Fully developed speckle patterns observed in coherent imagery are characterized by a Rayleigh-distributed envelope amplitude. Non-Rayleigh distributions are observed in many cases, such as when the number of scatterers in a resolution cell is small or scatterers are organized with some periodicity. Distributions resulting from the assumption of random scatterer phase (random walk models) have been used to describe the speckle amplitude in these cases, leading to K, Rician, and homodyned-K amplitude distributions. An alternative is to incorporate nonrandom phase implicitly by adopting models that directly describe the spatial placement of point scatterers. We examine the consequences of assuming that scattering is described in one dimension by a stationary renewal process in which the arrival times are the locations of ideal point scatterers, the interscatterer distances are drawn from a gamma distribution, and the scatterer amplitudes are allowed to be correlated in space. This model has been called the marked regularity model because variations of the model parameters can generate spatial distributions ranging from clustered to random to nearly periodic. We will demonstrate that all of the non-Rayleigh distributions generated by the previous random phase models can also be generated by the marked regularity model, and we show under what conditions the different distributions will result. We also demonstrate that the regularity model is inherently capable of describing certain sparse scattering conditions. Therefore, the model can represent many cases and provide an intuitively pleasing description of the spatial placement of the scatterers.