New space meets old physics

A few months ago, I attended a conference where a company introduced their new optical payload capable of producing a 4.7m Ground Spatial Distance (GSD) from a 410 km orbital height with an F/5.6 optical front-end and 300mm focal length. When companies promise medium-to-high resolution remote-sensing capabilities from space with an optical payload the size of a 330ml Coca-Cola can, it sends a cold shiver running down my spine.

Earth observation is currently in a perfect storm. The value chain is shifting from raw data to services and application, fuelled by the latest artificial intelligence and machine learning technologies. There’s a strong pressure to reduce infrastructure and operational costs. Getting into space has never been so easy. New space and disruptive initiatives are not only developed by start-ups but also by large web actors with a huge investment capacity. Both aim at transforming space into a commodity.

Commercial providers of Earth Observation imagery know full well that GSD is not the sole criterion for selecting geospatial data. In fact, it is a noticeably weak criterion. Depending on the specific application – revisit time, speed of delivery, radiometric and spectral quality, geolocation accuracy and acquisition capacity are all key performance factors that require attention.

Still, we are living in a GSD-crazy world. Remote-sensing payloads and platform providers are in a continuous race for sharper images and higher revisit rates. In the process, newcomers are testing the boundaries of physics and placing their bets on large numbers of small and low-cost satellites.

10m, 5m, 2m, 1m, 50cm and going down to 25cm GSD is the new benchmark. It seems that there’s a feeling in the market that Moore’s law drives the evolving resolution of earth observation satellites. Unfortunately, the diffraction limit of optics – as defined by Rayleigh’s Criterion – has the final say when it comes to the performance of an earth observation instrument.

The spatial resolution of an instrument describes its ability to capture geometrical dimensions and determines what type of objects it is able to image correctly. It is frequently defined by the GSD and ground resolved distance (GRD), of which GSD is the most commonly used.

GSD is a theoretical measure that only takes into account the geometry of the detector elements (pixel pitch), the focal length of the instrument and the distance from the target (orbital height):

GSD = Orbital Height x Pixel Pitch / Focal Length

GRD relates to the diffraction-limited resolution for a given wavelength and orbital height, bringing the power of the optical system into the equation. The resolution is inverse proportional to the aperture of the instrument:

GRD = 1.22 x (Wavelength / Aperture Diameter) x Orbital Height

On CubeSat platforms, the instrument aperture is in many cases determined by the available dimension and/or volume. For example, on a 3U/6U CubeSat, the optical aperture is limited by a 10cm x 10cm area and on a 12U/16U structure, it is limited to a 20cm x 20 cm area. The next table provides an overview of the expected GRD for various optical apertures as associated with the above mentioned structures for the visible wavelengths and an orbital height of 500km.

Structure < 3U 3U/6U 12U/16U
Aperture Diameter 54mm 95mm(xScape100) 190mm

(xScape200)

GRD @ 450nm 5.12m 2.89m 1.44m
GRD @ 550nm 6.26m 3.53m 1.77m
GRD @ 650nm 7.40m 4.17m 2.09m

When using Rayleigh’s Criterion to determine the optimal spatial resolution of an instrument, it shows that a GRD of between 3-4m is achievable with a 95mm aperture system from a 500km orbital height and a GRD of between 1.5 and 2m with a 190mm aperture system in the visible range.

Figure 1: The on-axis Point Spread Function at 450nm of the xScape100 OFE with the relative pixel sizes for Q=1 and Q=2 electro-optical systems indicated.

Rayleigh’s Criterion is very useful for an initial and simple estimation of the spatial resolution that is achievable with an optical instrument. Other parameters, such as the Modulation Transfer Function (MTF), are also very useful to specify the optical performance. In conjunction with the MTF, the Q of an electro-optical system and the spatial sampling frequency could be used to select the GSD and ultimately the pixel pitch, where

Q = Wavelength x F-number / Pixel Pitch
or
Q = GRD / (1.22 x GSD)

Figure 1 provides a graphical representation of the relation between Q, pixel pitch and the point spread function of an optical system. The Q value is frequently used to relate GRD to GSD but must not be used in isolation. There is a seemingly perspective that image quality improves with a decrease in GSD, but there is also a degradation of image quality due to decreased optical MTF relative to the smaller GSD, increased sensitivity to image smear, and decreased spatial Signal-to-Noise ratio (SNR). The complexity of the interactions often makes it difficult to clearly understand and quantify the risks of a higher Q system.

In the field of digital signal processing, the Nyquist Sampling Theorem is used to determine the required sampling frequency to reconstruct the original signal in the digital domain. A continuous signal can be reconstructed from its samples if the waveform is sampled over twice as fast as its highest frequency component.

This Theorem is also applicable in the spatial sampling domain. This means that the spatial sampling frequency must be twice as high as the smallest distinguishable feature that needs to be detected. Therefore, when using a 5.5μm (0.0055mm) pixel pitch sensor to sample the image, the highest spatial frequency that the system will be able to reconstruct the image at the focal plane is 1/(2 x pixel pitch) = 91 line pairs/mm.

For example, one will be able to identify a 2m wide car with an optical payload with 10m GSD but when two 2m wide cars are 2m apart it will be impossible to distinguish between the two. To be able to extract this level of detail from the image the GRD and GSD of the system needs to be at least 1m.

Figure 2: Simulated images showing the effect of various spatial resolution limits with (a) the original image (b) a decrease in optical performance (c) a decrease in the pixel sampling resolution and (d) a decrease in sampling and optical resolution.

To illustrate the effect of the various spatial resolution limits, simulated images are shown in Figure 2 with:

A. The original high resolution image (instrument and optical resolution), natural features, cars and road markings are clearly visible.

B. The optical resolution is reduce by added “blur” to the image with natural features, cars and road markings less visible, but still distinguishable. This simulates a factor 10 reduction in optical resolution.

C. The instrument resolution is reduced by reducing the number of pixels by a factor of 10. natural features, cars and road markings less visible, but still distinguishable. Note how the spectral content in the image assist in distinguishing in features.

D. Both optical and instrument resolutions are reduced to simulated a 10x overall reduction in spatial resolution.
The system engineer, optical designer and remote sensing specialists are tasked to decide if the spatial resolution of the optical payload will be limited by the characteristics of the optical aperture (diffraction limited) or by the geometrical dimensions of the instrument (instrument limited). Where the GSD becomes smaller than the GRD, the optical payload’s spatial resolution will be limited by the diameter of the optics and the spectral wavelength.

The system engineer, optical designer and remote sensing specialists are tasked to decide if the spatial resolution of the optical payload will be limited by the characteristics of the optical aperture (diffraction limited) or by the geometrical dimensions of the instrument (instrument limited). Where the GSD becomes smaller than the GRD, the optical payload’s spatial resolution will be limited by the diameter of the optics and the spectral wavelength.

Figure 3: The GSD and GRD (at 550nm) plots for the xScape100 system versus orbital height.

Due to the diffraction at the image plane, all optical systems act as a low pass filter with a finite ability to resolve detail and have a spatial cut-off frequency. This frequency quantifies the smallest object resolvable by an optical system. The spatial cut-off frequency for a perfect corrected optical system is given by:

𝑓o = 1 / (λ x F/#) cycles/mm

For the xScape100 and xScape200, these frequencies are 298 cy/mm and 345 cy/mm, respectively. It is important to note that this formula gives the best-case resolution performance and is valid only for a perfect optical system. The modulation transfer function (MTF) is frequently used by * to describe the low-pass filter effect of an optical system. The MTF is defined as the absolute value of the Optical Transfer Function which is the Fourier transform of the point-spread function (PSF). The MTF value indicated how much of an object’s contrast in the image as a function of spatial frequency.

Figure 4: The diffraction limited MTF of the xScape100 system.

Figure 3 shows the diffraction limited MTF of the xScape100 OFE, which is above 24% at the Nyquist sampling frequency of 91 cycles/mm.

Almost all new optical earth observation instruments are pushing the limits of what’s theoretically possible with an optical payload and GSD is the popular parameter to quote when claiming and specifying the spatial resolution performance of such systems. GSD can’t be used in isolation and if it is, it will certainly disappoint the users of the data. Other parameters to evaluate are the GRD, the pixel size relative to the PSF, optical cut-off frequency and MTF at the Nyquist sampling frequency.

The rule of thumb, and the main take-away from this blog, is: when someone quotes a GSD the first question you need to ask is, “what is the effective aperture diameter of the optical system and what will the diffraction limited performance be at the intended orbital height and spectral wavelength?”. Follow-up questions must focus on the SNR and spectral resolution but more on this later.