Yellowknife array (Northwest Territories, Canada) recordings of nuclear explosions detonated at French Tuamotu, South Pacific and Shagan River, Eastern Kazakhstan (USSR) test sites are used to derive t* values via the application of the spectral decay method. An important factor that limits the reliability of this widely used method is the degree of accuracy with which one is able to determine the signal spectral shape. For transient, pulse-like, short-period teleseismic phases, the conventional single-window spectral estimate methods may not be appropriate due to the trade-off betgween the leakage resistance and variance. A recently developed, multiple-window spectral analysis method is used in this study to effectively control spectral leakage, a capability that is especially important when analyzing seismic data characterized by a rapid high-frequency fall-off rate, such as the French Tuamotu explosion data. We compare the t* estimates obtained using a conventional single-window method with those obtained using the multiple-window method and show that the latter are more reliable. The t* (0.5 to 4.5 Hz) found by the multiple-window method for the Tuamotu-Yellowknife path is 0.66 sec. For the Eastern Kazakhstan-Yellowknife path, the multiple-window t* estimate is 0.42 sec in the 0.5 to 4.5 Hz range; a smaller value is obtained at higher frequencies (4.5 to 8 Hz).