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Due to the limited available bandwidth and dynamic channel, data rates are extremely limited in underwater acoustic (UWA) communications. Addressing this concern, in-band fullduplex (IBFD) has the potential to double the efficiency in a given bandwidth. In an IBFD scheme, transmission and reception are performed simultaneously in the same frequency band. However, in UWA-IBFD, because of reflections from the surface and bottom and the inhomogeneity of the water, a significant part of the transmitted signal returns back to the IBFD receiver. This signal contaminates the desired signal from the remote end and is known as the self-interference (SI). With an estimate of the self-interference channel impulse response (SCIR), a receiver can estimate and eliminate the SI. A better understanding of the statistical characteristics of the SCIR is necessary for an accurate SI cancellation. In this article, we use an orthogonal frequency division multiplexing (OFDM) signal to characterize the SCIR in a lake water experiment. To verify the results, SCIR estimation is performed using estimators in both the frequency and time domains. We show that, in our experiment, regardless of the depth of the hydrophone, the direct path of SCIR is strong, stable and easily tracked; however, the reflection paths are weaker and rapidly time-varying making SI cancellation challenging. Among the reflections, the first bounce from the water surface is the prevalent path with a short coherence time around 70 ms.