We present a method for forecasting the foF2 and hmF2 parameters using modal decompositions from measured ionospheric electron density profiles. Our method is based on Dynamic Mode Decomposition (DMD), which provides a means of determining spatiotemporal modes from measurements alone. Our proposed extensions to DMD use wavelet decompositions that provide separation of a wide range of high-intensity, transient temporal scales in the measured data. This scale separation allows for DMD models to be fit on each scale individually, and we show that together they generate a more accurate forecast of the time-evolution of the F-layer peak. We call this method the Scale-Separated Dynamic Mode Decomposition (SSDMD). The approach is shown to produce stable modes that can be used as a time-stepping model to predict the state of foF2 and hmF2 at a high time resolution. We demonstrate the SSDMD method on data sets covering periods of high and low solar activity as well as low, mid, and high latitude locations.