The radiation pattern of an arbitrarily shaped antenna system placed directly on the surface of a polar ice sheet (a configuration often used for ice-probing radars) may be determined at any depth within the ice sheet using a two-step process. The distribution of the radiated field just beneath the surface may be determined first using the method of moments, and geometric optics can then be used to calculate the effects of the exponential density gradient of the ice. Closed-form solutions of the ray-tracing equation, valid at any depth within the ice sheet, allow rapid calculation of the effective look angles and gain increases of any antenna mounted on the surface. These solutions asymptotically approach previously determined relations which are valid only at great depths. A numerical example demonstrates the use of this procedure in the design of a practical ice-probing radar antenna system.

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