We present methods for detecting NDE defect signals in correlated noise having unknown covariance. The proposed detectors are derived using the statistical theory of generalized likelihood ratio (GLR) tests and multivariate analysis of variance (MANOVA). We consider both real and complex data models. To allow accurate estimation of the noise covariance, we incorporate secondary data containing only noise into detector design. Probability distributions of the GLR test statistics are derived under the null hypothesis, i.e. assuming that the signal is absent, and used for detector design. We apply the proposed methods to simulated and experimental data and demonstrate their superior performance compared with the detectors that neglect noise correlation.