The environment of the New Jersey shelf is characterized by high spatial and temporal variability of water column properties caused by intrusions of warm, salty water from the continental slope. These intrusions cause fluctuations in the water column sound speed profile which can have significant effects on acoustic propagation in shallow water. In this work, a linearized perturbative inverse technique is applied to estimate range-dependent water column sound speed profiles. This method utilizes estimates of horizontal wave numbers to determine sound speed as a function of depth. This technique is appropriate for the range-dependent shallow-water environment as horizontal wave numbers can be measured semilocally (1-2 km aperture) and their values are a direct measurement of the local environmental parameters. Difficulty is encountered in application of the perturbative inverse technique because the wave number data are insensitive to some portions of the waveguide and, as a result, the solution can deviate wildly from true values. This issue is addressed by application of approximate equality constraints which force the solution to be close to likely values at prescribed locations.