The microstructure of the uppermost portions of a melting Arctic sea ice cover has a disproportionately large influence on how incident sunlight is reflected and absorbed in the ice/ocean system. The surface scattering layer (SSL) effectively backscatters solar radiation and keeps the surface albedo of melting ice relatively high compared to ice with the SSL manually removed. Measurements of albedo provide information on how incoming shortwave radiation is partitioned by the SSL and have been pivotal to improving climate model parameterizations. However, the relationship between the physical and optical properties of the SSL is still poorly constrained. Until now, radiative transfer models have been the only way to infer the microstructure of the SSL. During the MOSAiC expedition of 2019-2020, we took samples and, for the first time, directly measured the microstructure of the SSL on bare sea ice using X-ray micro-computed tomography. We show that the SSL has a highly anisotropic, coarse, and porous structure, with a small optical diameter and density at the surface, increasing with depth. As the melting surface ablates, the SSL regenerates, maintaining some aspects of its microstructure throughout the melt season. We used the microstructure measurements with a radiative transfer model to improve our understanding of the relationship between physical properties and optical properties at 850 nm wavelength. When the microstructure is used as model input, we see a 10-15% overestimation of the reflectance at 850 nm.This comparison suggests that either a) spatial variability at the meter scale is important for the two in situ optical measurements and therefore a larger sample size is needed to represent the microstructure or b) future work should investigate either i) using a ray-tracing approach instead of explicitly solving the radiative transfer equation or ii) using a more appropriate radiative transfer model.