In optical fiber communication, system identification (SI) for the nonlinear Schrödinger equation (NLSE) has long been studied mainly for fiber nonlinearity compensation (NLC). One recent line of inquiry to combine a behavioral-model approach like digital backpropagation (DBP) and a data-driven approach like neural network (NN). These works are aimed for more NLC gain; however, by directing our attention to the learned parameters in such a SI process, system status information, i.e., optical fiber parameters, will possibly be extracted. Here, we show that the model-based optimization and interpretable nature of the learned parameters in NN-based DBP enable transmission line monitoring, fully extracting the actual in-line NLSE parameter distributions. Specifically, we demonstrate that longitudinal loss and dispersion profiles along a multi-span link can be obtained at once, directly from data-carrying signals without any dedicated analog devices such as optical time-domain reflectometry. We apply the method to a long-haul (~2,080 km) link and various link conditions are tested, including excess loss inserted, different fiber input power, and non-uniform level diagram. The measurement performance is also investigated in terms of measurement range, accuracy, and fiber launch power. These results provide a path toward simplified and automated network management as another application of DBP.