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A new method for generating stochastic simulations of daily
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precipitation and air temperature
Kimberly Smith and Courtenay Strong
3
∗
Department of Atmospheric Sciences, University of Utah, Salt Lake City, Utah
Firas Rassoul-Agha
4
Department of Mathematics, University of Utah, Salt Lake City, Utah
∗
Corresponding author address: Courtenay Strong, Atmospheric Science, University of Utah, 135 S 1460
East Rm 819 (WBB) Salt Lake City, UT 84112-0110.
E-mail: court.strong@utah.edu
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ABSTRACT
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In this study, a stochastic weather generator (SWG) is introduced that simulates trended,
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nonstationary precipitation and temperature values directly, circumventing the conventional
8
approach of adding simulated standardized anomalies of temperature to a prescribed cy-
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clostationary mean. The model mean makes autocorrelated transitions between wet- and
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dry-state values, and its parameters are determined by optimizing harmonic and any trend
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terms. If the stochastic (“noise”) term is assumed to have constant amplitude, analytical
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results are available via maximum likelihood estimation (MLE) and are equivalent to least
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squares estimation (LSE). Where observations motivate a seasonally-varying noise coefficient,
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MLE becomes nonlinear, and we formulate an analytical solution via LSE. For illustration,
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the SWG is shown to produce realistic representations of daily maximum air temperature
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at a single site, which for the study is the Salt Lake City International Airport (KSLC), and
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can be generalized to multiple variables (e.g. minimum temperature and solar radiation) at
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multiple sites.
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1. Introduction
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While both statistically-based stochastic weather generators (SWGs) and dynamically-
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based global climate models (GCMs) are used in climate impacts studies, there are major
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differences between them. SWGs work on a point-scale, or on a point-scale expanded via
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multisite generalization to a basin-scale, whereas GCMs work on a broad regional scale and
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can be downscaled to the basin or smaller scale. GCMs have difficulty capturing detail in
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areas of complex terrain, including the Great Basin. SWGs also have a faster computational
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time than GCMs, which can take upwards of months to complete a single run. GCMs are
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very computationally expensive compared to SWGs, and thus, there are not many GCM runs
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available for analysis. GCMs also have difficulty capturing the very low-frequency (century-
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scale) connections between the Pacific Ocean and the Great Basin. The performance of
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state-of-the-art GCMs has been evaluated in terms of ability to capture the “extremes”
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in precipitation and temperature, and it has been found that GCMs poorly capture the
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extremes, though they perform better at temperature extremes than precipitation (Kiktev
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et al. 2007).
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SWGs alleviate some limitations of GCMs and were introduced as a way to overcome a
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lack of observational meteorological data and problems associated with missing data both
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temporally and spatially (Wilks and Wilby 1999; Wilks 2008). In addition, they have been
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used to better understand the uncertainties associated with future climate (e.g., Wilks 1992;
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Forsythe et al. 2014). These statistical models generate synthetic time series of precipitation
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and in some cases also air temperature and solar radiation, which statistically resemble the
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data used to force the model–usually daily observational weather data (Wilks and Wilby
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1999). There have been a multitude of early studies on SWGs that solely generate precip-
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itation occurrence and amount because air temperature and other meteorological variables
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are affected by whether precipitation occurred. The output of SWGs is a realistic statistical
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representation of what could happen and is not necessarily what is forecasted to happen.
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The first studies using stochastic simulators of weather data employ two-state, first-order
2
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Markov chain frameworks regarding precipitation (Bailey 1964; Richardson 1981; Roldàn and
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Woolhiser 1982), meaning that the probability of precipitation occurrence on a given day is
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only dependent on whether precipitation occurred or not on the previous day. Precipitation
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amount was modeled separately, and maximum/minimum temperatures and solar radiation
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were modeled as a function of precipitation occurrence. Other studies involving stochastic
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weather generators considered a two-state, second-order Markov chain process (Stern and
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Coe 1984; Wilks 1999a). Markov chains of higher order have been found to better capture
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dry spells than first-order Markov chains, thus providing more accurate results for most areas
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of the western US where dry spells are common, such as the semi-arid Great Basin.
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One limitation of the common stochastic weather generators is the ability to successfully
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capture nonstationary variability. In general, the use of the Markov chain analysis assumes
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stationarity. Previous studies have found that over the western US, El Niño results in a
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wetter Southwest and a drier Northwest, while La Niña results in the opposite (Ropelewski
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and Halpert 1986; Dettinger et al. 1998; Woolhiser 2008). In addition, the PDO also has
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significant impacts on precipitation in the western US. As discussed previously, the PDO
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is linked to ENSO, which in turn affects how the different phases of ENSO will impact the
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western US. Woolhiser (2008) introduces the idea of adding nonstationarity to the stochastic
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framework in order to rectify the effects these major oceanic oscillations have on western US
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precipitation. Essentially, perturbations given as time series of the oscillations (plus a trend)
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were linearly added to the probability of precipitation, and the coefficients associated with
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each perturbation give information on the sensitivity of each of the oscillations (Woolhiser
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2008). We employ this method in this study and also include a trend to account for the
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changing climate.
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In the SWG literature, simulation of daily maximum and minimum air temperature is
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usually conditioned on whether the day is wet or dry. The most widely used method for
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simulating temperature is the method used by Richardson (1981). This method involves
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generating the standardized residual time series of temperature (maximum and mininum
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temperature; the study also includes solar radiation) and using the multivariate generation
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model as described by Matalas (1967). These standardized residuals are assumed normally
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distributed, and the coefficients in the generating model are matrices containing the cross-
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correlations and auto-correlations between the residuals (Matalas 1967). After generating the
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synthetic residuals, the wet- or dry-state means and standard deviations that were initially
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removed are reintroduced to yield daily values of the variables. The means and standard
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deviations depend on whether the day was wet or dry; they are assumed to be cyclostation-
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ary and are determined by fitting harmonics of the annual cycle to observations (Richardson
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1981). A limitation of this model is that its mean and standard deviation switch abruptly
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between wet- and dry-state values prescribed in advance of the simulation. Here, we in-
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troduce a model whose mean makes autocorrelated transitions between wet- and dry-state
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values.
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2. Data and study area
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We chose to illustrate the proposed SWG using observations from the Salt Lake City
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International Airport (KSLC), which is located in the Great Basin. The Great Basin is a
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semi-arid region within the western U.S. characterized by its basin-and-range topography. Its
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precipitation depends largely on a combination of the state of El Niño-Southern Oscillation
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(ENSO) and the state of the Pacific Decadal Oscillation (PDO) (Wise 2010; Brown 2011).
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The precipitation and temperature data used to force the SWG are daily observational data
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recorded at KSLC (40.78◦ N, 111.97◦ W) from 1 January 1948 to 31 December 2010 via the
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Global Historical Climatology Network (GHCN-Daily) provided by the National Centers for
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Environmental Information (obtained from www.ncdc.noaa.gov). The domain map (see Fig.
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1) shows the location of KSLC within the eastern Great Basin and surrounding area.
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A day was considered “wet” and given value χ = 1 if the total precipitation that day
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reached at least 0.25 mm (approximately 0.01 inches). Otherwise, the day was considered
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dry and given value χ = 0. The χ vector was determined from the precipitation time series,
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and this provided the precipitation occurrence needed to model temperature with the SWG.
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In this study, we use and generate only maximum surface air temperature at a single site,
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and generalization to multiple variables at multiple sites is planned.
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3. Estimation of precipitation and maximum tempera-
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ture
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For stochastic simulation of daily precipitation occurrence, the model follows that intro-
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duced by Woolhiser (2008). The probability of precipitation occurrence is determined with
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a two-state (wet or dry), second-order Markov chain, which means that the probabiity of
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precipitation on a given day depends on the precipitation state on the previous two days as
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follows
pij0 (t) = P {xt = 0|xt−1 = j, xt−2 = i} ; t = 1, 2, . . . , 365M.
(1)
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If we assume cyclostationarity, then the pij0 terms are periodic, meaning pij0 (t + K365) =
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pij0 (t) for any integer K. Due to nonstationarity caused by oceanic forcings of ENSO, PDO,
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and a trend due to climate change, we define perturbed versions of (1)
(ij0)
p̃′ij0 (t) = p̃ij0 (t) + b0
(ij0)
+ b1
(ij0)
t + b2
(ij0)
S1 (t − τ1 ) + b3
S2 (t − τ2 ))
(2)
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where {b0 , b1 } enable a trend, S1 and S2 are oceanic forcing with periodicity of 3-7 years
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(ENSO) and 10-15 years (PDO), respectively, and the τ terms are positive lags (i.e., oceanic
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modes leads precipitation by τ months).
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For maximum temperature at a single site, the linear model is
Tk+1 = aTk + bk + ck ϵk ,
T 1 = b 0 + c 0 ϵ0 ,
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(3)
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where a is a coefficient assumed constant, and bk and ck are coefficients that depend on day
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k. Errors ϵk are independent and identically distributed (i.i.d.) random standard normals.
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The temperature on day k + 1 is dependent on the temperature on day k, where k ranges
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from 1 to K − 1 (K being the length of the simulation).
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a. Maximum likelihood estimation
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We begin with a simplified case where c does not depend on k. The temperature entries
T1 to TK are multivariate normals, and the joint density function is given by
f (T1 , ..., TK ) =
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−(DT − B)′ (DT − B)
1
exp
,
(2π)K/2 c
2c2
where D is the K × K matrix:
⎡
0 ··· 0
⎢ 1
⎢
⎢
⎢−a 1
⎢
⎢
⎢
... ...
D=⎢
⎢ 0
⎢
⎢ .
⎢ . ... ...
1
⎢ .
⎢
⎣
0 · · · 0 −a
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⎤
0⎥
⎥
⎥
0⎥
⎥
⎥
.. ⎥
.⎥
⎥
⎥
⎥
⎥
0⎥
⎥
⎦
1
and B and T are the K × 1 vectors:
⎡
⎤
⎡
⎤
⎢ b0 ⎥
⎢ T1 ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ b1 ⎥
⎢ T2 ⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢ . ⎥
⎢ . ⎥
⎢ . ⎥
. ⎥
B=⎢
⎢ . ⎥ ,T = ⎢ . ⎥ .
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
b
T
⎢ K−2 ⎥
⎢ K−1 ⎥
⎢
⎥
⎢
⎥
⎣
⎦
⎣
⎦
bK−1
TK
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(4)
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To restrict the model to a reasonable number of parameters, we give structure to the bk
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values by giving them a trend and harmonics
bk = γχk+1 + αk + βχk+1 cos(2πk/τ ) + βχ′ k+1 sin(2πk/τ ) + δχk+1 cos(4πk/τ ) + δχ′ k+1 sin(4πk/τ ),
(5)
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128
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where τ is the period, assumed to be 365 days.
We applied a maximum likelihood estimate (MLE) to the joint density function, which
involves maximizing (4) or minimizing its negative log
c−2 (DT − B)′ (DT − B) + 2K log c.
(6)
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We first minimize (DT − B)′ (DT − B) to get the MLEs for the D and B matrices. This
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returns the sum of squared errors
2
′
(DT − B) (DT − B) = (b0 − T1 ) +
K−1
'
(aTk + bk − Tk+1 )2 ,
(7)
k=1
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133
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where bk is given in (5).
Taking derivatives in (7) with respect to a and each of the parameters in bk and setting
them equal to zero gives the following twelve equations:
K−1
'
k=1
K−1
'
Tk (aTk + bk − Tk+1 ) = 0,
k(aTk + bk − Tk+1 ) = 0,
k=1
(b0 − T0 )1{χ1 = 0} +
(b0 − T0 )1{χ1 = 1} +
(b0 − T0 )1{χ1 = 0} +
(b0 − T0 )1{χ1 = 1} +
K−1
'
(aTk + bk − Tk+1 )1{χk+1 = 0} = 0,
k=1
K−1
'
(aTk + bk − Tk+1 )1{χk+1 = 1} = 0,
k=1
K−1
'
k=1
K−1
'
cos(2πk/τ )(aTk + bk − Tk+1 )1{χk+1 = 0} = 0,
cos(2πk/τ )(aTk + bk − Tk+1 )1{χk+1 = 1} = 0,
k=1
K−1
'
sin(2πk/τ )(aTk + bk − Tk+1 )1{χk+1 = 0} = 0,
k=1
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K−1
'
sin(2πk/τ )(aTk + bk − Tk+1 )1{χk+1 = 1} = 0,
k=1
(b0 − T0 )1{χ1 = 0} +
(b0 − T0 )1{χ1 = 1} +
K−1
'
k=1
K−1
'
cos(4πk/τ )(aTk + bk − Tk+1 )1{χk+1 = 0} = 0,
cos(4πk/τ )(aTk + bk − Tk+1 )1{χk+1 = 1} = 0,
k=1
K−1
'
k=1
K−1
'
sin(4πk/τ )(aTk + bk − Tk+1 )1{χk+1 = 0} = 0,
sin(4πk/τ )(aTk + bk − Tk+1 )1{χk+1 = 1} = 0,
k=1
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where 1 is an indicator function which takes the value of one if the condition in brackets is
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met and zero otherwise. This is a linear system of 12 equations and 12 unknowns, which we
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solve numerically.
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We then minimize (6) as a function of c. Taking a derivative in c yields
−2c−3 (DT − B)′ (DT − B) + 2Kc−1 .
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(8)
The derivative has a unique point at which it vanishes:
c=
(
K −1 (DT − B)′ (DT − B) ,
(9)
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which is both the MLE value and least squares estimation (LSE) value. However, constant
141
c tends to overestimate the variance in the summer and underestimate it in the winter (see
142
Fig. 2), motivating a seasonally-varying c denoted by ck , as in (3). The seasonally-varying
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ck makes the MLE nonlinear in the parameters, so we proceed by taking an LSE approach
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where linear analytical expressions can be obtained.
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b. Least squares estimation with varying ck
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When ck does not depend on k, the LSE for the parameters in bk and a is equivalent to
147
the MLE derived in Section 3.1, namely the set of equations on page 7. Now, we assume
8
148
that c2k has a cyclostationary structure similar to bk but without a trend. Its formulation is
149
given by
c2k,0 = ρ0 + ϵ0 cos(2πk/τ ) + ϵ′0 sin(2πk/τ ) + κ0 cos(4πk/τ ) + κ′0 sin(4πk/τ )
150
(10)
for dry days and
c2k,1 = ρ1 + ϵ1 cos(2πk/τ ) + ϵ′1 sin(2πk/τ ) + κ1 cos(4πk/τ ) + κ′1 sin(4πk/τ )
(11)
151
for wet days. Here, k varies from 0 to K − 1. However, because we assume that ck is
152
cyclostationary with no trend, it is sufficient to specify ck,0 and ck,1 only for k = 0, ..., τ − 1.
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Our strategy to estimate ck,0 and ck,1 is to align the data by day of year j = 0, ..., τ − 1
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and segregate it according to the precipitation sequence. This yields the MLE (and LSE)
155
estimators
ĉj,0 =
)
−1
Nj,0
(DT − B)′j,0 (DT − B)j,0
and ĉj,1 =
)
−1
Nj,1
(DT − B)′j,1 (DT − B)j,1 , (12)
156
where (DT − B)j,0 is the K/τ × 1 vector populated with (DT − B)k if χk+1 = 0 and with
157
zero if χk+1 = 1, where k = j, j + τ, j + 2τ, ..., j + K − τ . Similarly, (DT − B)j,1 is the
158
K/τ × 1 vector populated with (DT − B)k if χk+1 = 1 and with zero if χk+1 = 0, where
159
k = j, j + τ, j + 2τ, ..., j + K − τ . Nj,0 is the number of times χk+1 = 0, and Nj,1 is the
160
number of times χk+1 = 1.
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Once we have estimated cj,0 and cj,1 , we use the LSE method to estimate the parameters
in equations (10) and (11). Specifically, we minimize
τ −1
'
(ρ0 + ϵ0 cos(2πj/τ ) + ϵ′0 sin(2πj/τ ) + κ0 cos(4πj/τ ) + κ′0 sin(4πj/τ ) − ĉ2j,0 )2
(13)
τ −1
'
(ρ1 + ϵ1 cos(2πj/τ ) + ϵ′1 sin(2πj/τ ) + κ1 cos(4πj/τ ) + κ′1 sin(4πj/τ ) − ĉ2j,1 )2 .
(14)
j=0
163
and
j=0
164
Taking derivatives in each of the parameters in (13) and (14) and setting them equal to zero
9
165
yields equations that are familiar from Fourier analysis. For dry days, we have
ρ̂0 = τ −1
τ −1
'
ĉ2j,0 ,
j=0
ϵ̂0 =
2
τ
τ −1
'
ĉ2j,0 cos(2πj/τ ),
j=0
τ −1
ϵ̂′0 =
2' 2
ĉ sin(2πj/τ ),
τ j=0 j,0
τ −1
2' 2
κ̂0 =
ĉ cos(4πj/τ ),
τ j=0 j,0
τ −1
κ̂′0
166
2' 2
=
ĉ sin(4πj/τ ),
τ j=0 j,0
and for wet days, we have
ρ̂1 = τ
−1
τ −1
'
ĉ2j,1 ,
j=0
ϵ̂1 =
ϵ̂′1 =
2
τ
τ −1
'
ĉ2j,1 cos(2πj/τ ),
2
τ
τ −1
'
ĉ2j,1 sin(2πj/τ ),
j=0
j=0
τ −1
2' 2
κ̂1 =
ĉ cos(4πj/τ ),
τ j=0 j,1
τ −1
κ̂′1
2' 2
=
ĉ sin(4πj/τ ).
τ j=0 j,1
167
The parameters are inserted back into (10) or (11) to generate the synthetic temperature
168
series using the linear model (3). An example simulation with the seasonally-varying ck is
169
shown in Fig. 3. Note how seasonally-varying ck better captures the low variability in the
170
summer and high variability in the winter.
10
171
4. Comparison to the Richardson method
172
Because the Richardson method of simulating stochastic temperature is the most widely-
173
used in the field (referred to as the multivariate generation model), it is useful to compare it to
174
the model introduced here. The Richardson method is essentially an autoregressive process
175
that simulates standardized residuals; the details of this method can be found in Richardson
176
(1981) and Matalas (1967). The Richardson method prescribes the means and standard
177
deviations of the data (for wet and dry days) prior to simulation via a harmonic fit and then
178
reintroduces them after simulating standardized residuals. As noted in the Introduction,
179
this causes the model mean and standard deviation to abruptly switch between wet- and
180
dry-state values. The model we introduce here (3) also has wet- and dry-state harmonics
181
(bk ) and noise amplitudes (ck ) prescribed in advance, but the mean of the model (D−1 B)
182
and standard deviation make autocorrelated, and hence more realistic, transitions via the
183
parameter a in D.
184
We highlight the difference between the methods in Fig. 4, which compares the compos-
185
ite synthetic temperature simulated by the two models to the observational temperature for
186
precipitation occurrence sequences of dry-dry-wet-wet-dry-dry for each season. The obser-
187
vational temperature reflects a typical cold frontal passage in each season (e.g., Shafer and
188
Steenburgh 2008). In general, the observed maximum temperature increases shortly before
189
the frontal passage due to southerly flow and warm air advection; on the first day of precip-
190
itation, the maximum temperature decreases modestly. On the second day of precipitation,
191
the temperature continues to decrease, and it slowly rebounds following the precipitation
192
event. Our model tends to follow this same pattern. In contrast, the abrupt switching be-
193
tween wet- and dry-state means in the Richardson model results in an unrealistically large
194
decrease in temperature on the first day of precipitation, followed by minimal change on the
195
second day (actually zero change with large enough sample).
11
196
5. Discussion and conclusions
197
This study presents a new linear model for simulating stochastic temperature realiza-
198
tions, and the method was illustrated for maximum temperature at a single site within the
199
Great Basin. We first considered a simplified version of the model with a constant noise
200
coefficient, c, and applied MLE to obtain its parameters. However, this constant c compro-
201
mised between the variance in the summer and the variance in the winter, which resulted in
202
a simulation that did not adequately capture the seasonal variance found in the observations.
203
A seasonally-varying noise coefficient, ck , rendered the MLE nonlinear, and we presented an-
204
alytical solutions via LSE. The resulting temperature realization more closely matched that
205
of observations, with increased wintertime variance and decreased summertime variance.
206
Further realism may also be possible by relaxing assumptions used here. For example,
207
we assume the amplitude of noise, ck , to be annually cyclostationary but without trend.
208
We also assume that temperature depends only on itself and precipitation occurrence, but
209
precipitation amount and climate teleconnections that influence air mass trajectories may
210
be additionally important.
211
Even though this study is focused on only maximum temperature at a single site, the
212
method described can be generalized to include minimum temperature and solar radiation.
213
In addition, it is also possible to generalize the method for multiple sites because of the
214
linear formulation, extending ideas described in Wilks (1998) and Wilks (1999b), where the
215
sites themselves have spatial correlation but are generated independently of each other.
216
Acknowledgments.
217
This material is based upon work supported by the National Science Foundation under
218
grants EPS-1135482, EPS-1135483, EPS-1208732, and DMS-1407574. Any opinions, find-
219
ings, and conclusions or recommendations expressed in this material are those of the authors
220
and do not necessarily reflect the views of the National Science Foundation. Provision of
12
221
computer infrastructure by the Center for High Performance Computing at the University
222
of Utah is gratefully acknowledged.
13
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1.
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List of Figures
1
The study area: the eastern half of the Great Basin (which includes northern
290
and western Utah, extreme southwestern Wyoming, extreme southern Idaho,
291
and Nevada) and surrounding area. The star indicates the location of the Salt
292
Lake City International Airport (KSLC). The colorbar indicates elevation in
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meters above sea level.
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2
18
Observational temperature data (black line) and synthetic data (red line) with
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constant c. The depicted time period is 8 September 1961 to 4 June 1964.
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Note the too-high variability in the summer and relatively too-low variability
297
in the winter in the synthetic data.
19
20
298
3
As in Fig. 2 but with seasonally-varying ck on the stochastic term.
299
4
Composite observational temperature (black lines) and composite synthetic
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temperature for sets of days that follow the precipitation occurrence sequence
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dry-dry-wet-wet-dry-dry in each season. The red lines indicate the model
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presented here, and the blue lines indicate the Richardson model. The number
303
of samples in each set is approximately 125.
17
21
Fig. 1. The study area: the eastern half of the Great Basin (which includes northern and
western Utah, extreme southwestern Wyoming, extreme southern Idaho, and Nevada) and
surrounding area. The star indicates the location of the Salt Lake City International Airport
(KSLC). The colorbar indicates elevation in meters above sea level.
18
50
40
temperature (deg. C)
30
20
10
0
-10
-20
5000
5100
5200
5300
5400
5500
5600
day of simulation
5700
5800
5900
6000
Fig. 2. Observational temperature data (black line) and synthetic data (red line) with
constant c. The depicted time period is 8 September 1961 to 4 June 1964. Note the too-high
variability in the summer and relatively too-low variability in the winter in the synthetic
data.
19
50
40
temperature (deg. C)
30
20
10
0
-10
-20
5000
5100
5200
5300
5400
5500
5600
day of simulation
5700
5800
5900
6000
Fig. 3. As in Fig. 2 but with seasonally-varying ck on the stochastic term.
20
spring
temperature (deg. C)
21
34
19
33
18
32
17
31
16
30
15
29
14
28
13
27
12
26
fall
temperature (deg. C)
21
8
19
7
18
6
17
5
16
4
15
3
14
2
13
1
1
2
3
4
day
winter
9
20
12
summer
35
20
5
6
0
1
2
3
4
5
6
day
Fig. 4. Composite observational temperature (black lines) and composite synthetic temperature for sets of days that follow the precipitation occurrence sequence dry-dry-wet-wetdry-dry in each season. The red lines indicate the model presented here, and the blue lines
indicate the Richardson model. The number of samples in each set is approximately 125.
21