8.4. Application to data

. (8.14)

The data are reported as the number of fish observed in 2 feet intervals (total depth - 133 ft). To accommodate the discrete form of the data, equation (8.14) must be integrated:

. (8.15)

N is the total number of fish observed, and n_i is the number of fish observed in the ith vertical interval. I evaluated this integral numerically using Romberg integration (Press, et al. 1988).

To fit the model to the data, I use the following procedure. First, since the preferred depth, z* (corresponding to the preferred light intensity, I*), is the mode of the distribution, I selecte values of z* for the two groups based on the two local maxima of fish frequencies from the data. Also, I do not have information about the light intensity, which would have to be measured directly, or decay rate, which depends on factors such as turbidity. Since initial light intensity, I₀, can be factored out from the inner term of the exponential, and since the two parameters and occur as a ratio, I define a new parameter, , which is defined as

. (8.16)

This parameter is the ratio of chemotactic movement to diffusive movement scaled by initial light intensity. Thus, I need to estimate 4 parameters: ₁, ₂, , and w. I estimate these parameters with the maximum likelihood method based on a multinomial distribution (see Chapter 3). The maximum likelihood is determined numerically with the downhill simplex method (Press, et al., 1988).

I first apply the model to the composite data from the seven periods (April 22 - May 31) and estimated the parameters. I then use all these parameter estimates except w and apply the model to the weekly data. To fit these data, I only vary w, the weighting function that distinguishes between the groups of fish.

results

Table 8.1 Parameter estimates for equation (8.14) applied to daytime hydroacoustic data from Lower Monumental Dam for the composite data.
z*1	z*2	1	2		w	lik.
13.0	39.0	118.30	18.46	0.022	0.146	3.940

Table 8.1 contains values of the parameter estimates, and Figure 8.3. contains a plot of the model versus the data for the composite data. The correspondence between the data and the fitted model is excellent. Table 8.1 shows that the two groups have quite different preferred depths, 14 feet versus 40 feet, with approximately 15 per cent of the fish in the first group. Also, there is a large difference between the estimates of for the two groups. This indicates that relative to each other, the second group undergoes a great deal more random movement, and the first group's position is more dictated by the light intensity.

For the weekly data, the estimates of w and likelihoods are contained in Table 8.2 and plots of the model versus the data are in Figure 8.4. For all but the first week, the model and data are quite consistent. The values of w can be compared to observed passage timing of steelhead and yearling chinook on the Snake River (Fish Passage Center, 1987). Steelhead passage was shifted 10-15 days later than yearling chinook passage, which is consistent with an increasing portion of the higher swimming fish as the season progressed.

These results indicate that vertical distributions are quite constant through the season. Also, hydroacoustic data may be useful in distinguishing among species of salmonids.