INFERRING SPATIAL VARIATION OF SOLAR PROPERTIES FROM HELIOSEISMIC DATA

A common method to infer that solar properties vary with position is t o compare linear estimates of averages of those properties centered at different locations. If some of the confidence intervals for the aver ages do not overlap, one concludes that the property varies. In order for this conclusion to be statistically valid, the lengths of the inte rvals must be adjusted to obtain the correct ''simultaneous coverage p robability.'' We illustrate the notion of simultaneous coverage probab ility using coin tossing as an example. We present four methods for ad justing the lengths of confidence intervals for linear estimates, and a complementary approach to infer changes based on constructing a line ar estimator that is directly sensitive to changes. The first method f or constructing simultaneous confidence intervals is based on Bonferro ni's inequality, and applies generally to confidence intervals for any set of parameters, from dependent or independent observations. The se cond method is based on a chi(2) measure of fit to the data, which all ows one to compute simultaneous confidence intervals for any number of linear functionals of the model The third method uses a chi(2) distri bution in the space of estimates, which yields ''Scheff'' confidence i ntervals for the functionals. The fourth method, which produces the sh ortest confidence intervals, uses the infinity-norm in the space of es timates to construct ''maximum-modulus'' confidence intervals. We appl y the four methods to search for radial changes in averages of solar a ngular velocity, using data from Big Bear Solar Observatory (BBSO) ave raged for the 4 yr 1986, 1988-1990. Finally, we apply the new differen cing estimator to the BBSO data, finding strong evidence that the aver age solar angular velocity is lower near the poles than near the equat or over a range of depths, as is observed at the surface as well.