In soil erosion by overland flow, sediment discharge is driven by discharge and gradient against the resistance of the soil and vegetation. The form of the relationship, which is taken as a sum of power law terms, may then be inferred from several data sources: (1) ephemeral gully head location, (2) permanent channel head location, (3) slope profile form, (4) consistency with evolutionary models and (5) erosion plot data. The paper focuses on (1) to (4), in order to understand how far these data constrain the form of the dependence on discharge and gradient. Field data from mapping a set of ephemeral gully heads after storms show a weak inverse relationship between channel head gradient and catchment area, which flattens towards an upper threshold gradient of about 25%. These relationships are compared with those theoretically obtained for a range of sediment transport laws which have been previously used, and which are analysed to forecast both overall slope profile form and the location of stream heads. The theory predicts that the location of ephemeral gully heads produced in a single storm will differ from the long-term average stream head location. There is a fairly good relationship between theoretical and observed locations, associated with the form of the sediment transport law. For a single event, the location depends on storm magnitude which cannot be well defined from the data sets, creating a spread of data with an envelope curve defined for the largest events. The empirically acceptable sediment transport equations have also been used to analyse families of slope profiles in a small area, and used within a landscape evolution model to generate landscapes which are compatible with those observed in the field, and provide confirmation of the long-term viability of the expressions obtained to forecast the direction, style and rates of landscape evolution in response to land use or climate drivers.