Two methods for determining a best initial heading estimate for use in an AINS system, while recording position data values and in-motion data values, assigning incremental changes to indexed heading ψk, calculating a set of time indexed present AINS position values and a corresponding set of time indexed absolute horizontal position measurement residual values, recording the absolute maximum time indexed horizontal position measurement residual and the corresponding value of ψk returning to the step of assigning an incremental increase to a next value to indexed heading ψk up to 360 degrees, identifying the maximum time indexed horizontal position measurement residual and corresponding ψk, followed by identifying the ψk having the smallest absolute maximum horizontal position measurement residual as Ak Second method uses a circular array of AINS ENGINES each with a different ψk running in parallel and finding smallest horizontal position measurement residual for Ak.