Our research group has been measuring Extremely Low Frequency (ELF) magnetic fields across Japan. The ELF measurements are mixtures of signals associated with various natural or artificial phenomena. When focus on specific factor, the signals related to other factors distort analysis result. In order to get specific information accurately, we should estimate desired signals or eliminate undesired signals. We found that Image Space Reconstruction Algorithm (ISRA), one of the Nonnegative Matrix Factorization (NMF) algorithm, works better than independent component analysis to estimate the ELF background signal. However, ISRA sometimes failed to estimate the weight vector for the background signal. We considered that ISRA has weakness for outliers and sparse signals because ISRA is based on minimizing L2 (Frobenius) norm between input matrix and projected matrix from estimated matrices. In order to improve robustness, we developed new methods based on minimizing quasi-L1 norm (QL1-NMF). In the experiment using generated signals and ELF observed signals which had outliers, the proposed method estimate background signal more accurately than ISRA and other L1 norm based algorithms.