As a first example of the application of the travel time model to data, I will apply the model to data of the travel time of fish through a single reservoir. The reservoir is the John Day Pool, the reservoir between McNary and John Day Dams, and the fish observed here are yearling chinook salmon. This data set has several desirable features. First, the fish are traveling through a single relatively homogeneous reach - there are no intervening dams or major tributaries. Also, the study was repeated over four years, and in each year, releases occurred over many days, providing migration characteristics in a variety of conditions. Finally, the fish are active migrants collected from the river. Fish that are raised in hatcheries and then released usually undergo a period of delay before initiating active migration; this adds further complications to the travel time model. Essentially, this data set is a test of whether the simple travel time model can form a basis on to which further complexity can be added as needed.
The data consist of yearling chinook salmon collected at McNary Dam, freeze branded with a unique brand (on a daily basis) and then released back into the river below the dam. Approximately 1,000 fish were marked and released per day. Marked fish were sampled as they passed John Day Dam, 122.9 km downstream from the release point. Data were collected over five week periods in 1989, 1990, and 1991; in 1992, six weeks of data were collected. Fish collected and released for 5 days each week (Monday through Friday) were lumped together into weekly cohorts to achieve adequate sample sizes. Cohorts below 20 in sample size were excluded from the analysis. Week 1 of 1990 was excluded because a fire at John Day Dam precluded data collection, and week 5 of 1990 was excluded because the collection facilities were shut down before the groups of this cohort completely passed the dam. A total of sixteen cohorts over the four years were analyzed (Table 4.1). The first two years of the data set have been analyzed statistically by Stevenson and Olson (1991), and they provide a fuller description of the experimental design.
| Table 4.1 Descriptive information for the 16 cohorts used in the data analysis. The date of release is for the first release group of the cohort. See text for the procedure used to calculate the average flows for the cohorts. | ||||
|---|---|---|---|---|
| cohort release information | number sampled | ave. flow (kcfs) | ||
| # | year - week | date (Julian date) | ||
| 1 | 1989 - 1 | May 01 (121) | 27 | 263.5 |
| 2 | 1989 - 2 | May 08 (128) | 57 | 283.2 |
| 3 | 1989 - 3 | May 15 (135) | 48 | 258.9 |
| 4 | 1989 - 4 | May 22 (142) | 32 | 228.3 |
| 5 | 1990 - 2 | April 30 (120) | 36 | 233.9 |
| 6 | 1990 - 3 | May 07 (127) | 32 | 231.3 |
| 7 | 1990 - 4 | May 14 (134) | 24 | 196.3 |
| 8 | 1991 - 1 | April 22 (112) | 38 | 250.0 |
| 9 | 1991 - 2 | April 29 (119) | 20 | 236.7 |
| 10 | 1991 - 3 | May 06 (126) | 24 | 249.4 |
| 11 | 1992 - 1 | April 20 (111) | 85 | 178.3 |
| 12 | 1992 - 2 | April 27 (118) | 88 | 195.4 |
| 13 | 1992 - 3 | May 04 (125) | 88 | 206.4 |
| 14 | 1992 - 4 | May 11 (132) | 86 | 205.2 |
| 15 | 1992 - 5 | May 18 (139) | 119 | 202.7 |
| 16 | 1992 - 6 | May 25 (146) | 80 | 195.0 |
In this application, I apply the discrete time, two parameter, travel time model, equation (4.8), to these data. Parameters are estimated numerically using maximum likelihood based on the multinomial distribution and a downhill simplex fitting routine (Press, et al., 1988). Nonparametric 95 per cent confidence intervals are constructed using the bootstrap methods described in chapter 3. Model performance is assessed using Pearson's X2 statistic.
Plots of the data with the best fit model show that the model captures the general behavior of the observed travel time distributions (Figure 4.12). Parameter estimates and confidence intervals are provided in Table 4.2. Estimates of r range from 11.4 to 32.2 km/day, with 95 per cent confidence intervals ranging in width from 4.6 to 17.5 km/day. Estimates of 
ranged from 15.7 to 39.4 km/day1/2 with 95 per cent confidence intervals ranging in width from 7.3 to 40.6 km/day1/2. In three cases the model is rejected at the 
= 0.05 level; for four additional cases, the model is rejected at the = 0.10 level.
The difficulty in implementing the model will arise in choosing appropriate parameter values. Table 4.2 reveals variability among cohorts in estimates of r and . In the next chapter I will attempt to relate the variability in parameter estimates to observable factors such as river flow and temperature. Variability also arises from sampling error as demonstrated by the broad confidence intervals obtained in the bootstrap analysis. More studies with larger sample sizes would decrease this uncertainty.
| Table 4.2 Parameter estimates, confidence intervals and goodness-of-fit results for the sixteen cohorts. The units for r are km/day, and the units for are km/day½. For the goodness-of-fit test results, df refers to the degrees of freedom. The model is rejected for small p-values, e.g., p > , with often chosen as 0.05. | ||||||
|---|---|---|---|---|---|---|
| cohort | parameter estimation | goodness-of-fit | ||||
| # | year - week |  (95% C.I.) |
 (95% C.I.) |
 2 |
df | p |
| 1 | 1989 - 1 | 25.5 (18.3, 28.8) | 15.7 (10.9, 26.8) | 1.87 | 3 | 0.600 |
| 2 | 1989 - 2 | 28.1 (24.2, 32.8) | 35.1 (26.6, 42.7) | 4.97 | 6 | 0.548 |
| 3 | 1989 - 3 | 32.2 (26.6, 37.5) | 25.1 (18.7, 31.3) | 6.20 | 3 | 0.112 |
| 4 | 1989 - 4 | 26.3 (20.6, 38.1) | 39.4 (11.7, 52.3) | 8.63 | 4 | 0.071 |
| 5 | 1990 - 2 | 23.2 (16.8, 26.5) | 19.7 (15.4, 28.7) | 5.47 | 6 | 0.486 |
| 6 | 1990 - 3 | 19.3 (15.6, 29.5) | 28.4 (13.9, 32.4) | 4.99 | 5 | 0.417 |
| 7 | 1990 - 4 | 20.2 (16.5, 25.9) | 27.5 (20.4, 32.1) | 9.22 | 4 | 0.056 |
| 8 | 1991 - 1 | 17.8 (14.6, 23.5) | 29.3 (21.6, 34.2) | 6.54 | 7 | 0.479 |
| 9 | 1991 - 2 | 21.0 (17.4, 26.2) | 25.4 (18.6, 30.2) | 3.59 | 4 | 0.464 |
| 10 | 1991 - 3 | 25.9 (22.3, 30.4) | 22.1 (16.9, 26.0) | 5.01 | 3 | 0.171 |
| 11 | 1992 - 1 | 11.4 (9.8, 13.8) | 20.7 (15.3, 23.9) | 14.16 | 15 | 0.514 |
| 12 | 1992 - 2 | 16.2 (11.0, 22.4) | 28.0 (20.2, 33.7) | 51.21 | 16 | 0.000 |
| 13 | 1992 - 3 | 12.5 (10.7, 15.3) | 34.2 (30.3, 37.6) | 25.28 | 17 | 0.089 |
| 14 | 1992 - 4 | 17.6 (13.0, 20.3) | 22.1 (18.2, 28.1) | 31.52 | 12 | 0.002 |
| 15 | 1992 - 5 | 24.9 (20.8, 27.5) | 18.8 (14.2, 25.2) | 42.61 | 10 | 0.000 |
| 16 | 1992 - 6 | 22.3 (18.8, 30.9) | 39.6 (28.7, 45.4) | 16.61 | 10 | 0.084 |