Generation of an Hourly Meteorological Time Series for an Alpine Basin in British Columbia for Use in Numerical Hydrologic Modeling

Transcription

1 862 JOURNAL OF HYDROMETEOROLOGY Generation of an Hourly Meteorological Time Series for an Alpine Basin in British Columbia for Use in Numerical Hydrologic Modeling MARKUS SCHNORBUS AND YOUNES ALILA Department of Forest Resources Management, University of British Columbia, Vancouver, British Columbia, Canada (Manuscript received 7 October 2003, in final form 30 April 2004) ABSTRACT Spatially distributed numerical hydrologic models are useful tools for examining the long-term impact of forest harvesting in mountainous basins on streamflow regime properties. Such models require the input of longduration subdaily meteorological time series data that are not routinely available in mountainous headwater basins. A relatively simple method is presented for extending short-duration records by using a combined stochastic empirical technique, and the approach is demonstrated using the Redfish Creek in British Columbia, Canada. Synthetic hourly precipitation, precipitation gradient, air temperature, temperature lapse rate, wind speed, relative humidity, solar beam and diffuse irradiance, and downward longwave irradiance for two station locations are generated in a three-step process: 1) hourly precipitation is generated using a clustered rectangular pulse point process, 2) daily meteorology is generated using a multivariate first-order autoregressive process, and 3) final hourly nonprecipitation meteorology is derived by disaggregating daily meteorology. Seasonal and annual precipitation means are reproduced to within 10% of observed. The skill of the generated nonprecipitation meteorological data to reproduce the statistical properties and diurnal structure of the observed data ranged from good to poor, with bias ranging from 0% to 500% and efficiency ranging from 76 to Despite discrepancies in the generated meteorology, a comparison of annual hydrologic fluxes, spatial distribution of winter snow accumulation, flow duration, and average hydrographs, simulated using the Distributed Hydrology Soil Vegetation Model (DHSVM), indicates that model skill shows negligible response to the use of the generated subdaily meteorology. 1. Introduction Forestry activities conducted in the snow-dominated, mountainous basins of the British Columbia (BC), Canada, interior can affect the hydrological processes generating streamflow. These activities can subsequently produce a shift in the streamflow regime and alter the frequency of geomorphically significant discharge events, which in mountainous terrain are typically associated with peak flows of recurrence intervals of anywhere from 1 to 100 yr [see Besehta et al. (2000) and Church (2002) for reviews]. Unfortunately, the available empirical evidence derived mainly from paired-basin studies is limited in its ability to quantify impacts for peak streamflow events with recurrence intervals larger than about 10 to 20 yr. In order to fill knowledge gaps regarding the impacts of forest management activities upon peak streamflow, many investigators have espoused making greater use of long-term physically based hydrologic modeling (i.e., Thomas and Megahan 1998; Alila and Beckers 2001). Corresponding author address: Markus Schnorbus, 733 St. Patrick Street, Victoria, BC V85 4X6, Canada. mschnorbus@shaw.ca This approach has been adopted in BC, and a project is currently underway to simulate the hydrologic impact of forest harvesting in basins spanning varied physiographic domains (Whitaker et al. 2002, 2003; Thyer et al. 2003; Schnorbus and Alila 2004). The goal of said project is to utilize continuous numerical simulation to generate large samples (i.e., N 100) of pre- and postharvest peak flow events for various watersheds with which to assess forest harvesting impacts upon the peak flow regime. However, a major challenge for this simulation project is the collection of the required meteorological (met) forcing data. For instance, the hydrologic model used in this project, the Distributed Hydrology Soil Vegetation Model (DHSVM; Wigmosta et al. 1994), requires an extensive suite of nine met input variables provided as a time series at the temporal resolution of the model time step (1 h in this application). Met data of the necessary scope (number of variables) and resolution, let alone duration (100 yr), are nonexistent in the mountainous interior of BC. There are, however, several experimental headwater basins in BC at which met data of the required scope have been collected at an hourly resolution over varying periods, with one significant drawback: the data records are of short duration, typically spanning no more than 2004 American Meteorological Society

2 OCTOBER 2004 SCHNORBUS AND ALILA yr. Nevertheless, it is feasible to extend these observed records and generate a proxy long-term met time series of an hourly resolution suitable for use in driving physically based hydrologic models. The purpose of such an extended record is not to recreate or forecast actual weather events, but to generate a proxy data series that resembles, statistically, the weather during the calibration period. It is expected that if such proxy met data were used to force a physically based hydrologic model, then the simulated output of that model (i.e., discharge) will statistically resemble the output generated had actual (i.e., measured) met data been available for input. One such attempt at generating a proxy met record was undertaken in order to support the long-term numerical simulation of forest harvesting effects upon the peak discharge regime of Redfish Creek, a mountainous, snowmelt-dominated basin located in southeastern BC (Schnorbus and Alila 2004). This study addresses how well hourly met records can be extended from only a few years ( 10) of observation and the effect of generated met data upon hydrologic simulation. Although the method is site specific to Redfish Creek, the authors believe that, with some modification, the technique can be applied in a wider variety of landscapes. Additionally, the requirement for extending and generating hourly resolution met data is not limited to hydrology but has utility in, for instance, agricultural, ecological, and climatic studies. The paper begins with a brief description of the study area and collected data in section 2. The met generation method is described in section 3. The performance of the proposed method is assessed in section 4 and discussed in section 5, and concluding remarks are provided in section Study area and data The Redfish Creek watershed is located in the Selkirk Mountains of southeastern BC (Fig. 1, inset). The watershed has a drainage area of 26 km 2 and an elevation range of 700 to 2300 m, although 80% of the watershed lies above 1500-m elevation. Basin slopes are moderately steep, with a median gradient of 50%. Slopes are heavily forested, with the lower elevations falling within the Interior Cedar Hemlock (ICH) and the upper elevations within Engelmann Spruce Subalpine Fir (ESSF) biogeoclimatic zones. Above an approximate elevation of 1800 m, the ESSF zone transitions into subalpine parkland, which is a thinly wooded subzone that occupies roughly 40% of the basin. The study region has a climate that is best described as humid continental, being subject to the influence of both maritime and continental airstreams (Demarchi 2004). Winter is by far the most variable season in southeastern BC; a nearly constant progression of eastward moving Pacific disturbances brings warm, moist air. However, influxes of cold, dry Arctic air occur with varying frequency from year to year, commonly during November to February (Hare and Thomas 1979). The influence of the Pacific is strongly felt with respect to precipitation, particularly at high altitude, and substantial frontal precipitation is deposited along the west slope of the Selkirk Mountains, making the region one of the wettest outside the coast (Meidinger and Pojar 1991). Local convective precipitation is also frequent during the summer, with June being one of the wettest months in the year. Mean annual precipitation is influenced by orographic effects and trends from 1100 to 1600 mm at Burn (1290-m elevation) and Cabin (1730- m elevation) stations, respectively. Because of the decrease in temperature with height, the proportion of precipitation that falls as snow increases with elevation such that the gradient of snow water equivalent (SWE) is quite pronounced; mean annual peak SWE is 220 and 1060 mm at Burn and Cabin stations, respectively, and 1500 mm at the highest elevations (as recorded at the Alpine climate station at 2045-m elevation; see Fig. 1). Air temperature displays a continental regime; however, altitudinal effects moderate annual variability, and monthly average temperature at Burn station ranges from 6 C in December to 15 C in July. Seasonal temperature variability decreases with elevation, and mean annual temperature observed over a 5-yr period decreases from 4 C at Burn station to 2 C at Cabin station. Annual water yield is dominated by snowmelt (60% of total yield), and the generally humid climate generates an annual runoff ratio (ratio of discharge to precipitation) of about 60%. A mixture of radiation melt and rain-on-melting-snow processes generate peak annual streamflow in Redfish Creek, although the peak flow regime is still fundamentally governed by the snowmelt process. On average, peak annual streamflow from Redfish Creek typically commences at the beginning of April and peaks between mid-may and mid- June. Hydrograph recession ceases around the first week of August, and base flow is maintained from late summer through to early spring. DHSVM requires time series input of precipitation (P), air temperature (T a ), relative humidity (R), wind speed (U), solar beam irradiance (S b ), solar diffuse irradiance (S d ), downward longwave irradiance (L), temperature lapse rate (T lapse ), as well as a coefficient describing the increase in precipitation with elevation (P grad ), at one or more station locations within the study area. The original calibration and validation of Redfish Creek application of DHSVM was conducted using met time series data for Burn and Cabin climate stations spanning the period October 1992 to December 1997 (Whitaker et al. 2003). This initial dataset is referred to as the observed data and is used as the baseline for further discussion. During this period T a, R, U, and P have been continuously recorded at both Burn and Cabin stations. Hourly values of global solar radiation (S g ) have only been recorded since July of 1994 and September 1995, for Burn and Cabin stations, respec-

3 864 JOURNAL OF HYDROMETEOROLOGY FIG. 1. Redfish Creek study area. tively; therefore, prior to this period S g was estimated using the method of Bristow and Campbell (1984) (Whitaker et al. 2003; see also section 3d). Hourly values of T lapse were based on the observed hourly difference in T a between Burn and Cabin climate stations whereas P grad was based on the difference in monthly precipitation between Burn and Cabin stations. Values of L, S b, and S d are not collected in the study area and were estimated using the techniques described in section 3d. An additional three years of precipitation data (spanning 1998 to 2000) for Burn and Cabin stations, not used in the original observed dataset, were also available for this study. Hourly data from Alpine station was of too short a duration and contained too many gaps to be of use. Streamflow has been gauged by the Water Survey of Canada since 1993 and is published as daily average discharge. Hourly discharge data have also been collected by the BC Ministry of Forests since April 1993, but some data gaps exist prior to November 1995.

4 OCTOBER 2004 SCHNORBUS AND ALILA 865 FIG. 2. Schematic representation of the met generation process. Variables are as follows (hourly unless otherwise indicated): P precipitation (m); P grad relative change in precipitation with elevation (m 1 ); T a temperature ( C); R relative humidity (%); U wind speed (m s 1 ); T lapse temperature lapse rate ( C m 1 ); S b direct solar irradiance (W m 2 ); S d diffuse solar irradiance (W m 2 ); L downward longwave irradiance (W m 2 ); T m daily maximum temperature ( C); T n daily minimum temperature ( C); T d daily average dewpoint temperature ( C); U d daily average wind speed (m s 1 ); H daily atmospheric transmittance ( ); and K cloud modification factor ( ). 3. Met generation a. Overview The purpose of the met generation procedure is to generate an hourly time series of all the required DHSVM input met variables for the Burn and Cabin climate stations. As no turnkey solution currently exists, to the best of the authors knowledge, it became clear that the problem could be most efficiently addressed by amalgamating currently existing models into a component-based approach. From the outset the design of the met generation algorithm was based on the intent to use, when possible, simple and parsimonious component models and to minimize the number of component models. The final met generation model employed in this study is a three-step approach that the authors feel makes the most efficient use of proven techniques. The first step is the two-site stochastic generation of an hourly precipitation time series using a clustered rectangular pulses point process. The second step is the two-site stochastic generation of daily weather variables using a multivariate first-order autoregressive model. The final step in the met generation model involves disaggregation of daily data into the final hourly DHSVM input time series using various empirical models. The met generation algorithm is shown schematically in Fig. 2 and is described in procedural order in the following subsections. A significant limitation to this approach is the fact that only five (nonprecipitation variables) to eight (precipitation) years of observed data is available for calibration of the met generation model. Considering the shortness of the data-collection period, it was decided to employ the additional three years of precipitation data (1998 to 2000) in order to maximize the information content of the observed record. The generated time series can be considered a stationary representation of only this most recent climate period. Within DHSVM, hourly meteorology is extrapolated to each grid cell from the station input using inverse distance weighting (IDW) and is based on a grid-cell resolution of 30 m. The scaling of temperature and precipitation with elevation and the scaling of solar radiation with slope, aspect, and shading is also conducted within DHSVM based on topographic information derived from a digital elevation model (also 30-m resolution). b. Stochastic precipitation generation and precipitation gradient As indicated, the initial step in the met generation process is the direct generation of an hourly precipita-

5 866 JOURNAL OF HYDROMETEOROLOGY tion time series. It is desirable that not only should the generated hourly precipitation time series reproduce the stochastic structure of the observed hourly record, but that its associated daily time series also be statistically consistent. At the very least the generated daily time series had to adequately reproduce the daily precipitation occurrence structure of the observed data as the daily precipitation state is used to condition the generation of daily meteorological variables (see section 3c). Stochastic precipitation models based on the clustered rectangular pulses point process are capable of generating an adequate representation of the precipitation process at a range of temporal scales. Conceptually, storms of random interarrival times and duration are generated, and each storm is composed of a cluster of precipitation cells, approximated by a random occurrence of rectangular pulses, each of random intensity and duration. The total storm intensity at a point in time is obtained by summing the intensities of each rain cell active at that time. In this study hourly precipitation data were generated directly using the modified version of the Bartlett Lewis Rectangular Pulses Model (MBLRPM) as described by Rodriguez-Iturbe et al. (1988), which is based on a modification of the original work of Rodriguez-Iturbe et al. (1987). In the original Bartlett Lewis model storms are assumed to arrive randomly, following a Poisson process with rate parameter. Within each of these storms, a cell occurs at the storm origin and remaining cells arrive following a Poisson process with rate parameter, and the duration of each cell is exponentially distributed with parameter. The generation of cells within a storm terminates after a time that is exponentially distributed with parameter. The rainfall intensity of each cell follows an exponential distribution with mean value x. Rodriguez-Iturbe et al. (1988) modified the original model by assuming that the parameter varies randomly for each storm following a gamma distribution with shape parameter and scale parameter. The mean cell interarrival time and duration, namely 1 and 1,of each storm vary through the constant and dimensionless parameters / and /. This allows for different storms to retain a similar structure but to occur on different time scales, where the number of cells per storm has a geometric distribution with mean c 1 /. First- and second-order properties of the sixparameter (,,,,, and x ) MBLRPM are given in Rodriguez-Iturbe et al. (1987, 1988). Estimation of the six parameters of the MBLRPM is difficult and is typically accomplished by relating the analytical expressions for certain statistical moments to the corresponding observed values (Smithers et al. 2002). The resulting set of nonlinear equations is often solved simultaneously by minimizing a defined objective function, as unique roots rarely exist. Khaliq and Cunnane (1996) found that the MBLRPM most resembled historical data when more statistics than necessary TABLE 1. Monthly parameters of the modified Bartlett Lewis rectangular pulses model. Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec (h 1 ) * Fixed during parameter optimization. (h) 0.800* 0.800* 1.100* 1.100* 1.700* 1.500* 0.700* 0.800* 0.200* 2.100* * X (mm h 1 ) were used to estimate model parameters and suggested using 16 statistics: mean, variance, lag 1 autocorrelation, and dry probability at 1-, 6-, 12-, and 24-h levels of temporal aggregation. This statistic set was adopted for the current study, and parameters were optimized by minimizing a least squares objective function (Khaliq and Cunnane 1996) N o 2 Z min [F i(x) F i], (1) i 1 where F i (X) is the analytical expression for statistic i o computed using the parameter vector X, F i is the sta- tistic i estimated from the data, and N is the number of statistics used in parameter optimization. The rainfall generation process was assumed seasonally variable but stationary by month (72 parameters in total). The highly nonlinear analytical expressions for the first- and second-order moments of the MBLRPM renders them sensitive to any inconsistencies in the data, a situation that would become more acute in a short data record. As a result, it was found that the unconstrained optimization of all six parameters often resulted in physically unrealistic storm properties. Consequently, optimization for 11 months was conducted using a reduced parameter space by fixing the value of (Smithers et al. 2002). As storm and cell properties cannot be separated out from gauge data, no information on precipitation structure was available with which to constrain. Values in the range of 0.2 to 2.1 (which are within the range of values reported in the literature) were found to produce what the authors felt were qualitatively reasonable storm and cell properties, the final value being selected on the basis of the greatest minimization of the objective function. Generally, the value of was found to vary proportionally with average monthly precipitation. Optimized model parameters are given in Table 1, and derived precipitation properties, are shown in Fig. 3. Given that the horizontal separation between the Burn and Cabin climate stations is only 3000 m, it is likely that both locations are generally subject to the same

6 OCTOBER 2004 SCHNORBUS AND ALILA 867 FIG. 4. Double mass curve comparing Burn (P 1 ) and Cabin (P 2 ) daily precipitation (z 2 z m) for the months of Jan and Jun (for ). Plots show observed data (symbols) and fitted linear regression lines (lines). Regressions forced through origin. FIG. 3. Derived Burn station precipitation properties by month: (a) average storm frequency, storm duration, cell duration, and cells per storm and (b) average storm-precipitation depth and storm-precipitation intensity. storm events, such that temporal storm structure is roughly identical at both locations. The observed data do not refute this assumption, although the data do indicate that precipitation depth must be scaled for elevation between the two stations. Based on this assumption, the hourly precipitation time series for Cabin station was generated by scaling the Burn station precipitation time series using P grad (see below). This alleviated the need to employ a more complicated multisite stochastic technique (i.e., Cox and Isham 1994). At a subseasonal time scale the change of precipitation depth with elevation is highly variable, changes from storm to storm, and cannot be realistically calculated for each hourly time step. On a seasonal or monthly basis, however, precipitation in Redfish Creek shows a strong orographic influence with a clear trend of increasing accumulation with increasing elevation (Whitaker et al. 2003). Within DHSVM, precipitation depth at a location of higher elevation than a reference location is estimated by P2 [1 P grad(z 2 z 1)]P 1, (2) where P 2 is the precipitation depth at elevation z 2, P 1 is the precipitation depth at elevation z 1 (z 2 z 1 ), and P grad is a coefficient representing the relative increase of precipitation (relative to P 1 ) per unit increase in elevation (with units of m 1 ). The actual precipitation gradient (in units of m m 1 ) is represented by the quantity P 1 P grad. The value in square brackets in (2) (the ratio P 2 /P 1 ) was estimated on a monthly basis from the slope of the double mass curve of Burn and Cabin precipitation depth for the period 1992 to 2000, an example of which is given for January and June in Fig. 4. Given this empirical evidence, the assumed linear relationship advocated by (2) is considered reasonable. The difference in slopes is attributed to a shift from a predominance of cyclonic activity during the winter to a predominance of convective activity, which displays a smaller altitudinal effect, during the summer (Barry 1992). The monthly precipitation gradient calculated between Burn and Cabin stations was assumed to apply uniformly across the basin, and the hourly P grad for each station was set equal to the derived interstation monthly value (Fig. 2). This assumption, in conjunction with the IDW extrapolation employed by DHSVM, effectively means that basinwide precipitation is extrapolated solely from the generated Burn station precipitation time series. Current estimates of P grad are higher than those of Whitaker et al. (2003) because of the inclusion of the additional three years of precipitation data. The partitioning of precipitation into rain or snow is based on temperature thresholds and executed from within DHSVM. c. Stochastic daily met generation At the current stage of the met generation algorithm we now have an hourly precipitation time series for both Burn and Cabin climate stations (Fig. 2). The next stage involves the generation of the daily time series of nonprecipitation variables. Daily values of maximum temperature (T m ), minimum temperature (T n ), average dewpoint temperature (T d ), and average wind speed (U d ) were generated for Burn and Cabin stations based on the multivariate lag 1 autoregressive [MAR(1)] process originally proposed

7 868 JOURNAL OF HYDROMETEOROLOGY TABLE 2. Probability plot correlation coefficients for residual daily met variables. Italic values denotes rejection of hypothesis that residual is standard normal distributed ( 0.05). Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Burn station T T m T n T d U d Cabin station T T m T n T d U d by Matalas (1967) and further developed by Richardson (1981) and Parlange and Katz (2000). Observed values of T m and T n were set equal to the maximum and minimum T a, respectively, within a 24-h period (midnight to midnight). Observed T d is determined as the 24-h average of the hourly dewpoint temperatures, which were calculated from measurements of R and T a. Observed U d is simply the daily average of observed U. The MAR(1) process, originally developed as a singlesite model, is based on the generation of a time series of residual elements using the equation Zj AZj 1 B j. (3) Here, Z j denotes the column vector of dimension K (equal to 4 for a single station) whose elements are Z j (k), k 1, 2,..., K for the jth day, Z j 1 is the corresponding vector containing elements Z j 1 (k) for day j 1, A and B are K K parameter matrices, and j is a column vector of dimension K of error terms j (k); j has a standard normal distribution [N(0, 1)] and is uncorrelated with Z j 1. Extending the MAR(1) from a singleto a two-site model conceptually involves increasing the dimensionality of (3) from K to 2K, the details of which are described by Wilks (1999). The matrices A and B 1 are derived from A M 1 M and BB T 0 M 0 1 M 1 M T 0 M 1, where the matrices M i contain the cross correlations at lag i, whereby A and B are estimated using the sample correlations. The MAR(1) process is assumed stationary within months but variable between months; therefore, redimensionalization of the generated residuals is accomplished using monthly means and standard deviations. Generation of dimensional values is conditioned upon the precipitation status of the day, which requires separate means and standard deviations for wet and dry days. Model calibration and implementation is based on procedures described by Salas (1993) and Parlange and Katz (2000). The final twosite MAR(1) model preserves the serial and cross correlations of the daily met variables, both within a station and between the two stations. Use of the MAR(1) model requires that the residuals are normally distributed [N (0, 1)], which is generally assumed to hold true for T m, T n, and T d, but does not hold true for U d, which typically has a positively skewed distribution. Therefore, prior to model calibration, val- T ues of U d were square root transformed ( U d ; i.e., Parlange and Katz 2000), without making a distinction for seasonal variability or daily precipitation status. The applicability of the assumption of a normality to describe the distribution of daily residual met variables (transformed in the case of wind speed) for the study area is explored using the probability plot correlation coefficient goodness-of-fit test (Stedinger et al. 1993) for 0.05 (Table 2). For the winter months of January, February, and December the assumption of normality generally fails to capture the true distribution of the residual met variables. The situation is somewhat improved for the remaining months, and the probability plot correlation coefficients given in Table 2 indicate that the assumption of normality is accurate in approximately 60% of cases for both locations. A single transformation for wind speed produces results similar to the remaining untransformed variables. In the spirit of simplicity, the assumption of normality for T m, T n, T d, and T U d is retained. d. Empirical disaggregation At the current stage of the met generation algorithm we are left with an hourly precipitation time series and a daily time series for both Burn and Cabin climate stations (Fig. 2). The next stage involves the disaggregation of the daily time series into the final hourly time series of the appropriate met variables. 1) AIR TEMPERATURE AND LAPSE RATE Hourly temperature (T a ) was extrapolated from T m and T n independently at each station by assuming a periodic, quasi-sinusoidal diurnal variation that can be approximated by a Fourier series of the form (McCutchan 1979)

8 OCTOBER 2004 SCHNORBUS AND ALILA 869 T (t) A at b T cos( t/12) a c T sin( t/12) b T cos( t/6) 1 2 ctsin( t/6), (4) 2 where T a (t) is the hourly air temperature ( C) at time t (hours from midnight); A 0 is the annual mean of the Fourier series; a 1, b 1, b 2, c 1, and c 2 are regression coefficients; T is the daily mean temperature [ (T m T n )/2]; and T is the daily range of temperature ( T m T n ). Assessment of bias and root-mean-square error suggested that using monthly parameters in (4) produced negligible improvement in model accuracy versus the use of annual parameters, as such annual parameters were used in the met generation model. Hourly temperature lapse rate (T lapse ) was estimated using the extrapolated Burn and Cabin hourly air temperatures as follows: T lapse(t) [T a2(t) T a1(t)]/(z2 z 1 ), (5) where superscripts 1 and 2 denote Burn and Cabin stations, respectively. 2) RELATIVE HUMIDITY Although water vapor pressure (e) varies diurnally, the magnitude of e fluctuations are considerably less than those for temperature; therefore, it is assumed that e, and therefore T d, is constant throughout the day (Campbell and Norman 1998). Estimates of R at time t can then be calculated as the ratio of daily average e to hourly saturation vapor pressure (e s ), e s(t d) R(t) 100, (6) e s[t a(t)] where e s (daily average e) is calculated from T a (T d ) using the common Magnus formulation (Abbott and Tabony 1985) at e (T ) exp a s a, (7) b T a with T a (T d ) expressed in degrees Celsius, and vapor pressure expressed in millibars. The values of the coefficients a and b in (7) are a 17.38, b and a 22.4, b for evaporation over water and sublimation over ice, respectively. 3) WIND SPEED The spatial and temporal complexity of the wind field in mountainous topography renders its disaggregation to and quantification at a subdaily time scale problematic and complex (Barry 1992). In order to limit model complexity it is assumed that hourly wind speed is equal to the daily average wind speed [i.e., U(t) U d ]. 4) BEAM AND DIFFUSE SOLAR IRRADIANCE Hourly global irradiance (S g ) at the ground surface was estimated by calculating above-atmosphere potential irradiance ( S g ), using standard solar geometry (i.e., o Gates 1980), attenuated for atmospheric transmissivity (H), which is assumed to be constant throughout the day. The topography in the immediate vicinity of Cabin station (Fig. 1) produces observations of S g that are confounded due to shading effects. For this reason, H was derived for Burn station only and, assuming uniform atmospheric properties across the basin (i.e., Tarboton et al. 2000), applied directly to Cabin station (Fig. 2). Daily atmospheric transmissivity was estimated from the relationship (Bristow and Campbell 1984) c H H max [1 exp( B T )], (8) where T is the diurnal range in air temperature {T m (j) [T n (j) T n (j 1)]/2, where j is the day identifier}, H max is the maximum observed clear-sky atmospheric transmissivity, and B and C are empirical coefficients, optimized using S g data from the period July 1994 to December The use of a constant B in (8) represents a slight simplification over the method originally used by Whitaker et al. (2003) to derive S g estimates for the period October 1992 to July Inspection of the hourly radiation data for Burn station reveals that hourly H is, in fact, nonuniform throughout the daylight hours. Transmissivity is a function of solar angle and is minimum at dawn and dusk and maximum during midday, with the daytime H max at times reaching 10 times the minimum value. As a result, the relationship in (8) was optimized using an observed H estimated from radiation data for the midday period between 1000 to 1400 LST. Final parameter estimates for (8) are 0.89 for H max, the maximum observed midday average H, for B, locally optimized, and 2.4 for C, the value derived by Bristow and Campbell (1984) using data from western Washington. Hourly values of S b and S d were partitioned from S g using the following empirical formula (Black et al. 1991): Tt 4.900T 2 t 1.796T t 2.058T t, Tt 0.80 (9) 0.130, T S d 3 4 S g As only global solar irradiance was observed at Burn station, the selected technique of partitioning S g into S b and S d is quite arbitrary, and its performance can not be evaluated. The relationship proposed in (9) was selected based on its use in deriving S b and S d for the met time series used in the original calibration of DHSVM for Redfish Creek (Whitaker et al. 2003). Values of S b and S d were calculated assuming a horizontal surface, the influence of topography being accounted for explicitly within DHSVM. t

9 870 JOURNAL OF HYDROMETEOROLOGY FIG. 5. Observed and generated monthly probabilities that a given day is dry (P 0). 5) LONGWAVE SKY IRRADIANCE Hourly longwave irradiance, L, was calculated based on the Stefan Boltzmann law (Gates 1980), an effective clear-sky atmospheric emmissivity, and a cloud modification factor. The effective clear-sky atmospheric emmissivity was calculated separately for each station from c 0.70 [ e exp(1500/t a )], where e is calculated from (7) and T a is given as an absolute temperature (K) (Idso 1981). The cloud modification factor was given as K [(1 H/H max )/0.65]. The cloud modification factor was calculated for Burn station and extrapolated to Cabin station assuming uniform cloud cover over the basin. The adequacy of the proposed approach can not be assessed, as observations of L were not collected at either Burn or Cabin stations. Again, the given approach was selected as it was used to derive the original L time series used in model calibration (Whitaker et al. 2003). 4. Results a. Met generator performance 1) PRECIPITATION The stochastic generation of daily meteorology is based on the conditioning of the generated variables on the precipitation status of the day, either wet or dry. As such, it is informative to compare the generated marginal probability of the occurrence of a dry day for the observed and generated daily precipitation time series. As this statistic was used directly in model parameterization it is expected that it will be accurately reproduced, and this is for the most part verified by Fig. 5. Generated dry-day probabilities are within 10% of observed, with the exception of May, which has a 20% overconditioning error in dry-day probabilities (i.e., less frequent dry days). It is expected that errors due to under- or overconditioning for dry-day occurrence in the generation of daily met will be minimal. FIG. 6. Burn station monthly average observed ( ) and generated (15-yr run) precipitation and empirical monthly precipitation gradients. Seasonal precipitation, given as monthly averages, of the observed ( ) and generated time series (based on 15-yr run) is given in Fig. 6. Empirical P grad values are also included in Fig. 6 for comparison purposes. The seasonal precipitation pattern is well represented, including the two seasonal maxima during May June and October November, and this despite the overconditioning of precipitation occurrence in May. Relative errors in monthly precipitation are less than 10% for all months, with the exception of August, which has a relative error of 21%. On an annual basis, observed (1010 mm) and generated (1022 mm) precipitation differ by only 1%, and total generated winter precipitation [October February (ONDJF)] of 505 mm is only 2% lower than the observed value of 515 mm. MBLRPM accuracy was further assessed by comparing the observed and generated distributions of dry spell (P 0) durations, wet spell (P 0) durations, and conditional (wet spell) precipitation depth. The use of very large sample sizes in the current study makes it difficult to test hypotheses of statistical equality between two empirical distribution functions as any model departure, no matter how small, will be detected as significant. In such cases it is almost impossible to generate a statistically equivalent model of an observed distribution function (Martin-Löf 1974). Therefore, model fit is only assessed graphically by comparing plots of the empirical cumulative density function (ECDF). The comparisons for the month of July and December, which are representative of good and poor model performance, respectively, are shown in Fig. 7. In July the distribution of dry spell durations is accurately reproduced, although wet spell durations tend to be underestimated, and conditional precipitation depth is slightly overestimated. In December both dry and wet spells are overestimated dry spells substantially more so than wet spells and conditional precipitation depth is overestimated. However, regardless of discrepancies, the generated distributions of wet and dry spell durations and conditional precipitation depth for each month are structured in such

10 OCTOBER 2004 SCHNORBUS AND ALILA 871 FIG. 9. As in Fig. 8, but for Jun. are of slightly shorter duration and higher intensity in July. Average precipitation for both months is roughly identical. No seasonal pattern is apparent in model performance when examined over all months. In general, generated dry spell durations show the highest discrepancy, and wet spell durations show the lowest. FIG. 7. ECDF of hourly dry spell duration, wet spell duration, and conditional (P 0) precipitation depth for Jul and Dec. Note that the ECDF for wet spell duration is truncated at 1 h (i.e., minimum wet spell duration is 1 h). a way that mean seasonal precipitation accumulation is accurately maintained (Fig. 6). Additionally, seasonal changes in precipitation structure are also captured by the generated series: wet spells occur more frequently and are of shorter duration and low intensity in December, whereas wet spells occur much less frequently and FIG. 8. ECDF of Burn station daily maximum temperature, minimum temperature, average dewpoint temperature, and average wind speed for Jan. Observed distribution based on 5-yr series ( ); generated distribution based on 15-yr series. 2) DAILY MET DATA The observed and generated frequency distributions of T m, T n, T d, and U d are compared for Burn station for January and June in Figs. 8 and 9, respectively. The generation of the daily met time series is generally poor in January (Fig. 8), mostly attributable to the nonnormal distribution of all four met residuals (Table 2), although slight errors in the generated daily precipitation occurrence (Fig. 5) are also a factor. The observed distributions of T m, T n, and T d for January display distinct bimodal behavior (also present in February) that cannot be accounted for with a normal distribution; the observed distribution of U d is also more skewed than the generated distribution. In June (Fig. 9) the daily model performs quite well in reproducing the frequency structure of the daily met variables for Burn station, despite the nonnormal distribution of the observed residuals for T d and T U d (Table 2). The fact that the daily distributions are unimodal in June (and most months) likely contributes to the improved agreement between modeled and observed values. The stochastic generator performance generally correlates with the results of Table 2 in that poorer performance occurs for those months for which most of the residuals can not be adequately described by the standard normal distribution, specifically December, January, and February. Performance of the daily stochastic model at Burn station for the remaining months is similar to that of June, and results for Cabin station mirror those of Burn.

11 872 JOURNAL OF HYDROMETEOROLOGY TABLE 3. Model efficiency and bias for nonprecipitation variables. Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec E! B! T a T lapse U R S g T a T lapse U R S g ) HOURLY NONPRECIPITATION MET DATA An assessment of the overall accuracy of generated T a, T lapse, U, R, and S g was done by comparing ECDFs of the 5-yr observed and a 15-yr generated time series. The empirical frequency distributions are only compared graphically [owing to the very large sample sizes; see earlier discussion in section 4a(1)], yet they provide an efficient and effective means of showing the magnitude and nature of any model discrepancies. Agreement between the observed and generated met time series was assessed using the monthly bias statistic, B! GO /, where Gand Oare the monthly mean of the generated and observed hourly data, respectively, in which greater accuracy (lower bias) is represented by values closer to unity. The ability of the met generator to capture the correct diurnal structure of the nonprecipitation met variables was assessed using the mean hourly value for each month of the observed and generated time series (Waichler and Wigmosta 2003). The efficiency statistic of Nash and Sutcliffe (1970) 24 t t 1 24 t t 1 (O G t) 2 E! 1 (10) (O O ) 2 is used to assess model fit on the basis of the mean hourly value, where O t and G t are, respectively, the observed and generated monthly average value at time t. The E! statistic has a possible range of to 1, where greater accuracy is represented by values closer to unity; a value of E! less than zero indicates that the error variance is greater than the observation variance. The E! and B! statistics for hourly nonprecipitation variables are summarized in Table 3. The time series of S b, S d, and L were not compared owing to a lack of observed data. For most months, generated T a is generally in good agreement with observations (Figs. 10 and 11), E! is greater than 0.85, and B! is close to 1.0 (Table 3). The months of November through February are exceptions, t FIG. 10. ECDF of 5-yr observed and 15-yr generated hourly air temperature.

12 OCTOBER 2004 SCHNORBUS AND ALILA 873 FIG. 11. Average hourly values of 5-yr observed and 15-yr generated hourly air temperature. in which either E! is negative and/or B! is 1.0. The large B! between average generated and observed temperature in March (5.18) is due to the very low observed mean temperature ( 0.2 C), as by all other indications generated T a for this month shows good agreement with observations. Generated T lapse exhibits greater discrepancy with the observed data than does air temperature (Fig. 12). Lapse rates are generally underestimated (less negative) from November through January and overestimated from May through September. Inversions also occur more frequently in the generated time series during September through January. The generated mean hourly T lapse is greatly simplified compared to the observed profile; the minimum lapse rate consistently occurs at midday whereas the strongest lapse rates occur in late evening, entirely failing, for the most part, to capture observed mean hourly T lapse, which is highly variable between months (Fig. 13). As one would expect, the generated U data fail to accurately reproduce the ECDF and mean hourly averages of the observed data for most months (Figs. 14 and 15), the exceptions being October through December, in which observed U shows little variability throughout the day. As expected, however, mean wind speed is accurately captured in all months (as indicated by B! close to unity; Table 3), generating E! values near or equal to zero. Although some accuracy in the generation of U is due to inaccuracy in the generation of U d (Figs. 8 and 9), the major source of error is the assumption that U is constant throughout the day (Fig. 15). Generated monthly R distributions show slightly less variability than the corresponding observed data (Fig. 16). For April through September both the observed and generated mean hourly R show very close correlation with air temperature (Fig. 17), and corresponding E! is high (Table 3). Very large errors between the generated and observed data are apparent for October through March, as shown by large negative E! in Table 3. During the months of November through January R remains, FIG. 12. ECDF of 5-yr observed and 15-yr generated hourly temperature lapse rate.

13 874 JOURNAL OF HYDROMETEOROLOGY FIG. 13. Average hourly values of 5-yr observed and 15-yr generated hourly temperature lapse rate. on average, uniform throughout the day, suggesting that the absolute vapor content of the air varies diurnally. In all months B! is close to unity, indicating the monthly R mean is accurately reproduced throughout the year. Given that errors in reproducing the correct day length are negligible and, when discrepancies exists, they involve S g values that are relatively small compared to midday values, assessment of model by ECDF for S g is restricted to daytime (S g 0) values only. The generated monthly frequency distributions of conditional solar irradiance (S g ) show a poor model fit during October through April, consistently underestimating observed S g (Fig. 18). Model fit is substantially improved for the period of May through September, although S g is still underestimated. Hourly average S g reveals a similar story: in all months midday S g is underestimated, with the largest relative errors occurring in the winter months (Fig. 19). The B! between the observed and generated time series indicates that average S g is underestimated by as much as 50% in November (worst month) and 6% in August (best month). Model E! for average hourly S g is positive in all months and tends to be slightly higher in summer than in winter (Table 3). b. Hydrologic simulation An initial analysis of the coupling of full synthetic met time series with DHSVM is conducted by comparing the simulated average annual water balance using both observed (control; 5 yr) and entirely generated met data (15 yr), as shown in Table 4. The comparison is restricted to simulated water balance components, as insufficient data exists with which to derive an accurate determination of the actual water balance components for the period. The water balance results indicate that none of the individual components shows substantial change ( 10%) when simulated using the full synthetic met time series. Average annual accu- FIG. 14. ECDF of 5-yr observed and 15-yr generated hourly wind speed.

14 OCTOBER 2004 SCHNORBUS AND ALILA 875 FIG. 15. Average hourly values of 5-yr observed and 15-yr generated hourly wind speed. mulated streamflow, SWE, melt, and evapotranspiration (ET) simulated using both generated and observed met input time series are given in Fig. 20. In general, the simulated temporal distribution of major water balance components is little affected by the use of generated as opposed to observed met data; however, some discrepancies are apparent. The basinwide ET rate tends to be underestimated when using generated met data during winter and spring, such that cumulative ET differs by 31% as of 1 May. Conversely, the ET rate tends to be slightly overestimated during summer and fall such that by the end of the water year (30 October) cumulative ET differs by less than 2%. Simulated snow accumulation is overestimated when using the generated met data, and average 1 April SWE differs by 9% between the two series. This discrepancy is due to increased precipitation and reduced winter ET when using the generated met input. Basinwide melt and streamflow show little sensitivity to the use of generated met data. The spatial distribution of simulated peak SWE, assumed equivalent to 1 April SWE, was used to further evaluate the coupled models. The study basin was divided into 10 elevation bands based on equal elevation intervals of 167 m, and area-average simulated peak SWE was calculated for each band for each simulation year. The median SWE simulated using observed (control; 5 yr) and synthetic met data (generated; based on 15-yr time series) are plotted for each elevation band in Fig. 21. The SWE values simulated with the synthetic met data are shown with the 50% interquartile range. The area of each elevation band, as a proportion of total basin area, is plotted as a separate series in Fig. 21. The use of the synthetic met data overestimates 1 April SWE below and above 2034-m elevation, respectively. The largest discrepancy between generated and control SWE occurs in the elevation range of m, within which generated SWE is roughly 3 times that of the control value. Surprisingly, this is also the elevation range containing the Burn climate station (1290 m), from which basinwide precipitation is estimated. Re- FIG. 16. ECDF of 5-yr observed and 15-yr generated hourly relative humidity.

15 876 JOURNAL OF HYDROMETEOROLOGY FIG. 17. Average hourly values of 5-yr observed and 15-yr generated hourly relative humidity. gardless, within the elevation range of m, within which lies 85% of basin area, peak SWE simulated with the observed and synthetic met data compares to within 2%. The flow duration curves (FDC) developed from simulated streamflow using both observed (5 yr) and synthetic (15 yr) met data, and from observed streamflow (5 yr) are compared in Fig. 22. Because of gaps in the observed hourly streamflow record prior to 1995, data are presented for daily discharge. The FDCs are constructed using the annual-based FDC technique and are plotted as a median FDC with the 5% to 95% confidence intervals indicated for the observed FDC [using the method of Vogel and Fennessey (1994)]. The simulated FDCs are in close agreement for the full range of exceedance probabilities. Of greater note is that differences between the simulated FDCs and the observed FDC; the two simulated median FDCs fall within the 90% confidence region of the observed FDC curve only for exceedance probabilities Both simulated FDCs underestimate observed discharge for exceedance probabilities Overall, the difference between the two simulated FDCs is substantially less than the differences between each simulated FDC and the observed FDC. The average hourly discharge hydrographs for simulated (using observed and synthetic met data) and observed discharge are shown in Fig. 23. The average hydrographs indicate that little difference exists between all three streamflow series during the annual freshet peak (1 May to 1 August) and that the two simulated streamflow series are in close agreement for the entire hydrograph. However, the underestimation of observed fall and winter base flow by the two simulated streamflow series is again evident. The three average streamflow series were further compared using three statistics. The first statistic calculates the volume error V between control and test flows, where FIG. 18. ECDF of 5-yr observed and 15-yr generated hourly conditional (S g 0) global solar irradiance.

16 OCTOBER 2004 SCHNORBUS AND ALILA 877 FIG. 19. Average hourly values of 5-yr observed and 15-yr generated hourly global solar irradiance. Vtest Vcon V, (11) V con and the flow volume V (control or test) for a given period is calculated as the summation of flow rates Q over the period of simulation N as N i V Q. (12) i 1 The coefficient of determination, D!, relates how well the test hydrographs compares in shape to the control hydrograph and depends only on timing, not on volume (Whitaker et al. 2002): N i i (Qcon Q con)(qtest Q test) i 1 N N i 2 i 2 (Qcon Q con) (Qtest Q test) i 1 i 1 D!. (13) The third statistic is the efficiency statistic given by (10). All statistics are based on hourly discharge. Greater accuracy is represented by values of V close to zero and D! close to unity. Results are presented in Table 5. The V statistic shows that the volume error between the two simulated hydrographs is an order of magnitude less than that between each simulated hydrograph and the observed hydrograph. The D! statistic shows that all three hydrographs compare equally well (all have the same shape), whereas the E! statistic reveals that the two simulated hydrographs compare better to each other than to the observed hydrograph. It is clear that model error in replicating the observed average hydrograph is greater than any differences that exists between the two simulated hydrographs because of the use of synthetic input met data. 5. Discussion With regard to precipitation the authors had anticipated that storm structure would be an important mechanism for controlling evapotranspiration. However, within the climate of Redfish Creek, ET forms a very small component of annual water balance (Table 4) and has a negligible impact on peak flow (Fig. 20). As such, modeled streamflow results appear relatively insensitive to discrepancies between the generated and observed temporal structure of precipitation (although this is not conclusive, as ET is also sensitive to errors in U, R, and S g ). Given that ET is small, seasonal precipitation depth is more important than subdaily storm structure, and the MBLRPM is satisfactory in this respect. Hydrologic simulation of a coastal watershed dominated by frontal TABLE 4. Average simulated annual water balance using observed (control) and generated met. Here, P: Precipitation; ET: evapotranspiration; T: transpiration; EInt: evapotranspiration/sublimation from interception storage; Q: streamflow; and subscript o denotes overstory only. Met input Observed Generated P (mm) ET (mm) T o (mm) EInt o (mm) Water balance ET o (mm) Q (mm) Q/P ET/P Change (Proportion of control) Generated

17 878 JOURNAL OF HYDROMETEOROLOGY FIG. 20. Simulated accumulation of SWE, streamflow (Q), melt, and ET averaged over 5 and 15 yr using observed and generated met input, respectively. precipitation of long duration and low intensity was also found to be relatively insensitive to the subdaily time structure of precipitation input (Waichler and Wigmosta 2003). However, the ability to correctly model subdaily precipitation structure may prove sensitive for modeling the water balance over smaller time scales, in areas that experience more frequent convective precipitation (short duration, high intensity), in less humid climates, or in other applications (i.e., agricultural modeling). As such, performance of the model in reproducing the temporal structure of hourly precipitation could be improved by explicitly accounting for the structure of wet and dry spells during parameter optimization. Onof et al. (1994) describe an optimization procedure for the MPLRPM whereby the parameter is initially optimized by using observed and simulated values for mean dry spell duration and mean number of wet spells per month. With fixed, the remaining five parameters are then optimized in the usual manner. From a hydrologic perspective the spatial variation FIG. 22. Median FDCs of daily discharge comparing simulated discharge using synthetic (Gen Met; 15 yr) and observed (Obs Met) met data and observed discharge (5 yr). of precipitation in Redfish Creek is important only so far as it affects snow accumulation, which is well described using elevation as an explanatory variable. This may not be a defensible assumption in situations where basin scale is increased and/or the streamflow is derived mainly from rainfall. Such situations will likely require a more complicated multisite precipitation model to account for spatial variability in both precipitation occurrence and intensity. Additionally, the generation of precipitation that derives from complex (highly variable) precipitation fields will likely require a higher density of station inputs, particularly for mountainous topography. The use of a Fourier series to disaggregate hourly air temperature proves accurate for reproducing the monthly temperature statistics at each individual station for March through October, even when using annual (as opposed to monthly) parameters. However, this approach breaks down during the remaining months. The FIG. 21. Area-average 1 Apr SWE by elevation band (equal elevation interval) simulated using both observed (control) and synthetic (generated) met data. Also shown is the area of each elevation band as a proportion of the total basin area. FIG. 23. Average annual hydrographs for hourly discharge simulated using synthetic (Gen Met; 15 yr) and observed (Obs Met; 5 yr) met data and for hourly observed discharge (4 yr).