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18-Geom-A5 Remote Sensing and Image Analysis · May 2015

Question 4 of 5: From Digital Number to At-Surface Reflectance

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — May 2015 — 04-Geom-A5 Remote Sensing and Image Analysis. Closed-book; one approved Casio or Sharp calculator permitted. Format: five questions of equal value (20 marks each), all of which must be answered (total 100 marks). Questions 1–4 are essay-format; Question 5 is a short quantitative comparison of two covariance matrices. Radiometric and image-processing conventions follow standard North-American digital-image-processing practice (8-bit Landsat/ETM+ imagery).

Reference texts: J. R. Jensen, Introductory Digital Image Processing: A Remote Sensing Perspective (4th ed., Pearson, 2016); Lillesand, Kiefer & Chipman, Remote Sensing and Image Interpretation (7th ed., Wiley, 2015); J. A. Richards, Remote Sensing Digital Image Analysis (5th ed., Springer, 2013); J. R. Schott, Remote Sensing: The Image Chain Approach (2nd ed., Oxford, 2007).

Question 4: From Digital Number to At-Surface Reflectance (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Major processing steps (DN → at-surface reflectance)

The conversion proceeds through a fixed chain, each step undoing one physical effect that stands between the recorded count and the intrinsic surface reflectance.

  1. Radiometric calibration: DN → at-sensor (top-of-atmosphere) radiance. Apply the sensor's published gain and offset (bias) to convert each raw digital number to spectral radiance: $$L_\lambda = \text{gain}\times \text{DN} + \text{offset},$$ where $L_\lambda$ is the at-sensor spectral radiance. This removes the arbitrary, sensor-specific quantization scaling and puts every band in absolute physical units (W m$^{-2}$ sr$^{-1}$ µm$^{-1}$).
  2. Conversion to top-of-atmosphere (TOA) reflectance. Normalize the radiance by the incoming solar irradiance and the illumination geometry: $$\rho_{\text{TOA}} = \frac{\pi\, L_\lambda\, d^2}{E_{\text{sun},\lambda}\,\cos\theta_s},$$ where $d$ is the Earth–Sun distance (astronomical units), $E_{\text{sun},\lambda}$ the mean exo-atmospheric solar irradiance for the band, and $\theta_s$ the solar zenith angle. This step removes the effects of varying solar elevation, seasonal Sun–Earth distance, and per-band solar output, yielding a unitless, scene-comparable reflectance measured at the top of the atmosphere.
  3. Atmospheric correction: TOA → at-surface (bottom-of-atmosphere) reflectance. Remove the atmosphere's additive and multiplicative effects — subtract the path radiance $L_p$, divide out the upward (target-to-sensor) transmittance $T_{\uparrow}$ and the downward (Sun-to-target) transmittance $T_{\downarrow}$, and account for the diffuse downwelling skylight $E_{\text{down}}$ (and, for high-accuracy work, adjacency). Conceptually, $$\rho_{\text{surf}} = \frac{\pi\,(L_\lambda - L_p)}{T_{\uparrow}\,\big(E_{\text{sun},\lambda}\cos\theta_s\, T_{\downarrow}/d^2 + E_{\text{down}}\big)},$$ using either an image-based method (e.g. dark-object subtraction) or a radiative-transfer model (6S, MODTRAN, FLAASH) driven by aerosol and water-vapour estimates. The result is the intrinsic surface reflectance.
  4. (Supporting steps.) Geometric correction / orthorectification and, where needed, topographic (illumination) normalization and cloud/shadow masking accompany the radiometric chain so that the reflectance is delivered on a correct map grid free of terrain-shading artefacts.

Why convert DN to reflectance before retrieving biophysical parameters

Raw digital numbers are relative, sensor- and scene-specific counts with no physical meaning: the same surface yields different DNs on different dates, under different Sun angles, through different atmospheres, and on different sensors. Converting to at-surface reflectance is essential for four reasons:

• Physical meaning & model input. Biophysical-retrieval algorithms (LAI, chlorophyll, leaf water, albedo, vegetation indices such as NDVI) are calibrated in terms of surface reflectance, not DN; reflectance is an intrinsic property of the surface that these physical and empirical models require as input.
• Comparability across time and space. Reflectance removes the influence of solar elevation, Earth–Sun distance, and atmosphere, so images from different dates, seasons, and scenes become directly comparable — a prerequisite for change detection and multitemporal monitoring.
• Cross-sensor consistency. Because reflectance is normalized out of sensor gain/offset and solar irradiance, data from different sensors (Landsat, Sentinel-2, SPOT) can be combined and compared.
• Removal of atmospheric & illumination effects. Path radiance and transmittance otherwise vary from scene to scene and would masquerade as real surface change; correcting to surface reflectance strips these confounders so the retrieved parameter reflects the ground, not the air or the Sun angle.

In short, only after DNs are converted to at-surface reflectance do the pixel values become an absolute, atmosphere- and illumination-independent physical quantity that biophysical-parameter retrieval can trust.