Observations of Galactic H I gas for seven targeted regions at intermediate Galactic latitude are presented at 1 angular resolution using data from the DRAO Synthesis Telescope (ST) and the Green Bank Telescope (GBT). The DHIGLS data are the most extensive arcminute resolution measurements of the diffuse atomic interstellar medium beyond those in the Galactic plane. The acquisition, reduction, calibration, and mosaicking of the DRAO ST data and the cross calibration and incorporation of the short-spacing information from the GBT are described. The high quality of the resulting DHIGLS products enables a variety of new studies in directions of low Galactic column density. We analyze the angular power spectra of maps of the integrated H I emission (column density) from the data cubes for several distinct velocity ranges. Fitting power spectrum models based on a power law, but including the effects of the synthesized beam and noise at high spatial frequencies, we find exponents ranging from -2.5 to -3.0.. Power spectra of maps of the centroid velocity for these components give similar results. These exponents are interpreted as being representative of the 3D density and 3D velocity fields of the atomic gas, respectively. We find evidence for dramatic changes in the H I structures in channel maps over even small changes in velocity. This narrow line emission has counterparts in absorption spectra against bright background radio sources, quantifying that the gas is cold and dense and can be identified as the cold neutral medium phase. Fully reduced DHIGLS H I data cubes and other data products are available at www.cita.utoronto.ca/DHIGLS.
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