2, and L), (1)single reach modelThe single reach model determines the travel time distribution for a group of fish migrating through a single reach and is described elsewhere (Zabel and Anderson 1996; Zabel 1994). This model starts with a point release of fish at time = t0. The travel time distribution is based on the length of the reach (L) and two parameters: r, which determines the downstream migration rate, and
2, which determines the rate of population spreading.
pt = Prob(arriving during t-th time period, given values r,with t = (1,2,3,...). If there are N recaptured individuals in the cohort, then
The travel time distribution has a long right tail (Figure 1) and fits well to observed travel time distributions (Zabel and Anderson 1996).= expected number of individuals arriving during t-th time period. (2)
multiple reach modelThe multiple reach model moves a cohort of fish through a series of reaches. The model begins with a point release at t0. Based on the reach model, the arrival time distribution is determined at the next downstream site. This arrival distribution is then used as a departure distribution for the next reach. The model iterates through each departure time and distributes these fish at the next downstream site according to the reach travel time model. All fish that arrive at a site in the same time interval but departed during different time intervals from the upstream site are combined together in the arrival time distribution. The new arrival time distribution is then used as a departure distribution for the next reach. This cohort of fish is moved from reach to reach in this manner until the end of the river is reached.
The terms inside of the summation multiply the probability of a fish departing from the upstream site during the j-th time period by the probability of the fish taking exactly t-j time periods to travel through the reach. The summation is over all the possible combinations of departure times and travel times that result in fish arriving at the i-th site during the t-th time period.. (3)
where Ni is the total number of fish observed at the i-th site. The average travel time to the i-th site is computed as, (4)
where S is the length of the migratory season. These expected average travel times are compared to the observed ones in the statistical analysis., (5)
Migration rate modelTo implement the multiple reach model, migration rates (rt) must be provided on a per reach and per time step basis. The general model assumes fish velocity depends on a flow related and a flow independent term. Both terms are time dependent, reflecting the change in fish behavior through time. The general form of the full model is
The first term characterizes the flow independent component of migration which is related to the fish maturity relative to release time, TRLS. The second terms characterizes the flow dependent component of migration as expressed by the average water velocity (. (6)
) and a seasonal factor TSEASN that characterizes how the fish's behavior relative to water velocity changes over time.
model 1) The null model assumes that r is constant over time with an average value 
0:
Variation about the average rate is expressed by. (7)
t.
model 2) This model assumes a linear relationship between migration rate and river velocity:
. (8)
is the average velocity during the average migration period for each of the reaches. River velocity is assumed to be proportional to river flow through a dimensional argument:
where X is the cross sectional area of the reservoir., (9)
where t is the Julian date in the season. Equation (10) is based on the following four parameters:, (10)
0 - determines the flow independent migration (km/day); FLOW - determines the proportion of the river velocity used for downstream migration when the fish are full smolted (non-dimensional); 
- slope parameter that determines how quickly the flow effects shift from early season to late season behavior (1/days); Equation (11) introduces the following terms:. (11)
1 - slope parameter that determines rate of change of the experience effect (1/days); 0 and 1 - determine the magnitude of the flow independent migration rate (km/day). 0 and 1 are combined in the following way to determine the flow-independent contribution to migration rate:
MIN = 0 + 1/2 (minimum flow independent migration rate); MAX = 0 + 1 (maximum flow independent migration rate).1 = 0.15, which guarantees that 95 percent of the difference between MAX and MIN is attained within 25 days. This choice was based on the data analyzed in this application and ensured that unrealistically high migration rates would not be obtained when applying the model to lower reaches.
Model implementationThe models were run utilizing the Columbia River Salmon Passage (CRiSP) model, a management tool for evaluating the effect of Columbia River hydrosystem operations on juvenile salmon (Anderson et al. 1996). The CRiSP model imposes mortality due to dam passage and in the reservoirs due to predation and gas bubble disease. The modeled survival was calibrated to survival studies (Iwamoto et al. 1994; Muir et al. 1995; Muir et al. 1996) based on fish of similar origin and migrating through the same reaches as the ones we analyze below. In addition, the model keeps track of the numbers of fish collected at dams and barged through the system for release below the last dam on the river. These factors can cause slight changes in modeled travel times, so they were incorporated into the model predictions.
Statistical methodsThe objectives of the statistical analysis were to estimate parameters and standard errors, assess how well the models compare to the data, and determine the appropriate level of complexity for the migration rate models.
estimating parametersThe modeled average travel times are a function of the model chosen and the particular parameter values selected. The migration rate parameters are estimated by a least-squares minimization (with respect to the parameters) of the following equation:
where O is the total number of observation sites, C is the total number of cohorts,(12)
is the modeled average travel time to the i-th site by the k-th cohort, and 
is the observed average travel time to the i-th site by the k-th cohort. This equation is fit using a Levenberg-Marquardt routine (Fletcher 1990; Press, et al. 1994), with derivatives calculated numerically using a finite element method (Gill 1981; Seber and Wild 1989).
standard errorsApproximate standard errors of the parameters were calculated following procedures for nonlinear least squares regression (Bates and Watts 1988). The model function was linearized at the optimal parameter values, and then linear least squares calculations were implemented (Weisberg 1980). Approximate 95 per cent confidence intervals can be constructed by adding and subtracting twice the standard errors from the parameter values. Since these values are approximate, they are not used for inference but characterize the stability of the parameter estimates.
estimating rate of population spreadingWhile the purpose of this paper is to study the variability in migration rate, in order to implement the reach model, the rate of spreading (
2) of the cohorts must also be specified. The units for 2 are km2/day. Although 2 is also variable from cohort to cohort (Zabel and Anderson 1996), for this paper we treat it as constant among all cohorts to simplify the analysis. Also, the travel time model predictions are not as sensitive to variability in 2 as to variability in migration rate (Zabel 1994).
The estimate of is based on the spread of the travel time distribution; this information is lost when computing average travel times. Thus 2 is estimated separately after the migration rate parameters are estimated using equation (12). To estimate 2, a finer resolution of the data is used. The unit of comparison between the model and the data is the number of individuals from each cohort observed per time step at each of the observation sites. Since the variance associated with this measure is highly variable, generalized least squares (Draper and Smith 1981; Seber and Wild 1989) is used, with each element of the summation weighted by the variance. The equation to be minimized is
where nijt is the observed number of fish arriving at the ith site from the j-th cohort during the t-th time period and, (13)
is the expected number. Vijt is the variance (under a multinomial model see (Zabel 1994)) associated with this group and is calculated as
where nij is the number of fish from the jth cohort observed at the ith site. Equation (13) is also fit using a Levenberg-Marquardt routine, and the standard errors are calculated in the same manner as with the migration rate parameters., (14)
model comparisonsTo compare the performance of the models, a modified R2 value is reported as the sum of squares of each model as compared to the sum of squares of the mean model (model 1):
where SS1 is the sum of squares for the mean model, and SSA is the sum of squares for the more complex (alternative) model. The R2 values gives the percent reduction in the sum of squares for the alternative model as compared to the null model.(15)
DataPassive integrated transponder (PIT) tags are used to monitor individual fish. The tag, 12 mm long, is inserted in the fish's body cavity and contains a microchip that is programmed with individual fish identification codes (Prentice et al. 1990). The system yields information about passage times of individuals at interrogation sites. The tags do not seem to adversely affect the fish in terms of survival or swimming performance (Prentice et al. 1990).
The model was evaluated with run-of-the-river fish (hatchery and wild stocks) yearling chinook. The fish were captured tagged and released at the Snake River trap and observed at Lower Granite, Little Goose and McNary Dams over a migratory route of 277 km (Figure 2). Separate releases were made daily during the from early April to early May. Although these fish are classified as run-of-the-river fish, it is likely that the vast majority were yearling chinooks based on the distribution of lengths (most fish longer than 110 millimeters) and the timing of migration (early spring). This is consistent with other treatments of these fish (e.g., Fish Passage Center 1991). We did not analyze fish released after May 10 because reduced length distributions after this date indicate a possible presence of subyearling chinook which have different migratory behavior.
Release cohorts were formed by combining releases from up to three consecutive days to achieve sample sizes of at least 80 individuals observed at Lower Granite Dam. 78 cohorts are analyzed representing releases from 1989-1996. Table 1 contains release dates, average travel times to the three observation sites, and sample sizes for all the cohorts. Flow and temperature information was obtained from the Army Corps of Engineers, Portland, Oregon.
PredictionsPredicted cumulative passage distributions were generated at the three downstream observation sites for the 1996 fish. These predictions were derived from parameters estimated from the 1989-1995 data, so the predictions are independent of the 1996 data. The downstream passage distributions for the 1996 cohorts were combined to generate one distribution for all the cohorts at each of the observation sites. The predicted cumulative passage distributions were then compared to the observed cumulative passage distributions.
