Temperature

Random Field Generation

The entire 2D model estuary is enclosed in a rectangle that is discretized into a total of $N_x$ and $N_y$ points in the $x$ and $y$ directions, while the number of points used over the time interval (March 16th to November 14th) is $N_t$. This leads to a series of rectangles in space and time. Gaussian random fields are generated over this rectangular grid using a FFT algorithm (Dietrich and Newsam 1993) with filter:

$$\eta(x, y, t) = \xi(x, \sigma_x) \otimes \xi(y, \sigma_y) \otimes \xi(t, \sigma_t)$$ (A.1)

where

$$\xi(x, \sigma) = \frac{\exp\left(-\frac{x^2}{2 \sigma^2}\right)}{\pi^{1/4}\sigma^{1/2}}.$$ (A.2)

The correlation between locations are governed by the parameters $\sigma_x$, $\sigma_y$, and $\sigma_t$. Points separated by distances of more $\approx 3 \sigma$ are not very correlated. The number of discretization points ($N_x$, $N_y$ and $N_t$) are chosen so that $\sigma_i / \triangle_i > 2$ for $i = x, y, t$ where $\triangle_i$ is the constant distance between discretization points (Dietrich and Newsam 1993). The random fields are generated independently each year for the 245 days between March 16 and November 14th. No randomness is present in these environment variables for the remaining time period (November 15th to March 15) that the model runs over. Because the temperature in the estuary is low during this period, a crab's respiration and growth rates will also be low and the lack of spatial heterogeneity in environmental variables will not homogenize the crab population. Finally, the values of the generated random field are interpolated from the rectangular grid onto the finest triangulation of the estuary and exported to a file. As the crab model runs, the appropriate values are read in every 24 hours. Pre-generating these variables lessens the computational burden.