The Cosmic Infrared Background (CIB) is hidden behind veils of foreground emission from our own solar system and Galaxy. This paper describes procedures for removing the Galactic IR emission from the 1.25 - 240 micron COBE DIRBE maps as steps toward the ultimate goal of detecting the CIB. The Galactic emission models are carefully chosen and constructed so that the isotropic CIB is completely retained in the residual sky maps. We start with DIRBE data from which the scattered light and thermal emission of the interplanetary dust (IPD) cloud have already been removed. Locations affected by the emission from bright compact and stellar sources are excluded from the analysis. The unresolved emission of faint stars at near- and mid-IR wavelengths is represented by a model based on Galactic source counts. The 100 micron DIRBE observations are used as the spatial template for the interstellar medium (ISM) emission at high latitudes. Correlation of the 100 micron data with H I column density allows us to isolate the component of the observed emission that is associated with the ISM. Limits are established on the far-IR emissivity of the diffuse ionized medium, which indicate a lower emissivity per H nucleus than in the neutral medium. At 240 micron, we find that adding a second spatial template to the ISM model can greatly improve the accuracy of the model at low latitudes. The crucial product of this analysis is a set of all-sky IR maps from which the Galactic (and IPD) emission has been removed. We discuss systematic uncertainties and potential errors in the foreground subtraction process that may have an impact on studies seeking to detect the CIB in the residual maps.
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