We present quantitative results of forward modeling applied to a suite of travel time delays observed for teleseismic P wave first arrivals recorded in the Valles Caldera, New Mexico, in 1987. We used a ray tracing method to construct a two-dimensional model of the caldera's subsurface P wave velocity structure for the vertical cross section defined by the linear seismometer array geometry used to collect the data. Our initial trial models were based on observations of a systematic pattern in the recorded P wave delays, which indicated that an isolated, low-velocity zone (LVZ) exists in the Precambrian basement beneath the caldera's surface layers. Thus the primary goal of the forward modeling was to determine the size, shape, location, and velocity contrast of the LVZ. Due to incomplete ray coverage we could not uniquely constrain all parameters defining the LVZ, and our final model thus represents a first look at the caldera's midcrustal structure. We placed upper and lower bounds on most model parameters by investigating the ranges of models that fit the data well. We determined that the LVZ must be lens shaped, pinching out laterally on each side, and is located beneath the center of the caldera's resurgent dome. Its width is approximately 17 km from tip to tip, and the depth from the surface of its center is 10--13 km. we could not determine an upper bound for the LVZ's thickness, but we found it could be no smaller than 8 km from top to bottom. For this thickness, the P wave velocity was constrained to lie between 3.5 and 4.2 km/s. Compared with a homogeneous half-space model, the best LVZ model reduced the rms of the residual delays for 21 events by 70%, equivalent to a 91% decrease in the residual variance. The low P wave velocity suggests that the LVZ could be a region of partial melt, possibly related to the Bandelier ignimbrite or postcaldera rhyolite eruptions. ¿ American Geophysical Union 1991.