- In sonar or radar array processing, one of the most important objectives is to estimate the angles of incidence of distant source signals in a noisy environment. The standard eigenstructure-based method, which is known to yield high resolution spectral estimates for solving the direction finding problem, relies on the assumption that additive sensor noises are white Gaussian processes that are spatially uncorrelated between sensors. In the cases of correlated noise fields, non-Gaussian noises, highly correlated source signals, relatively low signal-to-noise (S/N) ratio, and/or very closely located source signals, direction finding performance based on the standard eigenstructure algorithm is severely degraded. For this study, the (ordinary) nonlinear second-order method (SOM) and revised nonlinear SOM, along with the generalized eigenstructure algorithm, which decorrelates correlated noise fields with known or estimated noise correlation coefficients, are discussed. Ordinary and revised SOMs produce new data sequences, or "second-order signals", by auto-convolution of the original data sequences that were corrupted by additive noises on the sensor array. By reference to appropriate algebraic calculations, an eigenstructure orthogonality relationship, which is similar to the classical first-order method (FOM) that deals directly with original data sequences, is derived. It is also demonstrated that the revised SOM can be used more effectively than the (ordinary) SOM in conjunction with an FOM, such as multiple-signal classification (MUSIC) or many of its subsequent variations, to accommodate troublesome cases such as closely located, multiple (coherent) sources, limited numbers of sensors, correlated Gaussian noises lacking information on the noise correlation coefficients, non-Gaussian noises, and a low S/N ratio. Based on computer simulation, examples of analysis and estimations for the direction finding problems in several environments are provided. Furthermore, the theoretical derivations of the S/N ratios for the FOM and SOM, as well as the threshold S/N ratios for very closely spaced multiple sources and a statistical consideration of the computer simulation results, are presented.