Sea clutter consists of three components: a mean Doppler spectrum, persistent spikes, and discrete spikes, with a random degree of relative power for each component. We propose a non-linear optimization technique designed to decompose noisy sea clutter into these three components plus a noise component using sparsity inducing norms and linear time-invariant (LTI) filtering in various domains. This novel approach is proposed for non-stationary clutter because it avoids any quasistationarity assumptions, unlike the currently proposed state-of-the-art detectors . The decomposition is applied to real South African sea clutter data provided by the Council for Scientific and Industrial Research (CSIR) . We additionally propose a secondary classifier stage for post-processing of potential target detections from the decomposition, and discuss some features that assist in classification between targets and persistent spikes beyond amplitude. Several such extensions are discussed in the conclusion.