Correlation of laser speckle fields generated by light of two different wavelengths, which illuminates successively a randomly rough surface, is solved theoretically within the framework of the scalar Kirchhoff theory of wave scattering from random rough surfaces. The Fresnel approximation is used in description of the scattered wave. The solution obtained is a contribution to the scalar Kirchhoff theory of electromagnetic wave scattering from randomly rough surfaces. Its utilization for surface roughness measurement by means of the digital processing of corresponding specklegrams is presented.