Part II, most non-parametric and semi-parametric weather generators simulate weather data by resampling historical data. One drawback to this approach is that it does not simulate any data outside of the observed record; it simply re-orders them. Modifications to the simple resampling approach have been used in some stationary studies (Prairie et al., 2006; Leander and Buishand, 2009) as mentioned in Part II, and can be made for climate change studies as well. Steinschneider and Brown (2013) investigate several methods on their semi-parametric weather generator. Since their generator does have some parameters (specifically, transition probabilities for a spatially averaged Markov chain model of precipitation amounts), these can be modified using the methods described by Wilks (1992). For the non-parametric part of the generator, Steinschneider and Brown (2013) modify the resampled data itself using a few different techniques.
The first two methods they explore are common in climate change assessments: applying scaling factors to precipitation data and delta shifts to temperature data. Using the scaling factor method, resampled data for variable i, xi, are simply multiplied by a scaling factor, a, to produce simulated weather under climate change, axi. Using delta shifts, resampled data, xi, are increased (or decreased) by a specified amount, δ, to produce simulated weather under climate change, xi + δ.
Another more sophisticated method is the quantile mapping approach. This procedure is generally applied to parametric CDFs, but can also be applied to empirical CDFs, as was done by Steinschneider and Brown (2013). Under the quantile mapping approach, historical data of the random variable, X, are assumed to come from some distribution, FX, under the current climate. The CDF of X under climate change can be specified by a different target distribution, FX*. Simulated weather variables xi under current climate conditions can then be mapped to values under the projected climate conditions, xi*, by equating their values to those of the same quantiles in the target distribution, i.e. xi* = F*-1(F(xi)).
While simple, these methods are effective approaches for top-down or bottom-up robustness analyses. Unfortunately, what one often finds from such analyses is that there is a tradeoff between meeting robustness criteria in one objective, and sacrificing performance in another, termed regret. Fortunately, this tradeoff can occasionally be avoided if there is an informative climate signal that can be used to inform water management policies. In particular, skillful seasonal climate forecasts can be used to condition water management plans for the upcoming season. In order to evaluate these conditioned plans, one can generate synthetic weather consistent with such forecasts by again modifying the parameters or resampling schemes of a stochastic weather generator. Methods that can be used to modify weather generators consistent with seasonal climate forecasts will be discussed in my final blog post on synthetic weather generators.
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