Migrating bed forms strongly influence hydraulics, transport, and habitat in river environments. Their dynamics are exceedingly complex, making it difficult to predict their geometry and their interaction with sediment transport. Acoustic instrumentation now permits high-resolution observations of bed elevation as well as flow velocity. We present a space-time characterization of bed elevation series in laboratory experiments of sand and gravel transport in a large 84 m long, 2.75 m wide flume. We use a simple filtering and thresholding methodology to estimate bed form heights and report that the shape of their probability density function (pdf) remains invariant to discharge for both gravel and sand and has a positive tail slightly thicker than Gaussian. Using a wavelet decomposition, we quantify the presence of a rich multiscale statistical structure and estimate the scale-dependent celerity of migrating bed forms, showing the faster movement of smaller bed forms relative to the larger ones. The nonlinear dynamics of gravel and sand bed forms is also examined, and the predictability time, i.e., the interval over which one can typically forecast the system, is estimated. Our results demonstrate that flow rate as well as bed sediment composition exert a significant influence on the multiscale dynamics and degree of nonlinearity and complexity of bed form evolution.