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Disordered nanostructures with correlations on the scale of visiblewavelengths can show angle-independent structural colors. These materials couldreplace dyes in some applications because the color is tunable and resistsphotobleaching. However, designing nanostructures with a prescribed color isdifficult, especially when the application -- cosmetics or displays, forexample -- requires specific component materials. A general approach to solvingthis constrained design problem is modeling and optimization: using a modelthat predicts the color of a given system, one optimizes the model parametersunder constraints to achieve a target color. For this approach to work, themodel must make accurate predictions, which is challenging because disorderednanostructures have multiple scattering. To address this challenge, we developa Monte Carlo model that simulates multiple scattering of light in disorderedarrangements of spherical particles or voids. The model produces quantitativeagreement with measurements when we account for roughness on the surface of thefilm, particle polydispersity, and wavelength-dependent absorption in thecomponents. Unlike discrete numerical simulations, our model is parameterizedin terms of experimental variables, simplifying the connection betweensimulation and fabrication. To demonstrate this approach, we reproduce thecolor of the male mountain bluebird (Sialia currucoides) in an experimentalsystem, using prescribed components and a microstructure that is easy tofabricate. Finally, we use the model to find the limits of angle-independentstructural colors for a given system. These results enable an engineeringdesign approach to structural color for many different applications.

Designing angle-independent structural colors using Monte Carlo simulations of multiple scattering