The impedance spectra of thin gold electrodeposits with 10−6−10−5 cm average columnar radius and 4×10−6−3×10−4 cm average height have been measured in acid solutions. Frequency dispersion has been observed for film thicknesses greater than 3 × 10−5 cm and electrolyte resistivities larger than 300 Ω cm−1. STM-SEM observations reveal a self-affine surface structure with a size distribution of pores ranging from 10−6 to 10−4 cm between the columnar elements. In diluted solutions the contribution of the smallest pores to the sampled electrode area decreases as ω increases explaining the observed frequency dispersion. This result confirms that self-affine surfaces exhibit a fractional-power-law impedance. However, no simple correlation was found between the α value and the D value of the self-affine surface measured by an independent method. The present results also show that the electrolyte resistivity plays a certain role in determining the α value.