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The antenna described in Kraus’s textbook is a half-wave slot antenna facing upward. The antenna is fed by a transmission line connected across the slot, but the transmission line only supplies power from a transmitter, and is not part of the antenna. The trench under the slot, which is a quarter wave deep, forms a resonant cavity. A slot antenna is a magnetic dipole, and, because of that, a slot oriented in the horizontal direction provides vertically polarized radiation. The radiation pattern is not omnidirectional, and has the shape one would expect for a horizontal dipole. This antenna truly has no structure above the surface of the earth.
The DDRR, which has a short vertical element and one or two horizontal loops, has all of its structure above the surface of the earth, and so is not a subterrainian radiator.