Ground sample distance

In remote sensing, ground sample distance (GSD) in a digital photo of the ground from air or space is the distance between pixel centers measured on the ground. For example, in an image with a one-meter GSD, adjacent pixels image locations are 1 meter apart on the ground.^[1] GSD is a measure of one limitation to spatial resolution or image resolution, that is, the limitation due to sampling.^[2]

GSD is also referred to as ground-projected sample interval (GSI) and is related to the ground-projected instantaneous field of view (GIFOV).^[3]

Formulas

The GSD can be calculated using the geometry of the imaging setup.

General case (oblique or slant view)

In the general case where the sensor may be imaging the ground at an oblique angle (i.e., not looking directly down), the GSD is given by:

${\displaystyle \mathrm {GSD} ={\frac {R_{S}\times p}{f\times \cos(\theta )}}}$ 

Where:

• ${\displaystyle \mathrm {GSD} }$  is the ground sample distance, e.g., in cm/px;
• ${\displaystyle R_{S}={\sqrt {d^{2}+h^{2}}}}$  is the slant range from the sensor to the point on the ground, e.g., in meters:
  • ${\displaystyle d}$  is the horizontal distance (or offset) from nadir, e.g., in meters;
  • ${\displaystyle h}$  is the height above ground level (AGL) of the sensor, e.g., in meters.
• ${\displaystyle p=P\div N}$  is the physical pixel size of the sensor, e.g., in micrometers:
  • ${\displaystyle P}$  is the physical width or height of the sensor, e.g., in millimeters;
  • ${\displaystyle N}$  is the number of total pixels in the same dimension as ${\displaystyle P}$ .
• ${\displaystyle f}$  is the focal length of the camera lens, e.g., in millimeters;
• ${\displaystyle \theta =\arctan \left(d\div h\right)}$  is the slant angle from nadir (which would correspond to 0°), e.g., in degrees.

The cosine of ${\displaystyle \theta }$  accounts for the oblique viewing angle, which increases the effective ground footprint of each pixel.

Nadir case (look-down view)

In the special case of a nadir view, i.e., when the sensor is looking directly downward, the formula is simplified since ${\displaystyle d=0}$ . Thus, ${\displaystyle R_{S}=h}$  and ${\displaystyle \theta =0}$ , the cosine of which is 1. Therefore, the formula becomes:

${\displaystyle \mathrm {GSD} ={\frac {h\times p}{f}}}$ 

Where all variables are defined as above.

Planar components derivative formula

Calculator

${\displaystyle d}$	1000 m
${\displaystyle h}$	2000 m
${\displaystyle p}$	2.9 μm
${\displaystyle f}$	600 mm
${\displaystyle \mathrm {GSD} }$	1.20833 cm/px

If the slant range ${\displaystyle R_{S}}$  and slant angle ${\displaystyle \theta }$  are to be derived from the horizontal and vertical components ${\displaystyle d}$  and ${\displaystyle h}$  thereof, after simplification, the formula becomes:

${\displaystyle \mathrm {GSD} ={\frac {d^{2}+h^{2}}{h}}\times {\frac {p}{f}}}$ 

Where all variables are defined as above.

Optimal off-nadir angle for maximal distance

Calculator

${\displaystyle \mathrm {GSD_{max}} }$	5 cm
${\displaystyle p}$	2.9 μm
${\displaystyle f}$	600 mm
${\displaystyle d=h\approx }$	5172 m

To maximize the horizontal imaging distance (${\displaystyle d}$ ) for a given optical system while adhering to a specified maximum ground sample distance (${\displaystyle \mathrm {GSD_{max}} }$ ) constraint, the optimal imaging geometry is achieved at a 45° off-nadir angle. This corresponds to a height above ground level (${\displaystyle h}$ ) equal to the horizontal distance between the target point (${\displaystyle d}$ ) and the sensor.

This configuration is useful for planning aerial or satellite imaging operations, for which both resolution and maximum coverable area are critical aspects. The maximum attainable ${\displaystyle d}$  under resolution constraint can be calculated as follows:

${\displaystyle d=h={\frac {\mathrm {GSD_{max}} }{2}}\times {\frac {f}{p}}}$ 

Where ${\displaystyle \mathrm {GSD_{max}} }$  is the desired maximum ground sample distance, and all other variables are defined as above.

Within this constraint, reducing the horizontal distance (${\displaystyle d}$ ) without lowering the height (${\displaystyle h}$ ) decreases the off-nadir angle and shifts the imaging closer to nadir, thereby improving the ground sample distance. Conversely, decreasing ${\displaystyle h}$  while keeping ${\displaystyle d}$  constant increases the off-nadir angle beyond 45°, which degrades the GSD. The 45° configuration provides the widest possible coverage while maintaining the specified GSD limit.

See also

• Geographic distance

References

1. NZ Aerial Mapping Ltd (2009). "Frequently Asked Questions: What Is Ground Sample Distance?". Archived from the original on 2018-11-29. Retrieved 2009-07-25.
2. Jon C. Leachtenauer and Ronald G. Driggers (2001). Surveillance and Reconnaissance Imaging Systems: Modeling and Performance Prediction. Artech House. pp. 30–31. ISBN 978-1-58053-132-0.
3. Ronald G. Driggers (2003). Encyclopedia of Optical Engineering. CRC Press. p. 1392. ISBN 978-0-8247-4251-5.