Error analysis for numerical estimates of space plasma parameters

Many papers estimate space plasma parameters from instrumentation data via a fitting method, such as reduced chi-square minimization of a model to the data; however, it is currently rare to see uncertainties for those estimates given in the form of error bars or a covariance matrix. This paper seeks to address this issue by providing a simple method that will provide the covariance matrix and therefore uncertainties with little extra computation, no matter how complex the model. Using established black box minimization codes will provide a best fit to the data but may not provide the covariance matrix, while others may provide the covariance matrix (providing their settings are tuned appropriately) but tend to locate a best fit vector to near-machine precision first. Our method allows the fitting to a physically sensible number of decimal places for the instrument yet also provides the covariance matrix with far fewer iterations required to locate the best fit values, greatly decreasing code run time for the fit procedure itself-a great benefit when there are years of data, or multiple spacecraft, to analyze. While the underlying method utilizing the Hessian matrix is not new, the application is currently rarely applied to spacecraft data and this approach is simple to implement. This paper reviews the basic technique and application to data and ends with simple pseudocode that anyone may employ to calculate the covariance matrix for the fitted parameters.