# Valley Height-Width (Vf) Ratio

The Valley Width-to-Height Ratio (Vf) allows comparison of erosional patterns between watersheds. The watershed in this case is the unit of measure; one Vf value per watershed.

The index was originally used to distinguish V-shaped valleys from U-shaped valleys (Bull and McFadden, 1977). V-shaped valleys are common in areas of active uplift and deep, linear stream incision (low Vf values, often close to 0). U-shaped valleys are representative of formerly glaciated or tectonically stable areas where stream valley bottoms tend to be wider (higher Vf values). When calculated for several streams draining a mountain range (or larger region), the index can reveal spatial variations in incision and differential uplift.

The four inputs to the Valley Width to Height Ratio equation.

The four inputs to the index are acquired from a DEM and/or aerial photos. Data is collected at a single cross section for each drainage. Where along the valley the cross section is located may vary. One option is at the halfway point between headwaters and outlet. At the point of maximum valley depth (ridgeline relief) is another. Above hard bedrock steps or other lithologic complications may also be appropriate (i.e., Pedrera et al., 2009). A consistent approach is best.

Vf = 2Vfw / [(Eld – Esc) + (Erd – Esc)]

Erd = Elevation of the river-right valley divide (ridgeline)
Eld = Elevation of the river-left valley divide (ridgeline)
Esc = Elevation of the valley floor (canyon)
Vfw = Width of valley floor

Example:
Erd = 1100m
Eld = 1000m
Esc = 500m
Vfw = 10m

Vf = 2(10) / [(1000 – 500) + (1100 – 500)] = 20 / (500) + (600) = 20/1100 = 0.018

Refs:
Bellin et al. (2014) E&PSL 390
Bull and McFadden (1977) Geomorphology in Arid Regions
Bull (2007) Tectonic Geomorphology of Mountains, 328 pgs.
Keller and Pinter (2002) Active Tectonics: Earthquakes, uplift, and landscape, 362 pgs.
Mahmood and Gloaguen (2012) Geoscience Frontiers 3
Pedrera et al. (2009) Geomorphology 105
Perez-Pena et al. (2009) Computers & Geosciences 35
Silva et al. (2003) Geomorphology 50