2.1. Data
Spatial performance evaluation of satellite imagery for thermal sensors via the knife-edge technique, also known as the Edge Method or Tilted Edge Method, has been successfully used in validating the spatial quality of satellite images across a wide range of channels on various platforms (e.g., [
12,
13,
14,
15,
16,
17,
18]). This technique relies on high contrast thermal boundaries (edges) to characterize the transition between the two sides of the edge. The edge targets shall be chosen such that they best represent the resolving power of the imaging system. In addition to the sharp thermal transition, this method relies on the thermal uniformity on either sides of the edge. Consistently high contrast between the two sides and thermal stability of these targets are more desirable for long-term monitoring of the spatial performance of the sensors than thermal events such as volcanic eruptions. Geometrically defined features, such as bridges and harbors (with enough thermal contrast with surrounding waters), provide lasting structural uniformity along the edge that the naturally formed edges (such as cliffs and coastlines) do not provide. With all of their imperfections, the naturally formed thermal edges are used for spatial characterization metrics for their high thermal contrasts, diversity of locations and orientations with respect to the orbital tracks of various platforms. Orbital motion of the platforms adds directional property to the scanning procedure. It should be noted that, in this work, along-track and across-track terms do not refer to scanning style differences (i.e., scanning from top to bottom or from side to side), but refer to the orientation of the edge image in the scanned scene. A north–south oriented edge image is used to characterize the across-track edge response and a perpendicular edge image to across-track direction, is used for along-track edge response characterization.
Table 1 lists our chosen edge targets in this work. High contrast edges (such as coastline) in desert areas are reliable candidates for the edge technique, especially those that sustain their thermal contrast for most of the year. Among the qualified coastlines with the aforementioned criteria, those with accommodating orientation for the near-polar orbits of Landsat 8 and 9 (at
inclination) were selected. The SAHA, OMAN, and YMEN sites were chosen by a study on spatial performance of Landsat 8 TIRS [
9]. LIBY was also chosen by [
5] for post-launch spatial performance assessment of Landsat 9 TIRS-2. Among the longer list of candidate scenes, these sites proved most thermally uniform, and seasonally reliable sites with relatively consistently high contrast coastlines for edge response measurements. The numbers separated by comma in the fifth column are associated with the total number of qualified images for Landsat 9 TIRS and Landsat 8 TIRS, respectively. If no cloud-free image existed in a 30 day interval, the cloud coverage threshold was gradually increased from zero until at least one usable image was found for a ∼30-day interval for the two instruments throughout 2022.
Figure 1 depicts the general location of those scenes. OMAN, YMEN, and SAHA have been previously chosen for spatial quality assessment of Landsat 8 TIRS, as these shorelines have near optimal angle of
from the sensors orbital orientation for across-track spatial characterization [
9]. LIBY in Northern Africa (Path/Row:186/38) also passed similar physical and positional criteria for along-track spatial performance characterization (shorelines with
orientation [
9]). Note that, even for the shared sites with past studies, no information about the exact sections of the shorelines that were used in their study was available. Our method of moving a measuring window along the shores and finding a section with consistently maximal thermal contrast and edge profile behavior over time was used to select the final sites among the candidate sites.
The top and bottom left panels of
Figure 2 show examples of the thermal maps of the Northern Africa and Western Sahara scenes as observed by the two TIRS instruments at B10 and B11.
The data used for this work are public and accessible through USGS’s Earth Explorer data repository. The TIRS imagery in both Landsat missions are Tier 1 L1-TP, 30 m resampled (using the Cubic Convolution method described in the Landsat Level-1 processing details
https://www.usgs.gov/landsat-missions/landsat-level-1-processing-details, accessed on 8 August 2024) TIF maps from the high quality (top-tier) Standard Terrain Corrected data products of collection 2. The data repository was queried over daytime nadir acquisitions by both TIRS instruments at both the B10 and B11 bands throughout 2022 (the first full-year overlap in their operations).
Table 1 lists the chosen scenes and their properties. SAHA and LIBY are the across- and along-track scenes that are used for spatial metric trends. For these two scenes, the goal was to find the least cloudy image in each ∼30-day interval throughout 2022 acquired by both sensors, in order to assess the longer-term consistency of the measured spatial metrics. In the months throughout which no cloud-free images were acquired, the acceptable cloud coverage range was gradually increased up to <
(and up to
in only two months for the Western Sahara site) until an image was found. Depending on the proximity of the cloud patch to our desired edge section, the chosen area for measurements would be moved by enough pixels away from the visible clouds to have a cloud-free section of the edge for our measurement. This was only necessary for one image in LIBY. The second set of along- and across-track sites (YMEN and OMAN) consisted of only
cloud coverage. Consequently, there are extended periods in 2022 during which no such image is available for one or both of these two sites. The idea behind choosing this limit for OMAN and YMEN was to have a set of “control” measurements in the along- and across-track directions, so that their metric fluctuations could not be speculated to have cloud-related reasons.
Before the final selection of the sites listed in
Table 1, more coastlines in Western Australia, the western coast of Mexico, and the western coast of Greenland were examined. Those sites did not provide as high contrast and sharp boundaries as the chosen sites. Those sites also did not have as many <
cloud covered scenes as SAHA and LIBY to make it possible to measure the trend of metrics during 2022. The numeric approach to find a perpendicular transect through the angled edge requires interpolation between crossing pixels. Combination of this numeric treatment with a lower contrast between the edge sides in some candidate scenes imposes a more selective approach in choosing the acceptable transects; a disadvantage that was more pronounced for the geometrically defined structures that are sharply angled with respect to the near-polar ground tracks of TIRS instruments (the Bay area bridges and Long Beach harbor were assessed before final selection). Our control scene, OMAN, happens to contain two geometrically defined structures in near-perfect perpendicular orientations with respect to each other and scan direction (see the Port of Duqm in
Figure 3). The measured metrics from these structures were used to compliment the spatial performance assessment of the sensors in the absence of coastline edge imperfections. The selection of the dataset is further discussed in the context of the results in
Section 4.
2.2. Measurement
The Edge Spread Function (ESF) is typically derived from analyzing the pixel values in each row of the chosen section of the image that includes the boundary between two relatively uniform yet significantly different sensor readings. Enough pixels on either side of the edge results in higher signal-to-noise edge profile (see Equation (
1)). ESF describes how the imaging system spreads the intensity of a sharp edge in its scanned image. Mathematical representation of the distribution of intensities along the individual transects crossing the edge can quantify this spread. As the slanted edge with respect to the focal plane axes is scanned by the sensor, the intensity values are measured at various positions along the transect that passes through the edge. The slight tilt of the edge in the image is a key feature in this analysis. Transects passing through a very straight edge with respect to the image frame would all have similar intensity distribution profile, which results in an undersampled edge profile created from the individual transect profiles, while a tilted edge causes different transects to pass through the edge center at different spots, resulting in an oversampled edge profile with a heightened ability to record the true spatial characteristics of the imaging sensor from that image.
Some conventionally measured spatial characterization metrics that are derived from ESF include: (i) An edge slope that is measured at 40–60% of the normalized ESF. Higher edge slopes mean sharper edges. (ii) a Relative Edge Response (RER), which is a measure of the steepness of the edge profile, and is the measured as the drop in ESF amplitude when moving within pixel of the center of the edge. RER reveals how an imaging system responds to a change in contrast over one pixel. A higher RER indicates a steeper edge and sharper image. Blurry images have higher RERs. (iii) Line Spread Function (LSF) and its Full Width at Half Maximum (FWHM), which signifies the width of the ESF at half of its maximum value. LSF is the first order derivative of the the ESF, and represents the extent of the edge spread. Smaller FWHMs indicate sharper edges and better spatial quality. (iv) The Edge Extent that is defined as horizontal distance between and of the edge response (lower and upper edge points), and measures the contrast transition distance on either side of an edge. And (v) the Modulation Transfer Function (MTF), which is the Fourier Transformation of the LSF. The spatial frequency response evaluation method is typically developed in accordance with the International Standard ISO 12233:2000. MTF characterizes the imaging system’s ability to reproduce different spatial frequencies or details in the image. Since MTF is a relative measure, it quantifies the ratio of the output modulation (i.e., contrast) to the input modulation for each spatial frequency. Higher MTF values at sampling frequency indicate a better ability to reproduce fine details and preserve contrast, while a lower MTF values indicates a reduction in detail reproduction and contrast. To quantify the shape of the MTF curves, three MTF shape parameters for each curve are measured: “MTF50”, which is the spatial frequency at which the MTF drops to of its maximum value; “”, which is MTF at half Nyquist sampling; and “”, which represents MTF at the Nyquist frequency.
The next step after finding one qualified image per month for both TIRS instruments is to identify the optimal section of the edge for the metric measurement.
was calculated for every
pixel array along the full stretch of a slightly tilted coastline with respect to the image frame (re square in
Figure 2). This was performed for all of the qualified images of Landsat 8 TIRS at B10. The goal was to identify a section of the edge that consistently yields
throughout the year. Initially, a threshold of
was chosen, as it has been reported to be a relatively effective signal level [
9]. Through numerous measurements of this factor from images captured under wide range of observing conditions and dates/times, this threshold was found to be sufficiently high to flag low-quality imagery, and not overwhelming high to unreasonably disqualify a reasonably sharp contrast. Equation (
1) shows the definition of
for the purpose of this study.
After the most consistently high-quality section of the coastline in the scene was identified, images of the same section of the coast in all of the TIRS images of the coast throughout 2022 at both B10 and B11 were gathered. There were only a few cases where the optimum measurement section needed to be moved by a few pixels to bypass a thermal anomaly (i.e., either a cluster of abnormal pixel values due to moisture in the land or in the atmosphere over the area) in the thermal map of the scene. Two examples of this optimally chosen section of the coastline (indicated by a red square) are shown in the top- and bottom-left-most panels of
Figure 2).
As described earlier, each row of the chosen 50 by 50 pixel array creates an edge profile. Those extracted profiles are then fit with a mathematical representation to locate the exact location of the edge with sub-pixel accuracy. A modified Fermi function (
) following the behavior of the linear portion of the edge profile where the edge center is located:
where “
” represents the mean value on the dark side of the edge, “
b” is the mean value on the bright side, “
s” represents the slope of the linear portion of the profile (since the conventional definition of the edge slope as a spatial metric is limited to a smaller section around the edge center, the best-fit parameter “
s” is not reported as the edge slope) and “
e” the edge center location. The shifted profiles by the best-fit value of the edge center,
e, create an oversampled edge profile that is then used to measure the spatial metrics defined earlier. An example of the aligned edge profiles in an along- and cross-track edges are shown in red in the top and bottom third panels of
Figure 2. The aligned stack of normalized edge spread responses are then resampled using linear interpolation to a uniform pixel spacing of 0.1–0.01 for slope determination of its linear section (finer spacing than 0.1 did not improve the fit, nor key parameter determination). The slope is determined from the 40 to 60 percent section (0.4–0.6) of the normalized edge response. The same procedure is applied for the Edge Extent determination (defined as 10 to 90 percent of the normalized response). To remove biases from our results, measurements from aliased or blurred images were removed before the performance evaluation. A modified form of an image classifier,
, to flag low-quality images was also measured in for the edge image (limited to the section where the edge profile is extracted).
Q essentially connects the width of representative diffraction-limited PSF of an imaging sensor’s produced image to its characteristic sampling distance. Q’s modified definition,
, includes the detector, optics, electronics, motion effects, etc. that contribute to the shape of the LSF [
19,
20]:
Using this definition, images with
are considered blurry, and those with
are considered aliased. A
of 100 m is applied for thermal imagery of Landsat 8 and 9. Prior to averaging all of the measurements of a particular spatial metric for each band and each sensor, any measurements associated with aliased and blurry images were removed from the set (see
Section 3). Motivated by multiple published works on Edge Method analyses that had found
alone to be helpful, but not enough, for weeding out the low quality images, the temporal variation of all of the measured spatial metrics (edge slope, Edge Extent, half edge response, FWHM of LSF, and MTF shape parameters) were examined together with the trend of their associated Q and
(early examples of these trends were presented in [
21]). For brevity,
Figure 3 only shows the edge slope trend with these tow factors per image in both B10 and B11 bands. There are clearly edges with a high
that have yielded a low edge slope (i.e., less sharp edge,) and the Q factor of that edge categorized the image as blurry, which justifies the low edge slope. Enough such cases led us to look into a combination of
and
in our vetting process for qualified images to used for metric averaging (presented in Figures 8–10).
2.3. Uncertainties
A definition of the uncertainties of the monitored spatial metrics can differ between the responsible teams in charge of verification and validation of the mission requirements. But, conceptually, the main contributors to the observed uncertainties are largely agreed upon: (i) optical system mirrors and lenses introduce aberrations, diffractions, and distortions that affect the shape of the LSF and MTF; (ii) atmospheric turbulence, scattering, and absorption can also impact the sharpness and spatial response of the imaging system by affecting the edge profile and the derived LSF and MTF; (iii) uncertainties in the radiometric calibration process can propagate into the calibrated digital numbers (DN) in the final map and affect the derived parameters, including the edge slope and width; (iv) inherent errors from the processing algorithms for radiometric and geometric corrections and resampling steps also propagate into spatial response characteristics; and (v) the characteristics of the imaged scene, such as the presence of complex textures, high-frequency details, or areas with low contrast, can also influence the measured edge slope and width. Spectrally variable or complex scenes and, in the case of thermal images, scenes that include thermally variable/unstable areas, challenge accurate edge detection, and contribute to uncertainties in the LSF and MTF measurements.
In our approach adopted here, the metrics for each of the individual transects that pass through the edge (rows or columns) are calculated. The standard deviation of the calculated metrics from individual transects is reported as the uncertainty of the measured metric. The metric itself is calculated from the full stack of the aligned transects.