Short note: Relief displacement expression

Relief displacement is the radial shift in the position of the image of a point on a vertical photograph caused by its elevation above or below the datum. Points above the datum are displaced away from the principal point (which is also the nadir point on a vertical photo); points below the datum are displaced towards it. Points lying on the datum are not displaced.

          O  (camera)
         /|\
      f / | \
       /  |  \ H
    a'  a |   \
   -------n-----  photo plane
            \ h  A
 ground   --A0--n'--     datum

Expression

Let A be the top of an object of height $h$ above the datum, A₀ its base on the datum, $H$ the flying height above the datum, and $R$ the horizontal ground distance from the nadir point to A. Let $a$ and $a_0$ be the images of A and A₀, and $r$ and $r_0$ their radial distances on the photo from the nadir point (the principal point of a vertical photo).

By similar triangles in the camera:

$$r = \frac{fR}{H-h},\qquad r_0 = \frac{fR}{H}$$

Relief displacement $d = r - r_0$:

$$d = fR\left[\frac{1}{H-h}-\frac{1}{H}\right] = \frac{fRh}{H(H-h)} = \frac{r\,h}{H}$$

since $r = fR/(H-h)$. So

$$d = \frac{r\,h}{H}$$

Conclusions: the displacement increases with the radial distance $r$ and the height $h$, and decreases as the flying height $H$ increases; it is zero at the principal point.