2.1.6 Earth curvature and atmospheric refraction

For a sufficiently large aerial block, the assumptions that light rays are straight and that the object coordinate system is Cartesian become imperfect. Atmospheric refraction bends a ray as air density changes with height. Earth curvature means that a level surface cannot be represented by one plane over an unlimited area. Both effects are radial in the image and grow toward the image edge.

BINGO can account for these effects by correcting the photo coordinates before forming the observation equations. This appendix documents the models implemented by the current BINGO computation engine.

caution

These corrections require a meaningful height datum and consistent units. They should be enabled only when the assumptions match the project. For many projects it is more rigorous to transform the object coordinates into a suitable local Cartesian system before adjustment. See PARA for the program controls.

Symbols and units

Let

• $x'$ and $y'$ be the photo coordinates after the preceding camera corrections;
• $r^2 = x'^2 + y'^2$ be the squared radial image distance;
• $c$ be the camera constant;
• $Z_0$ be the camera-station height;
• $Z$ be the object-point height; and
• $R$ be the Earth radius.

The coordinates $x'$, $y'$, and $c$ must use the same unit. Heights and $R$ must use the same object-coordinate unit. The BINGO default is $R = 6{,}380{,}000\ \mathrm{m}$. When the object coordinates use another unit, define the corresponding constants with CONS.

Atmospheric refraction

BINGO uses a height-dependent radial model. It first normalizes the camera and object heights to kilometers for a reference Earth radius of 6,380 km:

$$z_0 = 0.001\,Z_0\frac{6{,}380{,}000}{R}, \qquad z = 0.001\,Z\frac{6{,}380{,}000}{R}.$$

The refraction coefficient used for the observation is

$$k = 0.00241\left( \frac{z_0}{z_0^2-6z_0+250} - \frac{z^2}{z_0\left(z^2-6z+250\right)} \right).$$

With

$$f = \left(1+\frac{r^2}{c^2}\right)k,$$

the corrections applied to the photo coordinates are

$$\Delta x'_{\mathrm{ref}}=-x'f, \qquad \Delta y'_{\mathrm{ref}}=-y'f.$$

This is the image-coordinate refraction correction controlled by PARA. It is distinct from the refraction coefficient in a CONS statement used for terrestrial zenith-distance observations.

Earth curvature

BINGO provides two Earth-curvature models. Both produce a scale factor $e$ and apply the radial corrections

$$\Delta x'_{\mathrm{curv}}=x'e, \qquad \Delta y'_{\mathrm{curv}}=y'e.$$

Absolute-height model (Mikhail)

The absolute-height model uses the object-point height in the local Earth radius:

$$e_{\mathrm{Mikhail}} = \frac{1}{2(R+Z)} \frac{r^2(Z_0-Z)}{c^2}.$$

This is BINGO's default model when Earth-curvature correction is enabled.

Relative-height model (Kraus)

The alternative Kraus model uses a radius $R$ and includes the object height explicitly:

$$e_{\mathrm{Kraus}} = \frac{1}{2R} \left( \frac{r^2(Z_0-Z)}{c^2}-2Z \right).$$

The two formulations therefore respond differently to the height datum. Their difference can become relevant in hilly or mountainous areas and can propagate into the estimated exterior orientations.

Using the corrections

The corrections are disabled by default. Before enabling either one:

1. Confirm that all object heights and the Earth radius use consistent units.
2. Confirm that the vertical datum is appropriate for the selected Earth-curvature model.
3. Avoid applying a correction that has already been included during image preprocessing.
4. Compare a controlled adjustment with and without the correction, including changes in residuals and exterior orientation.
5. Record the selected model in the project documentation.

For the theoretical background, see /Kraus 1994, 1996/.