Advanced light detecting and ranging (LIDAR) sensors are the primary sensing modality for autonomous vehicles and are seeing increasing adoption in consumer and commercial vehicles for robust advanced driver assist systems. LIDAR returns from the environment are typically predicted using elastic LIDAR models, which can help emulate the performance of LIDAR sensors in environments with multiple returns or heavy obscurants. We derive the first elastic LIDAR model for a random modulated continuous wave LIDAR system using a homodyne receiver and show good agreement with experimental measurements.