Reflectivity inversion methods based on a stationary convolution model are essential for seismic data processing. They compress the seismic wavelet, and by broadening the bandwidth of seismic data, they assist the interpretation of seismic sections. Unfortunately, they do not apply to realistic nonstationary deconvolution cases in which the seismic wavelet varies as it propagates in the subsurface. Deep learning techniques have been proposed to solve inverse problems in which networks can behave as the regularizer of the inverse problem. Our goal is to adopt a semisupervised deep learning approach to invert reflectivity when the propagating wavelet is considered unknown and time-variant. To this end, we have designed a prior-engaged neural network by unrolling an alternating iterative optimization algorithm, in which convolutional neural networks are used to solve two subproblems. One is to invert the reflectivity, and the other is to estimate the time-varying wavelets. In general, it is well known that, when working with geophysical inverse problems such as ours, one has limited access to labeled data for training the network. We circumvent the problem by training the network via a data-consistency cost function in which seismic traces are honored. Reflectivity estimates also are honored at spatial coordinates in which true reflectivity series derived from borehole data are available. The cost function also penalizes time-varying wavelets from varying abruptly along the spatial direction. Experiments are conducted to find the effectiveness of our method. We also compare our approach to a nonstationary blind deconvolution algorithm based on regularized inversion. Our findings reveal that the developed method improves the vertical resolution of seismic sections with noticeable correlation coefficient improvements over the nonstationary blind deconvolution. In addition, our method is less sensitive to initial estimates of nonstationary wavelets. Moreover, it needs less human intervention when setting parameters than regularized inversion.