The multiple time-window linear waveform inversion is applied to study the source process of the 1999 Kocaeli, Turkey, earthquake using strong-motion data. Two peculiar observations characterize this event. The first one is a conflicting observation of the long surface ruptures and short source duration resulting from the observed strong-motion waveforms. The second is an anomalously short S-P time observed at a station SKR, considering its hypocentral distance. A supershear rupture propagation (Ellsworth and Celebi, 1999) and the P-wave triggering of an asperity (Anderson et al., 2000) were both proposed to explain the record at SKR. Our waveform inversion study is aimed to examine these possibilities in order to clarify the relation between the spatial distribution of the surface ruptures and coseismic slips. We divide the assumed fault plane into three parts and look for the first time-window front propagation velocity (propagation velocity of triggering front of the first time window at each subfault) in each part by performing numerous inversions, changing the first time-window front propagation velocity separately in each part, and selecting the best first time-window propagation velocities that give largest variance reduction. At the same time, appropriate smoothness of spatiotemporal slip distribution is searched for by performing inversions with different strength of smoothing constraints against the observational equations and judging the relative appropriateness of inversion results in terms of the Akaike's Bayesian Information Criterion. The best source model is characterized by an asperity that is about 35 km east of the epicenter and near a segment boundary at Sapanca Lake, triggered by the first time-window front with 5.8 km/sec, close to P-wave velocity. The time progression of the rupture suggests a scenario where a P wave arriving from the hypocenter or somewhere around the hypocenter triggered the second largest asperity near the junction of the segments at Sapanca Lake, which is a preferable interpretation. The first time-window propagation velocities west of the hypocenter and east of SKR are expected to be 3.0 km/sec.