# Help me figure out the profile likehihood given this covariance function

I'm taking a spatial stats class, and I'm on the road so I can't ask the prof for help. Would appreciate help understanding what is going on here.

The problem is set up with $Y = X_s'\beta + e$ (with $s$ indexing coordinates). $e$ is assumed to be second-order stationary, with the variogram

$$\gamma(h) = \left\{\begin{array}{l r} 0 & h=0\\ \tau^2 + \sigma^2\left(\frac{3h}{2\rho}-\frac{h^3}{2\rho^3}\right) & 0 < h \leq \rho\\ \tau^2 + \sigma^2 & h > \rho\\ \end{array}\right.$$ where $h$ is distance between two points. For second-order stationary processes, apparently it is true that $C(h) = lim_{h' \rightarrow \infty} \gamma(h') - \gamma(h)$, where $C(h)$ is a covariance function, which is only a function of distance, because the processes is second-order stationary. This means that the covariance function is

$$C(h) = \left\{\begin{array}{l r} \tau^2 + \sigma^2 & h=0\\ \sigma^2\left(1-\left(\frac{3h}{2\rho}-\frac{h^3}{2\rho^3}\right)\right) & 0 < h \leq \rho\\ 0 & h > \rho\\ \end{array}\right.$$ which makes intuitive sense.

Now for the trickier part (for me anyway): we are told to assume that $e$ is a gaussian process. We are asked if we might calculate a profile likelihood as a function only of $\rho$.

The likelihood function is that of a standard multivariate normal.

My question: how do I go about the algebra to (either) find a profile likelihood that is only a function of $\rho$, or perhaps a function of $\rho$ and $\tau$? If the latter, how do I deal with the fact that the parameter $\tau$ will show up on the diagonals of the covariance matrix, but not on the off diagonals? It seems like it makes the algebra intractable.

Hints appreciated.

So, the answer to my own question occurred to me after sleeping on it, as it often does. The covariance function up there in the question is equivalent to $$C(h) = \tau^2I + \sigma^2\left(1-\left(\frac{3h}{2\rho}-\frac{h^3}{2\rho^3}\right)\right)$$