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A statistical treatment of speckle correlations as a function of the position of a moving object is shown to provide access to object information through thick and heavily scattering random media. Experiments for a patchlike object of varying size and for varying degree of background scatter are explained using a model, and an experimental study allows evaluation of key attributes. Given a sufficient signal-to-noise ratio, adequate coherence, and developed field statistics, measured speckle intensity patterns from a set of object positions can allow high-resolution imaging deep into an obscuring medium and the medium's scattering strength can be gauged quantitatively with calibration. This enables new opportunities in application domains such as optical sensing, material inspection, and deep tissue in vivo imaging.