It is challenging to inspect austenitic welds nondestructively using ultrasonic waves because the spatially varying elastic anisotropy of weld microstructures can lead to the deviation of ultrasound. Models have been developed to predict the propagation of ultrasound in such welds once the weld stiffness heterogeneity is known. Consequently, it is desirable to have a means of measuring the variation in elastic anisotropy experimentally so as to be able to correct for deviations in ultrasonic pathways for the improvement of weld inspection. This paper investigates the use of external nonintrusive ultrasonic array measurements to construct such weld stiffness maps, representing the orientation of the stiffness tensor according to location in the weld cross section. An inverse model based on a genetic algorithm has been developed to recover a small number of key parameters in an approximate model of the weld map, making use of ultrasonic array measurements. The approximate model of the weld map uses the Modeling of anIsotropy based on Notebook of Arcwelding (MINA) formulation, which is one of the representations that has been proposed by other researchers to provide a simple, yet physically based, description of the overall variations of orientations of the stiffness tensors over the weld cross section. The choice of sensitive ultrasonic modes as well as the best monitoring positions have been discussed to achieve a robust inversion. Experiments have been carried out on a 60-mm-thick multipass tungsten inert gas (TIG) weld to validate the findings of the modeling, showing very good agreement. This work shows that ultrasonic array measurements can be used on a single side of a butt-welded plate, such that there is no need to access the remote side, to construct an approximate but useful weld map of the spatial variations in anisotropic stiffness orientation that occur within the weld.