Vertically propagating shear waves reflected from a thin reservoir permeated by a single oriented system of open vertical fractures contain information about the fracture density. The outer perimeter of the hodogram constructed from shear waves polarized in the fast and slow directions in the natural coordinate system is approximately elliptical.The polarization is defined as the angle which the major axis of the ellipse makes with the natural coordinate. Over a range of layer thickness which extends well above and below the tuning frequency, the polarization does not deviate from its value for an infinitely thick layer, psi = tan (super -1) R 2 /R 1 , by more than a few degrees. R 1 and R 2 are the reflection coefficients in the fast and slow directions. R 1 is independent of the fracture density, but psi , R 2 , and the anisotropy eta all depend on the fracture density. R 1 , R 2 , and eta are related by[R 1 + (1 + 2/eta )] [R 2 - (1 + 2/eta )] = 1 - (1 + 2/eta ) 2 .Analysis of the constant-anisotropy hyperbolas derived from this equation reveals conditions under which psi is extremely sensitive to variations in anisotropy. As eta increases by 20 percent, the change in psi approaches 135 degrees for small negative R 1 and 45 degrees for small positive R 1 .The ellipse is also characterized by an aspect ratio which is significantly different from zero over a range of layer thickness that extends above and below the tuning frequency. Aspect ratio versus polarization (ARP) curves can be constructed from multi component field data. Each curve is constructed for fixed R 1 and tau 1 (the two-way traveltime across the layer for the fast polarization). R 1 and tau 1 are independent of lateral changes in fracture density. Consequently, points on ARP curves correspond to different values of the anisotropy. If tau 1 is known from well measurements, R 1 can be determined and a value of anisotropy assigned to each point where there is a field measurement.