Uncertainties in Solar Synoptic Magnetic Flux Maps

Abstract

Magnetic flux synoptic charts are critical for a reliable modeling of the corona and heliosphere. Until now, however, these charts were provided without uncertainty estimates. The uncertainties are due to instrumental noise in the measurements and to the spatial variance of the magnetic flux distribution that contributes to each bin in the synoptic chart. We describe here a simple method to compute synoptic magnetic flux maps and their corresponding magnetic flux spatial variance charts that can be used to estimate the uncertainty in the results of coronal models. We have tested this approach by computing a potential-field source-surface model of the coronal field for a Monte Carlo simulation of Carrington synoptic magnetic flux maps generated from the variance map. We show that these uncertainties affect both the locations of source-surface neutral lines and the distributions of coronal holes in the models.

Appendix: Estimated Variance in Magnetic Flux Synoptic Charts

Using the notation described in Section 2.2, one can generalize Equation (3) to compute the variance $\sigma_{k}^{2}$ of a heliographic bin k in the synoptic chart given the contribution of N individual observations. That is,

$$\begin{aligned} \sigma_k^2 = &\frac{\tilde{w}_1 W_1+\cdots+\tilde{w}_N W_N}{ (\tilde{w}_1 W_1+\cdots+\tilde{w}_N W_N )^2 - (\tilde{w}^2_1 W^{\prime}_1+\cdots+\tilde{w}^2_N W^{\prime}_N )} \Biggl[\sum _{i}^{N_1}\tilde{w}_1 w_{i,1}(B_{i,1} - B_k)^2 \\ &{}+ \cdots + \sum_{i}^{N_N} \tilde{w}_N w_{i,N}(B_{i,N} - B_k)^2 \Biggr] \\ = & \frac{\sum_{j}^{N} \tilde{w}_j W_{j}}{ (\sum_{j}^{N} \tilde{w}_j W_{j} )^2 - \sum_{j}^{N} \tilde{w}^2_j W^{\prime}_{j}} \bigl[\tilde{w}_1W_1 \bigl(|B|_1^2 + B_k^2 - 2B_1B_k\bigr) + \cdots \\ &{} + \tilde{w}_NW_N\bigl(|B|_N^2 + B_k^2 - 2B_NB_k\bigr) \bigr] \\ = & \frac{\sum_{j}^{N} \tilde{w}_j W_{j}}{ (\sum_{j}^{N} \tilde{w}_j W_{j} )^2 - \sum_{j}^{N} \tilde{w}^2_j W^{\prime}_{j}} \sum_j^{N} \tilde{w}_j W_j\bigl(|B|_j^2 + B_k^2 - 2B_jB_k\bigr). \end{aligned}$$

With some manipulation, the previous formula can be written as

$$ \sigma_k^2 = \frac{ (\sum_{j}^{N} \tilde{w}_j W_{j} )^2}{ (\sum_{j}^{N} \tilde{w}_j W_{j} )^2 - \sum_{j}^{N} \tilde{w}^2_j W^{\prime}_{j}} \biggl(\frac{\sum_j^{N} \tilde{w}_j W_j|B|_j^2}{\sum_{j}^{N} \tilde{w}_j W_{j}} - B_k^2 \biggr), $$

(8)

or simply,

$$ \sigma_k^2 = \frac{1}{1 - [\sum_{j}^{N} \tilde{w}^2_j W^{\prime}_{j}/ (\sum_{j}^{N} \tilde{w}_j W_{j} )^2 ]} \biggl(\frac{\sum_j^{N} \tilde{w}_j W_j|B|_j^2}{\sum_{j}^{N} \tilde{w}_j W_{j}} - B_k^2 \biggr). $$

(9)

The standard deviation is simply the square root of the variance above.