I do not remember reading many papers or books that go into great detail on the subject, but it is complex, and likely to vary with the circumstance as well as river discharge levels. Normally water sampling for sediment concentration suggests sampling at numerous locations across the cross section, and lower sample from surface to bottom and back at each location. Streams with more turbulence can mix easier. Rivers that tend to run or glide along at relative constant or slow velocity are not mixing as fast. If a pollutant enters from the bank of a wide river, it may take substantial distances to become a relative homogenous concentration of a pollutant. If one goes a short distance downstream of the pollutant source and samples from the same side of the river as the point source, the concentration will be substantially higher than if one takes a sample along the opposite bank. One might just go a few miles or kilometers downstream and hope for mixing to be complete, but if the pollution source enters as a slug, one must have some idea on when to sample. In my coauthored paper on the Spruce Budworm project, there is a section on the Willow Creek spill that discloses the timing of the spill reaching different points downstream, and a concentration curve at the major monitoring site before the Heppner Oregon reservoir. In small mountain streams, mixing is not the question, but there is time of travel and dilution of concentration as it travels. If analysis of the contaminant is especially costly, compositing samples taken at each cross section may help. You may note in the Spuce Budworm report that we added Rhodamine B dye into the aerial insecticide treatments to help determine using a fluorometer, which samples to send to the lab for analysis. I used a similar process of adding fluorescent dye when we were aerially spraying portions of the National Forest for vegetation (Weed) control. When dye showed up in samples, those were the ones to send the lab to test for herbicide. As I remember, the fluorometer was able to detect dye in the low ppb range.

I might add that like the Spruce Budworm project, if toxics are being discharged, you might be able to use aquatic insects above the discharge and at varying locations downstream. This becomes difficult for large and deep rivers. Much like the canary in the mine, some may use aquatic organisms placed at varying locations downstream. You may have to run tests on contaminant levels that cause effects or mortality from one or several pollutant discharge events.

@ Williams F. Hansen, thanks for the insight.