It is generally believed that the enhanced backscattering of light from a highly reflecting, multiply-scattering, random surface is due to the coherent interference of each multiply- reflected optical path with its time-reversed partner, and is already present in the double- scattering approximation. If enhanced backscattering is indeed a multiple-scattering effect we should see it from any random surface that can multiply scatter light. The statistical properties of the surface, such as whether the surface profile function is a Gaussianly-distributed random variable or not, whether it is stationary or not, and the form of the surface height correlation function, should therefore be of secondary importance in determining whether enhanced backscattering occurs or not. In this paper we study the role played by the form of the surface height correlation function in the existence of enhanced backscattering and in the dependence on the scattering angle of the contribution to the mean differential reflection coefficient from the incoherent component of the scattered light. We consider the scattering of p- and s- polarized beams of light incident normally onto a random, one-dimensional metallic surface, when the plane of incidence is perpendicular to the generators of the surface. Four different forms of the surface height power spectrum g(Q) are considered: (a) g(Q) equals (pi) a exp(-Qa); (b) g(Q) equals (pi) 1/2a exp(-Q2a2/4); (c) g(Q) equals 2a[1 - (Qa/(pi) )](theta) ((pi) - Qa); and (d) g(Q) equals a(theta) ((pi) - Qa), where we have presented them in the order of increasing rate of decay to zero with increasing Q. In these expressions (theta) (Q) is the Heaviside unit step function. For each form of g(Q) we have calculated the mean value of , the distance between consecutive peaks and valleys on the surface and the variance of this quantity, (sigma) d equals [2> - 2]1/2. The values of these quantities are correlated with the rate of decay of g(Q) with increasing Q. For a fixed values of the wavelength of the incident light we then calculate the contribution to the mean differential reflection coefficient from the incoherent component of the scattered light when the value of a in case (b) above is varied in such a way that (lambda) / increases systematically, with the rms slope held constant. Enhanced backscattering is observed in each case. The width of the enhanced backscattering peak is related to the value of (lambda) /, as is the occurrence of first order subsidiary maxima. The latter disappear when (lambda) / has increased to about 0.6. This is interpreted as due to the inability of the incident light to resolve the structure of the surface responsible for these subsidiary maxima. Similar calculations are carried out for a band-limited fractal surface, characterized by g(Q) equals ((pi) a/tan-1Qoa)(theta) (Qo - Q)/(1 + a2Q2), and similar results are obtained. No evidence of second order subsidiary maxima is seen in the differential reflection coefficient. This result is believed to be due to the magnitude of (sigma) d/ for each of the forms of g(Q) considered. We conclude that while the detailed form of the mean differential reflection coefficient depends on the form of the surface height correlation function, the existence of enhanced backscattering does not, as long as the surface remains multiply reflecting.