Abstract. The author defines the concept of the intensity of crossings of a given level by a homogeneous field as the average number of points of level surface that hit the expanding space. It is shown that irrespective of the position of the center of expanding space, the problem of finding the intensity reduces to counting the level surfaces per unit volume. The author formulates the possibility of finding the number of level surfaces as a characteristic that depends on birth-surface points. A differential equation is found that relates the intensities of points of local maxima and local minima with the desired intensity of level surfaces. The accuracy of the results is verified for the stationary Gaussian process. The results completely coincide with the expression found by Rice for the first time.

Keywords: up-crossings (down-crossings) of the fixed level by random field, expected number of excursion sets for the random field, distribution of absolute maximum of random field.