University of California at Los Angeles

*J. Geophys. Res., 96,* 22, 741-22, 752, 1991

(Received: June 11, 1991;
accepted: October 1, 1991)

Copyright 1991 by the American Geophysical Union.

Paper Number 91JE02506.

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In order to readily compare phase angle distributions such as those plotted in Figures 6 - 8 we have elected to fit the histograms with a sinusoid, using a least squares analysis. This allows us to reduce the information in each histogram to a single phase angle. However, the histograms also convey information about the degree of confidence in the determination of the phase angle. For example, the relative phase histogram in Figure 6 is much smoother than that shown in Figure 8. This implies that in a statistical sense we should have more confidence in the phase angle as determined in Figure 6, in the absence of any consideration of other factors, such as contamination by interference. To this end, we carry out a probability and error analysis using methods for curvilinear regression, such as discussed by Pollard [1977].

Each histogram is divided into 10° bins covering a range of
180°. We shall denote the bin angles by *x _{i}* and the corresponding number of samples in a bin by

where is the mean of the *y _{i}*, and for convenience we use

with *y _{0}* = (

The coefficients in (A1) are determined through minimization of the residual sum of squares

giving

In order to determine the significance of the coefficients
given by (A*3a* and A*3b*) we compare the ratio of the variance of the data due to regression over the residual variance to that expected for two chi-square random variables [Pollard, 1977]. The variance ratio of the chi-square distributions follows the F distribution. With the particular functional form we have chosen the test statistic can be calculated using the relationship

where the left-hand side of (A4) gives the residual sum of squares,
which we denote by *S _{r}^{2}*, and the second term on the right-hand side is the regression sum of squares, which we denote by

There are 18 degrees of freedom in the data, and we calculate
three regression parameters, including the mean. Since we are
only testing for the coefficients *y _{s}*, and

Rather than simply compute the test statistic *F*, and compare that with a specific value of the *F _{2,15}* probability distribution (e.g., the upper 5%), we calculate the probability that

See, for example, (26.6.4) of Abramowitz and Stegun [1965]. In the figures and tables of the main text we give *P* as a percentage.

The residual variance can also be used to give confidence
limits on the coefficients *y _{s,c}* denoted by

Since we use (A2) to specify the fit, we wish to express the error as an angular measure. Noting that *y _{s,c}*, does not depend on the magnitude of the individual coefficients

and from this we calculate an augular error given by

The functional form used to define the angular error in (A7) is
somewhat arbitrary but has the advantage of only depending on
the test statistic *F*, which we use to calculate the
probability. The actual error depends on the confidence limit
required, as shown in Figure A1. With the form given by (A7), the limiting error is 67.5° when *F* = 0, independent of the confidence limit. The figure shows that the error is reduced by roughly a factor of 2 if we use a 70% confidence limit, rather than the 95% limit actually employed in the main text. This is to be expected since these confidence limits correspond to roughly one and two standard deviations respectively for a normal probability distribution.

Figure A1. Probability and angular error as a function of the test statistic. The probability is given by the single curve that approaches 100% for high values of the test statistic. The angular error depends on the degree of confidence desired, as indicated by the percentage labels. The horizontal dashed lines give the angular error for different confidence limits, assuming a test statistic that is 80% probable. In this paper we use 95% confidence limits when determining the error on the fit.

It is not clear that (A7) is the best form to be used for assigning an error to the fit. For example,
Figure A1 shows that the 80% confidence limit is around 34° when the fit is just significant at the 80% level. An error of 45° seems more appropriate. Consequently, we might consider a form such as *x _{0}* = 0.5 sin

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