as if all observations are taken at noon local time and assuming the process is stationary in longitude and time, the covariance matrix for the observations can be embedded in a block circulant matrix, greatly reducing the computational effort needed for multiplying the covariance matrix by a vector. Using (\[ascore\]) and a factorized sparse inverse preconditioner \[@kolo\], we are able to compute an accurate approximation to the MLE for a simple model that captures some of the main features in the OMI data, including the obvious movement of ozone from day to day visible in Figure \[figozone\] that coincides with the prevailing westerly winds in this latitude band.

Variance of stochastic approximation of the score function {#sec2}
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This section gives a bound relating the covariance matrices of the approximate and exact score functions. Let us first introduce some general notation for unbiased estimating equations. Suppose $\theta$ has $p$ components and $g(\theta)=(g_1(\theta
),\ldots,
g_p(\theta))'=0$ is a set of unbiased estimating equations for $\theta$ so that $E_\theta g(\theta)=0$ for all $\theta$. Write $\dot{g}(\theta)$ for the $p\times p$ matrix whose $ij$th element is $\frac{\partial}{\partial\theta_i}g_j(\theta)$ and $\operatorname{cov}_\theta
\{g(\theta)\}$ for the covariance matrix of $g(\theta)$. The Godambe information matrix \[@varin\], $$\mathcal{E}\bigl\{g(\theta)\bigr\}= E_\theta\bigl\{\dot{g}(\theta)\bigr
\} \bigl[\operatorname{cov}_\theta\bigl\{g(\theta)\bigr\} 