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A ship's exhaust gas contains both hot gas molecules, which emit infrared radiation at specific wavelengths (line emitters), and soot particles which emit broad-banded, like a black body. Our modeling shows that the observed radiance from these emissions falls at different rates with distance. The attenuation of intensity is caused by absorption and scattering of the emitted radiation in the atmosphere. The hottest part of the exhaust plume is spatially confined to a relative small volume. Usually, a ship's hull and its superstructure have a higher temperature than the sky or sea background. The temperature difference is generally not very large. However, the ship has a spatial extent that is much larger than the plume's. In this work we study how both the emitted radiation from the plume and the ship's total signature decrease with increasing distance. This study is based on experimental data that was collected during a measurement campaign at the southwest coast of Norway. Shore-based digital IR cameras, both LWIR and MWIR, recorded image sequences of ships as they sailed away from close to shore (~ 1 km) in a zigzag pattern out to about 10 km. We used a statistical method to identify the gas cloud pixels and used their integrated radiance as a measure for the plume intensity. The ship signature is defined here as the integrated radiance over all the ship's pixels in the imagery. From infrared spectroscopic data, collected using a Fourier Transform Infrared spectrometer aimed at the ship's plume when the ship is close to shore, a model is obtained for the composition of the exhaust gas. This model was used to perform FASCODE simulations to study numerically the attenuation with distance of the plume radiance. Our work shows that this approach may be well suited to explain the observed signal decay rate with distance.