Inferring spatial variation of solar properties from helioseismic data

A common method to infer that solar properties vary with position is to compare linear estimates of averages of those properties centered at different locations. If some of the confidence intervals for the averages do not overlap, one concludes that the property varies. In order for this conclusion to be statistically valid, the lengths of the intervals must be adjusted to obtain the correct ''simultaneous coverage probability.'' We illustrate the notion of simultaneous coverage probability using coin tossing as an example. We present four methods for adjusting the lengths of confidence intervals for linear estimates, and a complementary approach to infer changes based on constructing a linear estimator that is directly sensitive to changes. The first method for constructing simultaneous confidence intervals is based on Bonferroni's inequality, and applies generally to confidence intervals for any set of parameters, from dependent or independent observations. The second method is based on a chi(2) measure of fit to the data, which allows one to compute simultaneous confidence intervals for any number of linear functionals of the model The third method uses a chi(2) distribution in the space of estimates, which yields ''Scheff'' confidence intervals for the functionals. The fourth method, which produces the shortest confidence intervals, uses the infinity-norm in the space of estimates to construct ''maximum-modulus'' confidence intervals. We apply the four methods to search for radial changes in averages of solar angular velocity, using data from Big Bear Solar Observatory (BBSO) averaged for the 4 yr 1986, 1988-1990. Finally, we apply the new differencing estimator to the BBSO data, finding strong evidence that the average solar angular velocity is lower near the poles than near the equator over a range of depths, as is observed at the surface as well.