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Random fields are useful models of spatially variable quantities, such as those occurring in environmental processes and medical imaging. The fluctuations obtained in most natural data sets are typically anisotropic. The parameters of anisotropy are often determined from the data by means of empirical methods or the computationally expensive method of maximum likelihood. In this paper, we propose a systematic method for the identification of geometric (elliptic) anisotropy parameters of scalar fields. The proposed method is computationally efficient, nonparametric, noniterative, and it applies to differentiable random fields with normal or lognormal probability density functions. Our approach uses sample-based estimates of the random field spatial derivatives that we relate through closed form expressions to the anisotropy parameters. This paper focuses on two spatial dimensions. We investigate the performance of the method on synthetic samples with Gaussian and Matern correlations, both on regular and irregular lattices. The systematic anisotropy detection provides an important preprocessing stage of the data. Knowledge of the anisotropy parameters, followed by suitable rotation and rescaling transformations restores isotropy thus allowing classical interpolation and signal processing methods to be applied.