Resolution

When Leah presented the data sources, a board member asked the obvious question: why not simply buy the sharpest images available? The answer took the rest of the meeting. The service area has about 1,100 fields, many of them quarter sections of 64 ha under center pivots, but also narrow strips along the coulees, small vegetable plots and the canal itself, which is 24 m wide at the water surface. The program needed to see crops change within days, distinguish irrigated from dry fields by subtle differences in moisture and temperature, and keep the cost of data and processing within the budget. Every one of these needs pulls on a different kind of resolution. This lesson explains the four resolutions of a remote sensing system, spatial, spectral, radiometric and temporal, how they are defined and measured, and why they cannot all be maximized at once.

Spatial resolution

Spatial resolution describes the size of the smallest feature that can be detected or measured. It is commonly quoted as the ground sample distance (GSD), the spacing between pixel centers on the ground, but GSD is only part of the story. The instantaneous field of view (IFOV) of a detector is the angle it subtends, and at nadir its ground footprint is approximately \(H \cdot \text{IFOV}\), where \(H\) is the altitude. For Landsat's Operational Land Imager at 705 km, an IFOV of about 42.5 microradians gives a 30 m footprint. Pixels at the edge of a wide swath look obliquely and cover more ground; for a sensor with a ±55° scan, such as the weather satellite imagers, edge pixels are several times larger than nadir pixels unless the design compensates.

A sharper description is the point spread function (PSF), the image of an infinitely small bright point. Optics, detector size, platform motion, electronics and the atmosphere all blur the image, so each pixel collects some light from its neighbors. The modulation transfer function (MTF), the Fourier transform of the PSF, gives the contrast with which patterns of different spatial frequency are reproduced; a well-designed sensor has an MTF of 0.2 to 0.3 at its Nyquist frequency, the pattern with a period of two pixels. Two sensors with the same GSD can therefore differ in effective sharpness, and resampling an image to a smaller pixel size does not create detail that the sensor did not record.

A related idea is the difference between detection and identification. A feature smaller than a pixel can be detected if it contrasts strongly with its surroundings: a hot fire covering a few square meters raises the temperature of a 1 km thermal pixel enough to trigger a fire detection, and a narrow canal shows clearly in a 30 m near infrared image because water is so dark. Identifying what a feature is, or measuring its size, needs several pixels across it. A common rule of thumb is that measuring a feature's area to within 10 % needs it to span at least several tens of pixels.

Mixed pixels

Most pixels in an agricultural landscape contain more than one cover: field edges, farmyards, roads, shelterbelts, ditches and canal banks all create mixed pixels whose reflectance is a weighted combination of the covers inside them. At 30 m, a pivot field has a ring of mixed pixels around its edge and a corner of dryland or grass in each square quarter section; at 250 m, few pixels are pure. Mixed pixels blur boundaries, cause misclassification at edges and bias area estimates if they are assigned whole to one class. The linear mixture model of Lesson 10 treats a mixed pixel's spectrum as the area-weighted sum of pure endmember spectra and estimates the fractions.

Spectral resolution

Spectral resolution describes the number, width and placement of the bands. A panchromatic band covers much of the visible and near infrared in one broad band; a multispectral sensor has a handful to a dozen or so bands, each typically 20 to 100 nm wide; a hyperspectral sensor has 100 to 300 or more contiguous bands, each about 5 to 10 nm wide. The width of a band is usually given as its full width at half maximum, the interval over which the spectral response exceeds half its peak. Narrow, well-placed bands can capture features such as the red edge, the chlorophyll absorption or the clay absorption at 2.2 µm that broad bands average away. Placement matters as much as width: Landsat's SWIR bands were chosen to measure leaf and soil moisture, and its cirrus band to detect thin cloud.

Narrow bands collect less light, so for a given detector and integration time they have a lower signal-to-noise ratio, or they need larger pixels to collect enough. This is one reason hyperspectral satellites commonly have 30 m pixels while multispectral sensors reach 10 m and panchromatic bands reach sub-meter resolution. Pansharpening, covered in Lesson 6, exploits the trade by combining a sharp panchromatic band with coarser color bands.

Radiometric resolution

Radiometric resolution describes how finely a sensor distinguishes differences in brightness. It is often quoted as the number of bits per pixel: an 8-bit sensor records 256 levels, a 12-bit sensor 4,096 and a 14-bit sensor 16,384. Landsat 8 records 12 bits and Landsat 9 14 bits, and Sentinel-2 records 12 bits. Bits alone do not guarantee sensitivity; what matters is the signal-to-noise ratio at typical radiance levels and the noise-equivalent change in reflectance or temperature. A sensor with many bits but high noise records its noise very precisely. For thermal sensors the corresponding measure is the noise-equivalent differential temperature (NEΔT), about 0.1 K or better for modern thermal satellite instruments.

Radiometric resolution matters most over dark targets such as water, where the signal occupies only the lowest levels, and for detecting small changes, such as the slight drop in NIR reflectance that precedes visible stress. The older 8-bit sensors often saturated over bright snow and cloud and lacked detail in water; 12-bit and 14-bit data make it possible to map water quality and shadows with confidence. Radiometric accuracy, the agreement of calibrated values with the truth, is a separate property that depends on calibration rather than on the number of levels.

Temporal resolution

Temporal resolution is the interval between successive observations of the same place. For a single satellite it is set by the repeat cycle, but in practice it is reduced by combining satellites and increased by cloud. A program that needs a clear image of every field within a two-week window in June needs several opportunities in that window, because the probability that any one overpass is clear is well below one. Satellites with pointable sensors can revisit more often than their repeat cycle, at the cost of oblique views. For radar, cloud does not matter, but repeat geometry does: interferometry requires images from the same orbit track, so the useful interval is the exact repeat cycle.

Temporal resolution must match the speed of the process being watched. Canal levels and flooded fields change in hours; crops change in days; the coulee banks move a few millimeters a month; land use changes over years. Matching the observation interval to the process avoids both missing events and paying for redundant data.

import math
# Spatial: ground sample distance from the instantaneous field of view
for name, h_km, ifov_urad in (("Landsat OLI", 705, 42.5), ("Sentinel-2 MSI 10 m bands", 786, 12.7), ("MODIS 250 m bands", 705, 354.0)):
    print(f"{name:26s} IFOV {ifov_urad:6.1f} urad at {h_km} km -> GSD {h_km * 1000 * ifov_urad * 1e-6:6.1f} m")

# Mixed pixels: a square field 400 m on a side on a grid of pixel size p (field edges not aligned)
print()
print("pixel  pixels touching field  pure field pixels  share of field area in pure pixels")
side = 400.0
for p in (3.0, 10.0, 30.0, 250.0):
    n_touch = math.ceil(side / p + 1) ** 2
    n_pure = max(0, math.floor(side / p - 1)) ** 2
    print(f"{p:5.0f} m  {n_touch:10d}            {n_pure:8d}           {n_pure * p * p / side**2 * 100:6.1f} %")

# Radiometric: quantization levels and the smallest reflectance step
print()
for bits in (8, 12, 14, 16):
    lv = 2 ** bits
    print(f"{bits:2d} bits: {lv:6d} levels, step {1.0 / (lv - 1):.6f} over a 0-1 reflectance range")

# Temporal: chance of at least one clear view in a 10-day window
print()
p_clear = 0.40      # probability that a single overpass is clear over the field in June
for name, revisit in (("Landsat 8/9 (8 d)", 8), ("Sentinel-2 pair (about 3 d at this latitude)", 3), ("daily small-satellite fleet", 1)):
    n = 10 // revisit
    print(f"{name:44s} {n:2d} looks, P(at least one clear) = {1 - (1 - p_clear) ** n:5.3f}")

# Data volume for one 100 km x 100 km scene
print()
for name, p, nb, bits in (("10 m, 4 bands, 12 bit", 10, 4, 12), ("30 m, 11 bands, 12 bit", 30, 11, 12), ("30 m hyperspectral, 220 bands, 14 bit", 30, 220, 14)):
    px = (100000 / p) ** 2
    print(f"{name:40s} {px * nb * bits / 8 / 1e6:8.0f} MB")

Landsat OLI                IFOV   42.5 urad at 705 km -> GSD   30.0 m
Sentinel-2 MSI 10 m bands  IFOV   12.7 urad at 786 km -> GSD   10.0 m
MODIS 250 m bands          IFOV  354.0 urad at 705 km -> GSD  249.6 m

pixel  pixels touching field  pure field pixels  share of field area in pure pixels
    3 m       18225               17424             98.0 %
   10 m        1681                1521             95.1 %
   30 m         225                 144             81.0 %
  250 m           9                   0              0.0 %

 8 bits:    256 levels, step 0.003922 over a 0-1 reflectance range
12 bits:   4096 levels, step 0.000244 over a 0-1 reflectance range
14 bits:  16384 levels, step 0.000061 over a 0-1 reflectance range
16 bits:  65536 levels, step 0.000015 over a 0-1 reflectance range

Landsat 8/9 (8 d)                             1 looks, P(at least one clear) = 0.400
Sentinel-2 pair (about 3 d at this latitude)  3 looks, P(at least one clear) = 0.784
daily small-satellite fleet                  10 looks, P(at least one clear) = 0.994

10 m, 4 bands, 12 bit                         600 MB
30 m, 11 bands, 12 bit                        183 MB
30 m hyperspectral, 220 bands, 14 bit        4278 MB

The code quantifies the trade-offs. A 400 m field is 98 % pure at 3 m and 95 % at 10 m, but only 81 % at 30 m and not at all at 250 m, where every pixel is mixed. With a 40 % chance of a clear sky on any one pass, a single Landsat look in a 10-day window succeeds only 40 % of the time, three Sentinel-2 looks 78 % of the time and a daily constellation almost always. Data volume grows with the square of the inverse pixel size and in proportion to the number of bands and bits, so a 30 m hyperspectral scene with 220 bands is over 20 times the size of a 30 m multispectral scene.

Trade-offs and choices

The four resolutions compete. Finer spatial resolution means smaller pixels, which collect less energy, so narrower bands or more bits become harder to achieve. A wide swath, which gives frequent coverage, means more detectors or larger pixels and higher data rates. Data downlink capacity and onboard storage limit how much of everything can be recorded. The table summarizes how the needs of the district's products mapped to resolutions.

Leah's conclusion for the board was that no single sensor could serve every product, and that the sharpest images were the right choice only for a few small tasks. Sentinel-2 at 10 m offered the best balance for field-level crop and irrigation products, Landsat thermal data at 100 m were adequate for water use if interior pixels were used, and radar was needed for the banks and floods. Field size also guided the choice: most fields in the service area were large enough for 10 m and even 30 m pixels, but the small vegetable plots near the town would need commercial imagery or would be reported as a mixed class.