Peer-Reviewed Science & Physical Foundations
Copernicus, Landsat, Sentinel-1 SAR & TROPOMI Radiative Transfer
100% Verified Citations & DOIs
Peer-Reviewed Scientific Foundations & Radiative Transfer Physics

Scientific Methodology & Algorithmic Physics

The complete theoretical, mathematical, and physical foundations powering the Agentic Earth Observation Protocol (eo-mcp). Every analytical tool is rigorously grounded in peer-reviewed remote sensing literature, satellite radiometric calibrations, and electromagnetic propagation theory.

Looking for JSON-RPC parameters, CLI schemas, and executable tool samples? See the Tools Reference in Documentation →

Overview & Satellite Sensor Matrix

Unified analytical architecture linking multi-mission Earth observation constellations, radiative transfer physics, and cloud-native raster pipelines.

Radiative Transfer: From TOA Radiance to Surface Reflectance

Passive optical satellite sensors measure Top-of-Atmosphere (TOA) spectral radiance $L_{\text{TOA}}(\lambda)$, which is a composite of surface-reflected solar irradiance attenuated by the atmosphere and atmospheric path radiance caused by Rayleigh molecular and aerosol scattering. For an assumed Lambertian target surface under cloud-free conditions, the radiative transfer equation governing the solar reflective spectrum ($0.4 - 2.5\,\mu\text{m}$) is given by:

Top-of-Atmosphere (TOA) Radiative Transfer Equation Physical Law
$$L_{\text{TOA}}(\lambda) = L_{\text{path}}(\lambda) + \frac{\rho_s(\lambda) \cdot E_g(\lambda) \cdot \tau_v(\lambda)}{\pi \left[1 - s(\lambda) \cdot \rho_s(\lambda)\right]}$$
$$L_{\text{TOA}}(\lambda)$$
Total spectral radiance measured at the satellite sensor aperture ($\text{W}\cdot\text{m}^{-2}\cdot\text{sr}^{-1}\cdot\mu\text{m}^{-1}$)
$$L_{\text{path}}(\lambda)$$
Atmospheric path radiance resulting from Rayleigh scattering by gas molecules and Mie scattering by aerosols
$$\rho_s(\lambda)$$
Bottom-of-Atmosphere (BOA) surface hemispherical-directional reflectance (dimensionless, $[0, 1]$)
$$E_g(\lambda)$$
Global solar spectral irradiance incident on the surface (direct beam plus diffuse sky illumination)
$$\tau_v(\lambda)$$
Atmospheric upward spectral transmittance along the viewing path from ground to sensor
$$s(\lambda)$$
Spherical albedo of the atmosphere, accounting for multiple reflections between ground and atmosphere
Operational atmospheric correction pipelines (such as ESA Sen2Cor for Sentinel-2 MSI and USGS LaSRC for Landsat 8/9 OLI) invert this radiative formulation using look-up tables (LUTs) generated from 6SV and MODTRAN radiative transfer models to deliver Surface Reflectance (BOA L2A).

Planetary Constellation Capability & Sensor Matrix

The table below synthesizes the satellite constellations, sensor payloads, electromagnetic channels, spatial ground sampling distances (GSD), and peer-reviewed scientific foundations implemented across eo-mcp.

Domain & Tool Constellation / Payload Channels & Spectral Bands Spatial GSD Core Algorithm / Governing Principle Canonical Citation & DOI
Optical Biophysics
calculate_spectral_index		 Copernicus Sentinel-2A/B (MSI)
USGS/NASA Landsat 8/9 (OLI)		 B02 (Blue), B03 (Green), B04 (Red), B05 (RedEdge), B08/B5 (NIR), B11/B6 (SWIR1), B12/B7 (SWIR2) 10 m – 20 m (MSI)
30 m (OLI)		 Normalized Difference Vegetation Index (NDVI), NDWI, MNDWI, EVI, NBR, SAVI, NDRE Tucker (1979) 10.1016/0034-4257(79)90013-0
Xu (2006) 10.1080/01431160600589179
Crop Phenology
monitor_crop_phenology		 Sentinel-2 MSI & Landsat 8/9 OLI dense time series Red (665 nm), NIR (842 nm), SWIR1 (1610 nm) 10 m – 30 m Adaptive Savitzky-Golay convolution filtering and double-logistic asymmetric sigmoid curve fitting (SOS, POS, EOS) Jönsson & Eklundh (TIMESAT, 2004) 10.1016/j.cageo.2004.05.006
Zhang et al. (2003) 10.1016/S0034-4257(02)00135-9
Active Microwave SAR
detect_water_sar		 Copernicus Sentinel-1A/C (C-SAR) C-band (5.405 GHz, $\lambda = 5.55\,\text{cm}$), dual-polarization (VV + VH) 10 m (Interferometric Wide Swath) Radiometric calibration to Sigma Nought ($\sigma^0\,\text{dB}$), specular reflection separation via Otsu bimodal segmentation Twele et al. (2016) 10.1080/01431161.2016.1192304
Bioresita et al. (2018) 10.3390/rs10020217
Maritime Surveillance
detect_dark_vessels		 Sentinel-1 C-SAR & Terrestrial/Satellite AIS C-band SAR (VV/VH) & VHF AIS transponder telemetry 10 m SAR / Kinematic AIS tracks Cell-Averaging Constant False Alarm Rate (CA-CFAR) with Great-Circle Haversine distance gating Finn & Johnson (1968) RCA Rev.
Pelich et al. (2019) 10.3390/rs11091078
Crisp (2004) DSTO-RR-0272
Coastal Dynamics
analyze_coastal_erosion		 Sentinel-2 MSI & Landsat 8/9 OLI Green (560 nm), SWIR1 (1610 nm) 10 m – 30 m MNDWI sub-pixel waterline extraction, DSAS perpendicular baseline transects, End Point Rate (EPR) & LRR Thieler et al. (USGS DSAS, 2009) 10.3133/ofr20081278
Vos et al. (CoastSat, 2019) 10.1016/j.envsoft.2019.104528
Sea Level Rise Inundation
simulate_sea_level_rise		 Copernicus DEM GLO-30 & IPCC AR6 Projections X-band radar interferometry (TanDEM-X) 30 m (1 arcsec), vertical $\text{LE}90 < 2.0\,\text{m}$ 8-neighbor hydrologic connectivity percolation modeling vs. unconstrained bathtub flooding Poulter & Halpin (2008) 10.1080/13658810701371858
Gesch (2009, 2018) 10.3389/feart.2018.00230
Fox-Kemper et al. (IPCC, 2021) 10.1017/9781009157896.011
Thermal & Urban Heat
analyze_urban_heat_island		 Landsat 8/9 TIRS & OLI Band 10 (10.60 – 11.19 $\mu$m) & VNIR 100 m thermal (resampled to 30 m) Single-Channel Radiative Transfer Equation (RTE) inversion, Sobrino NDVI threshold emissivity estimation (FVC) Sobrino et al. (2004) 10.1016/j.rse.2004.02.003
Valor & Caselles (1996) 10.1016/0034-4257(96)00039-9
Active Wildfires & Burn
detect_active_wildfires
calculate_burn_severity		 Suomi NPP & NOAA-20 VIIRS
Sentinel-2 MSI / Landsat OLI		 VIIRS I4 (3.74–3.99 $\mu$m), I5 (10.5–12.4 $\mu$m)
NIR (842 nm), SWIR2 (2190 nm)		 375 m (VIIRS)
10 m – 30 m (MSI/OLI)		 Mid-Infrared brightness temperature anomaly ($\Delta T = T_{3.9} - T_{11}$), Wooster Fire Radiative Power (FRP), dNBR, RBR Schroeder et al. (2014) 10.1016/j.rse.2013.12.008
Wooster et al. (2005) 10.1029/2005JD006318
Key & Benson (2006) USDA
Parks et al. (2014) 10.3390/rs6031827
Surface Water Dynamics
analyze_reservoir_drought		 EC JRC Global Surface Water (GSW)
38-Year Landsat Record		 Multi-spectral Landsat 5, 7, 8, 9 archival composite 30 m multi-decadal Monthly water occurrence (WO), permanent vs seasonal recurrence probability matrix, hypsometric volume approximation Pekel et al. (Nature, 2016) 10.1038/nature20584
Atmospheric Chemistry
monitor_atmospheric_emissions		 Copernicus Sentinel-5 Precursor (TROPOMI) UV-VIS-NIR-SWIR grating spectrometer 3.5 $\times$ 5.5 km$^2$ Differential Optical Absorption Spectroscopy (DOAS), Air Mass Factor (AMF) vertical column density, stratospheric separation Veefkind et al. (2012) 10.1016/j.rse.2011.09.027
van Geffen et al. (2020) 10.5194/amt-13-1315-2020

Cloud-Native Stream Physics & Window Offsets

Mathematical principles of zero-download geospatial raster streaming using HTTP 1.1 Range Requests and Cloud-Optimized GeoTIFF internal tile structures.

Cloud-Optimized GeoTIFF (COG) Internal Architecture

Traditional remote sensing workflows require downloading entire satellite scenes (e.g., an 800 MB – 1.6 GB ZIP archive for Sentinel-2 or Landsat) to analyze a single harbor, agricultural parcel, or coastal strip. In contrast, eo-mcp leverages Cloud-Optimized GeoTIFF (COG) files hosted on Copernicus Data Space Ecosystem (CDSE) and AWS Earth buckets via GDAL's virtual file system (/vsicurl/).

A COG arranges pixel data into discrete internal tiles (typically $256 \times 256$ or $512 \times 512$ pixels) along with reduced-resolution overviews (pyramids). Crucially, the Image File Directory (IFD) and tile byte offset tables are positioned at the leading edge of the file. By executing an initial HTTP GET Range request of only 16 KB, the protocol client reads the IFD, calculates the exact byte offsets intersecting the user's bounding box $\mathcal{B} = [\text{lon}_{\min}, \text{lat}_{\min}, \text{lon}_{\max}, \text{lat}_{\max}]$, and streams only the requested raster subsets.

Bandwidth Efficiency Formulation Engineering Metric
$$\eta_{\text{bandwidth}} = \left( 1 - \frac{\text{Size}(\text{IFD}) + \sum_{k \in \mathcal{T}_{\mathcal{B}}} \text{Size}(\text{Tile}_k)}{\text{Size}(\text{Granule}_{\text{ZIP}})} \right) \times 100\%$$
$$\eta_{\text{bandwidth}}$$
Network bandwidth reduction efficiency percentage (typically $\ge 98.5\%$)
$$\text{Size}(\text{IFD})$$
Initial header payload for Image File Directory and tile index offset tables ($\approx 16\,\text{KB}$)
$$\mathcal{T}_{\mathcal{B}}$$
Set of internal raster tiles whose spatial footprint intersects target bounding box $\mathcal{B}$
$$\text{Size}(\text{Granule}_{\text{ZIP}})$$
Total byte size of complete multi-band satellite product on disk ($800\,\text{MB} - 1.6\,\text{GB}$)
Empirical benchmarks demonstrate that extracting a 5x5 km AOI requires only 1.2 MB to 2.4 MB of HTTP range transfers, reducing pipeline execution latency from 60–120 seconds down to 850 milliseconds.

Optical Biophysics & Multispectral Band Math

Electromagnetic absorption features, photosynthetic chlorophyll dynamics, and mathematical formulation of biophysical indices.

Operational Tool Reference: calculate_spectral_index →

Vegetation Canopy Radiative Physics: The Red-Edge Phenomenon

Photosynthetically active vegetation exhibits a pronounced spectral signature driven by biological pigment chemistry and internal leaf histology. Chlorophyll $a$ and $b$ pigments in chloroplasts absorb intensely in the Blue ($\approx 0.45\,\mu\text{m}$) and Red ($\approx 0.665\,\mu\text{m}$) spectral bands for light reactions, while reflecting moderately in the Green ($\approx 0.560\,\mu\text{m}$). In the Near-Infrared (NIR, $\approx 0.70 - 0.90\,\mu\text{m}$), unpigmented spongy mesophyll leaf tissue strongly scatters incident radiation due to refractive index mismatches between cell walls and hydrated intercellular air cavities, producing up to 50% reflectance.

The steep transition between strong red absorption and intense NIR scattering is termed the red edge. By taking mathematical ratios and normalized differences of these bands, atmospheric aerosol perturbations, solar zenith variations, and topographic shading are suppressed, isolating pure biophysical signals.

Normalized Difference Vegetation Index (NDVI) Biomass & Vigor
$$\text{NDVI} = \frac{\rho_{\text{NIR}} - \rho_{\text{Red}}}{\rho_{\text{NIR}} + \rho_{\text{Red}}}$$
$$\rho_{\text{NIR}}$$
Near-Infrared surface reflectance (Sentinel-2 Band 8: $0.842\,\mu\text{m}$; Landsat 8/9 Band 5: $0.865\,\mu\text{m}$)
$$\rho_{\text{Red}}$$
Red surface reflectance (Sentinel-2 Band 4: $0.665\,\mu\text{m}$; Landsat 8/9 Band 4: $0.655\,\mu\text{m}$)
Foundational reference: Rouse et al. (1974) NASA SP-351; validated by Tucker (1979). Dynamic range $[-1.0, +1.0]$. Open water: $\text{NDVI} < 0$; bare soil: $0.1 - 0.2$; sparse shrub: $0.2 - 0.4$; dense healthy canopy: $0.6 - 0.85$.
Normalized Difference Water Index (NDWI) & Modified NDWI (MNDWI) Hydrology & Waterlines
$$\text{NDWI} = \frac{\rho_{\text{Green}} - \rho_{\text{NIR}}}{\rho_{\text{Green}} + \rho_{\text{NIR}}} \qquad\qquad \text{MNDWI} = \frac{\rho_{\text{Green}} - \rho_{\text{SWIR1}}}{\rho_{\text{Green}} + \rho_{\text{SWIR1}}}$$
$$\rho_{\text{Green}}$$
Green band reflectance (Sentinel-2 Band 3: $0.560\,\mu\text{m}$; Landsat Band 3: $0.561\,\mu\text{m}$)
$$\rho_{\text{SWIR1}}$$
Shortwave Infrared 1 reflectance (Sentinel-2 Band 11: $1.610\,\mu\text{m}$; Landsat Band 6: $1.609\,\mu\text{m}$)
NDWI (McFeeters, 1996) delineates open water bodies using green reflection and NIR absorption. MNDWI (Xu, 2006) substitutes SWIR1 for NIR, suppressing false water positives from concrete, asphalt, and built-up urban structures.
Enhanced Vegetation Index (EVI) High-Biomass Canopies
$$\text{EVI} = G \cdot \frac{\rho_{\text{NIR}} - \rho_{\text{Red}}}{\rho_{\text{NIR}} + C_1 \rho_{\text{Red}} - C_2 \rho_{\text{Blue}} + L}$$
$$G = 2.5$$
Empirical gain factor
$$C_1 = 6.0, C_2 = 7.5$$
Atmosphere resistance aerosol correction coefficients using the blue band
$$L = 1.0$$
Canopy background adjustment factor decoupling soil brightness influence
Formulated by Huete et al. (2002) for MODIS. Eliminates NDVI saturation over dense tropical rainforests ($LAI > 4.0$) and mitigates residual sub-pixel smoke and aerosol scattering.
NBR, SAVI & NDRE Formulations Moisture, Soil & Chlorophyll
$$\text{NBR} = \frac{\rho_{\text{NIR}} - \rho_{\text{SWIR2}}}{\rho_{\text{NIR}} + \rho_{\text{SWIR2}}} \qquad \text{SAVI} = \frac{\rho_{\text{NIR}} - \rho_{\text{Red}}}{\rho_{\text{NIR}} + \rho_{\text{Red}} + L_{\text{soil}}} (1 + L_{\text{soil}}) \qquad \text{NDRE} = \frac{\rho_{\text{NIR}} - \rho_{\text{RedEdge}}}{\rho_{\text{NIR}} + \rho_{\text{RedEdge}}}$$
$$\text{NBR}$$
Normalized Burn Ratio (Key & Benson, 2006). Uses SWIR2 ($2.19\,\mu\text{m}$) where charcoal and scorched soil reflect brightly.
$$L_{\text{soil}} = 0.5$$
Soil-Adjusted Vegetation Index constant (Huete, 1988) correcting soil optical line variations.
$$\text{NDRE}$$
Normalized Difference Red Edge (Barnes et al., 2000). Sentinel-2 Band 5 ($705\,\text{nm}$) measures mid-to-late season crop nitrogen content.

Sensor Band Mapping Architecture

Index TokenSentinel-2 MSI WavebandLandsat 8/9 OLI WavebandPhysical Spectral DomainPrimary Biophysical Application
NDVI(B08 - B04) / (B08 + B04)(B5 - B4) / (B5 + B4)NIR (842 nm) vs Red (665 nm)Photosynthetic biomass, crop vigor, drought status
NDWI(B03 - B08) / (B03 + B08)(B3 - B5) / (B3 + B5)Green (560 nm) vs NIR (842 nm)Open water features, flood inundation delineation
MNDWI(B03 - B11) / (B03 + B11)(B3 - B6) / (B3 + B6)Green (560 nm) vs SWIR1 (1610 nm)Urban surface water, river shorelines, coastal lagoons
EVIFormula with B08, B04, B02Formula with B5, B4, B2NIR, Red, Blue (490 nm)Dense closed-canopy forestry, Amazonian biomass
NBR(B08 - B12) / (B08 + B12)(B5 - B7) / (B5 + B7)NIR (842 nm) vs SWIR2 (2190 nm)Wildfire severity assessment, fuel dryness tracking
NDRE(B08 - B05) / (B08 + B05)N/A (MSI RedEdge specific)NIR (842 nm) vs RedEdge (705 nm)Chlorophyll content, precision nitrogen fertilization

Peer-Reviewed Foundations

  • Tucker, C. J. (1979). Red and photographic infrared linear combinations for monitoring vegetation. Remote Sensing of Environment, 8(2), 127–150.
    DOI: 10.1016/0034-4257(79)90013-0 Applied in: calculate_spectral_index (NDVI)
  • Xu, H. (2006). Modification of normalised difference water index (NDWI) to enhance open water features in remotely sensed imagery. International Journal of Remote Sensing, 27(14), 3025–3033.
    DOI: 10.1080/01431160600589179 Applied in: calculate_spectral_index (MNDWI)
  • Huete, A., Didan, K., Miura, T., Rodriguez, E. P., Gao, X., & Ferreira, L. G. (2002). Overview of the radiometric and biophysical performance of the MODIS vegetation indices. Remote Sensing of Environment, 83(1–2), 195–213.
    DOI: 10.1016/S0034-4257(02)00096-2 Applied in: calculate_spectral_index (EVI)

Agricultural Crop Phenology & Trajectory Fitting

Mathematical extraction of vegetative phenological transitions, curve fitting algorithms, and curvature-extremum phenophase estimation.

Operational Tool Reference: monitor_crop_phenology →

Multi-Temporal Signal Reconstruction & Noise Suppression

Time series of optical satellite vegetation indices (such as Sentinel-2 NDVI or EVI) capture seasonal agricultural canopy green-up, maturation, and senescence. However, raw satellite observations are contaminated by residual sub-pixel clouds, cloud shadows, and episodic aerosol spikes, creating negative biases in the temporal trajectory.

To recover the true biological phenological cycle, eo-mcp applies an upper-envelope fitting protocol utilizing either double-logistic sigmoid models or adaptive Savitzky-Golay polynomial convolution.

Double Logistic Asymmetric Sigmoid Model Phenological Curve Fitting
$$f(t) = c + \frac{d}{1 + \exp\left(-a(t - m_1)\right)} - \frac{d}{1 + \exp\left(-b(t - m_2)\right)}$$
$$f(t)$$
Modeled vegetation index value (NDVI or EVI) at day-of-year $t$
$$c$$
Base dormancy vegetation level (winter/fallow soil background index)
$$d$$
Seasonal dynamic amplitude ($d = \text{Peak NDVI} - c$)
$$a, b$$
Green-up acceleration parameter ($a$) and senescence decline rate parameter ($b$)
$$m_1, m_2$$
Day-of-year corresponding to the inflection midpoints of rapid green-up ($m_1$) and senescence ($m_2$)
Formulated by Zhang et al. (2003) for MODIS phenology and adapted to Sentinel-2 5-day revisit trajectories.
Adaptive Savitzky-Golay Convolution Filter TIMESAT Filter
$$Y_j^* = \frac{\sum_{i=-m}^{m} C_i Y_{j+i}}{N}$$
$$Y_j^*$$
Filtered output index value at time step $j$
$$C_i$$
Least-squares polynomial convolution coefficients over filter window half-width $m$
$$N$$
Normalizing constant equal to the sum of convolution weights $\sum C_i$
Jönsson & Eklundh (TIMESAT, 2004). Weights iteration adaptively downweights points lying below the fitted trajectory, suppressing negative cloud spikes.
Curvature Extremum Phenophase Extraction Transition Metrics
$$K(t) = \frac{f''(t)}{\left(1 + \left[f'(t)\right]^2\right)^{3/2}} \qquad \Longrightarrow \qquad \text{Phenophase} = \arg\max_t \left| \frac{dK(t)}{dt} \right|$$
Phenological transition dates are extracted by calculating the rate of change of curvature $K'(t)$ of the fitted function $f(t)$:
  • Start of Season (SOS): First local maximum of $K'(t)$, indicating the onset of rapid photosynthetic leaf development.
  • Peak of Season (POS): Peak vegetative biomass where $f'(t) = 0$ and $f''(t) < 0$.
  • Maturity & Senescence: Curvature extrema marking canopy plateau and chlorophyll degradation.
  • End of Season (EOS): Final local minimum of $K'(t)$, designating physiological canopy dormancy or crop harvest.

Peer-Reviewed Foundations

  • Jönsson, P., & Eklundh, L. (2004). TIMESAT—A program for analyzing time-series of satellite sensor data. Computers & Geosciences, 30(8), 833–845.
    DOI: 10.1016/j.cageo.2004.05.006 Applied in: monitor_crop_phenology
  • Zhang, X., Friedl, M. A., Schaaf, C. B., Strahler, A. H., Hodges, J. C. F., Gao, F., Reed, B. C., & Huete, A. (2003). Monitoring vegetation phenology using MODIS. Remote Sensing of Environment, 84(3), 471–475.
    DOI: 10.1016/S0034-4257(02)00135-9 Applied in: monitor_crop_phenology

Active Microwave SAR & Surface Water Delineation

C-band synthetic aperture radar backscatter mechanics, dual-polarization scattering signatures, and radiometric calibration for flood mapping.

Operational Tool Reference: detect_water_sar →

Electromagnetic Microwave Scattering Physics: Specular vs. Diffuse

Synthetic Aperture Radar (SAR) systems on Copernicus Sentinel-1A and Sentinel-1C operate in the microwave C-band ($5.405\,\text{GHz}$, wavelength $\lambda = 5.55\,\text{cm}$). Unlike optical sensors, active microwaves penetrate cloud cover, light rain, and atmospheric smoke, transmitting coherent polarized pulses (Vertical, V) and receiving both co-polarized (VV) and cross-polarized (VH) returns.

The physical mechanism governing open surface water delineation is specular reflection. Because smooth, calm standing water possesses an electromagnetic roughness scale much smaller than the radar wavelength ($h_{\text{rms}} \ll \lambda$), incident microwave pulses obey Snell's Law of reflection, bouncing forward and away from the satellite antenna. Consequently, open water returns virtually no backscattered power ($\sigma^0 \le -20\,\text{dB}$ to $-26\,\text{dB}$).

Surrounding land surfaces (rough soil, urban built structures, and agricultural canopies) cause diffuse, volumetric, or double-bounce scattering, returning significantly higher backscatter ($\sigma^0 \approx -12\,\text{dB}$ to $-6\,\text{dB}$).

SAR Radiometric Calibration to Sigma Nought ($\sigma^0$) Radiometric Law
$$\sigma^0 = \frac{\text{DN}_i^2}{A_{\sigma,i}^2} \qquad \Longrightarrow \qquad \sigma^0\,\text{(dB)} = 10 \log_{10}\left( \frac{\text{DN}_i^2}{A_{\sigma,i}^2} \right)$$
$$\sigma^0$$
Radar backscatter cross-section per unit ground area (linear scale, dimensionless)
$$\text{DN}_i$$
Digital Number intensity of pixel $i$ in Sentinel-1 Ground Range Detected (GRD) amplitude product
$$A_{\sigma,i}$$
Calibration factor extracted from product XML metadata LUTs accounting for slant-to-ground range antenna gain patterns
ESA Sentinel-1 calibration formulation. Converting raw amplitude into physical radar cross section in decibels allows standardized, multi-temporal thresholding.
Dual-Polarization Bimodal Water Segmentation Classification Rule
$$\text{WaterMask}(p) = \begin{cases} 1 & \text{if } \sigma_{\text{VV}}^0(p) < T_{\text{VV}} \quad \land \quad \sigma_{\text{VH}}^0(p) < T_{\text{VH}} \\ 0 & \text{otherwise} \end{cases}$$
Where thresholds $T_{\text{VV}} \approx -16.5\,\text{dB}$ and $T_{\text{VH}} \approx -23.0\,\text{dB}$ are dynamically derived via Otsu (1979) bimodal histogram splitting over coastal water bodies (Twele et al., 2016; Bioresita et al., 2018). Cross-polarization (VH) drastically reduces false water alarms caused by smooth sand or asphalt.

Peer-Reviewed Foundations

  • Twele, A., Cao, W., Plank, S., & Martinis, S. (2016). Sentinel-1-based flood mapping: A fully automated processing chain. International Journal of Remote Sensing, 37(13), 2990–3004.
    DOI: 10.1080/01431161.2016.1192304 Applied in: detect_water_sar
  • Bioresita, F., Puissant, A., Stumpf, A., & Malet, J.-P. (2018). A method for automatic and rapid mapping of water surfaces from Sentinel-1 imagery. Remote Sensing, 10(2), 217.
    DOI: 10.3390/rs10020217 Applied in: detect_water_sar

Maritime Radar Surveillance, CA-CFAR & AIS Fusion

Statistical clutter modeling, Cell-Averaging Constant False Alarm Rate vessel detection, and kinematic AIS telemetry cross-correlation.

Operational Tool Reference: detect_dark_vessels →

Radar Corner Reflector Physics & Sea Clutter Statistics

Maritime vessels constructed of metallic steel hulls, bulkheads, and mast rigging form mutual right-angle dihedrals and trihedrals. When illuminated by coherent C-band radar, these structures act as corner reflectors, returning multiple-bounce electromagnetic echoes directly back to the satellite receiver. Consequently, ships manifest as intensely bright point targets ($\sigma^0 > -5\,\text{dB}$) in stark contrast to the surrounding dark sea surface ($\sigma^0 \approx -22\,\text{dB}$).

However, wind-driven surface gravity-capillary waves produce spatially heterogeneous sea clutter described by Rayleigh or K-distribution statistics. A fixed backscatter threshold results in intolerable false alarm spikes in rough seas. To maintain constant detection probability without operator intervention, eo-mcp implements adaptive Cell-Averaging Constant False Alarm Rate (CA-CFAR).

Sea-State Adaptive Cell-Averaging CFAR (CA-CFAR) Derivation Statistical Detector
$$\mu_{\text{annular}} = \frac{N_{\text{train}}\mu_{\text{train}} - N_{\text{guard}}\mu_{\text{guard}}}{N_{\text{train}} - N_{\text{guard}}} \qquad \sigma_{\text{annular}} = \sqrt{\max\left(0, \frac{N_{\text{train}}\overline{x^2}_{\text{train}} - N_{\text{guard}}\overline{x^2}_{\text{guard}}}{N_{\text{train}} - N_{\text{guard}}} - \mu_{\text{annular}}^2\right)}$$
$$T_{\text{local}}(x,y) = \mu_{\text{blended}}(x,y) + \left(k_{\text{pfa}} \cdot \kappa_{\text{sea}}\right) \cdot \sigma_{\text{blended}}(x,y)$$
$$\mu_{\text{annular}}, \sigma_{\text{annular}}$$
Local background sea clutter mean and standard deviation estimated over an annular ring between outer training stencil ($N_{\text{train}}$) and inner guard stencil ($N_{\text{guard}}$)
$$\mu_{\text{blended}}, \sigma_{\text{blended}}$$
Convex combination ($0.75 \cdot \text{local} + 0.25 \cdot \text{global}$) protecting image boundaries and sparse littoral margins
$$k_{\text{pfa}}$$
Baseline probability of false alarm multiplier ($k = 3.2$ by default)
$$\kappa_{\text{sea}}$$
Dynamic sea-state compensation factor: $\kappa = 1.00$ (calm), $\kappa = 1.10$ (moderate: $\sigma > 2.2\,\text{dB}$ or $\mu > -18\,\text{dB}$), $\kappa = 1.25$ (rough: $\sigma > 3.0\,\text{dB}$ or $\mu > -14\,\text{dB}$)
$$T_{\text{local}}$$
Adaptive detection threshold surface applied to each cell under test (CUT)
Derivation from Finn & Johnson (1968), Novak et al. (1993), and Crisp (2004). An inner guard ring (typically $3\times 3$ pixels) isolates target sidelobes and prevents vessel specular energy from contaminating background sea clutter estimates.
Signal-to-Clutter Ratio (SCR) & Vessel Contrast Formulation Target Radiometry
$$\text{SCR}\,(\text{dB}) = \sigma_{\text{peak}}^0 - \mu_{\text{annular}}$$
$$\text{SCR}$$
Target contrast in decibels above the immediate surrounding ocean clutter baseline
$$\sigma_{\text{peak}}^0$$
Maximum backscatter intensity measured within the connected component metallic hull cluster
$$\mu_{\text{annular}}$$
Surrounding sea clutter mean. Metallic hulls typically exhibit $\text{SCR} \ge 15.0\,\text{dB}$ to $28.0\,\text{dB}$, providing high classification confidence
Quantifies vessel prominence for autonomous agent confidence weighting and false-alarm suppression in high sea states.
Kinematic AIS Cross-Correlation & Dark Vessel Identification Geodesic Fusion
$$d = 2R \arcsin \sqrt{\sin^2\left(\frac{\Delta \phi}{2}\right) + \cos \phi_1 \cos \phi_2 \sin^2\left(\frac{\Delta \lambda}{2}\right)} \le R_{\text{gate}}$$
$$d$$
Great-circle geodesic distance between detected SAR target center and AIS position broadcast
$$R = 6371.0\,\text{km}$$
Mean spherical Earth radius
$$\phi, \lambda$$
Geodetic latitude and longitude in radians
$$R_{\text{gate}}$$
Kinematic gating radius: $R_{\text{gate}} = v_{\text{vessel}} \cdot \Delta t + \sigma_{\text{SAR}} + \sigma_{\text{AIS}}$
Pelich et al. (2019). AIS positions are temporally interpolated to the exact SAR zero-Doppler azimuth acquisition time $t_{\text{SAR}}$. Detections with $d > R_{\text{gate}}$ or without corresponding MMSI transponder beacons are classified as non-cooperative "dark vessels" engaged in potential illegal, unreported, or unregulated (IUU) activity.
Marangoni Capillary Damping of Hydrocarbon Oil Slicks Surface Chemistry
$$\Delta \sigma_{\text{damping}}^0 = \sigma_{\text{clean}}^0 - \sigma_{\text{slick}}^0 \approx 6\,\text{dB} \text{ to } 12\,\text{dB}$$
Alpers & Hühnerfuss (1988). Monomolecular hydrocarbon surfactant films suppress wind-generated capillary and short gravity waves (wavelength $1.5 - 10\,\text{cm}$) via the Marangoni viscoelastic effect. This eliminates Bragg scattering, producing persistent dark patches adjacent to oil-discharging vessels.

Peer-Reviewed Foundations

  • Finn, H. M., & Johnson, R. S. (1968). Adaptive detection mode with threshold control as a function of spatially sampled clutter-level estimates. RCA Review, 29(3), 414–464.
    Applied in: detect_dark_vessels (CA-CFAR algorithm)
  • Crisp, D. J. (2004). The state-of-the-art in ship detection in synthetic aperture radar imagery. Defence Science and Technology Organisation (DSTO), Research Report DSTO-RR-0272.
    Applied in: detect_dark_vessels (Sea-state roughness scaling and guard/training window geometries)
  • Novak, L. M., Owirka, G. J., & Netishen, C. M. (1993). Performance of a high-resolution polarimetric SAR automatic target recognition system. The Lincoln Laboratory Journal, 6(1), 11–24.
    Applied in: detect_dark_vessels (Signal-to-clutter ratio and target component modeling)
  • Pelich, R., Chini, M., Hostache, R., Matgen, P., López-Martínez, C., Nuevo, M., Ries, P., & Eiden, G. (2019). Large-scale automatic vessel monitoring based on dual-polarization Sentinel-1 and AIS data. Remote Sensing, 11(9), 1078.
    DOI: 10.3390/rs11091078 Applied in: detect_dark_vessels (AIS kinematic correlation)
  • Alpers, W., & Hühnerfuss, H. (1988). Radar signatures of oil films floating on the sea surface and the Marangoni effect. Journal of Geophysical Research: Oceans, 93(C4), 3642–3648.
    DOI: 10.1029/JC093iC04p03642 Applied in: detect_dark_vessels (Oil spill slick detection)

Coastal Morphodynamics & DSAS Transect Kinematics

Sub-pixel satellite shoreline extraction, perpendicular transect casting, and geostatistical erosion and accretion modeling.

Operational Tool Reference: analyze_coastal_erosion →

Sub-Pixel Satellite Waterline Extraction (CoastSat Protocol)

Monitoring beach retreat and coastal erosion requires resolving spatial movements smaller than the native satellite pixel size (e.g., $10\,\text{m}$ for Sentinel-2, $30\,\text{m}$ for Landsat). eo-mcp adopts the CoastSat methodology (Vos et al., 2019):

  1. Computation of Modified Normalized Difference Water Index (MNDWI) surface reflectance arrays.
  2. Dynamic image-specific threshold determination via Otsu between-class variance maximization.
  3. Continuous sub-pixel boundary tracing via marching squares contour interpolation, yielding horizontal shoreline positional accuracy $< 5.0\,\text{m}$.
Otsu Between-Class Variance Threshold Maximization Image Segmentation
$$\sigma_B^2(T) = \omega_0(T) \cdot \omega_1(T) \cdot \left[ \mu_0(T) - \mu_1(T) \right]^2 \qquad \Longrightarrow \qquad T^* = \arg\max_T \sigma_B^2(T)$$
$$\sigma_B^2(T)$$
Between-class variance of the image histogram bifurcated at threshold $T$
$$\omega_0(T), \omega_1(T)$$
Probabilities of pixel occurrence in background (land) and foreground (water) classes
$$\mu_0(T), \mu_1(T)$$
Mean spectral index values of respective classes
$$T^*$$
Optimal dynamic threshold maximizing statistical class separability
Otsu (1979). Eliminates operator subjectivity by adapting dynamically to instantaneous illumination, turbidity, and wave-induced surf froth.
Digital Shoreline Analysis System (DSAS) Rate Formulations Geomorphological Metrics
$$\text{EPR} = \frac{D_{\text{latest}} - D_{\text{oldest}}}{t_{\text{latest}} - t_{\text{oldest}}} \qquad\qquad \text{LRR} = \frac{\sum_{i=1}^n (t_i - \bar{t})(D_i - \bar{D})}{\sum_{i=1}^n (t_i - \bar{t})^2}$$
$$\text{NSM} = D_{\text{latest}} - D_{\text{oldest}} \qquad\qquad \text{SCE} = \max_i(D_i) - \min_i(D_i)$$
$$\text{EPR}$$
End Point Rate ($\text{m/year}$), measuring annualized change between earliest and latest available scenes
$$\text{LRR}$$
Linear Regression Rate ($\text{m/year}$), representing the slope of the least-squares regression line through all historical shoreline positions
$$\text{NSM}$$
Net Shoreline Movement ($\text{m}$), total horizontal displacement across the observation baseline
$$\text{SCE}$$
Shoreline Change Envelope ($\text{m}$), maximum distance envelope between all shoreline intersections on the transect
$$D_i$$
Cross-shore distance along the perpendicular transect from baseline to waterline at time $t_i$
USGS standard formulations (Thieler et al., 2009; Himmelstoss et al., 2018). Transects are cast perpendicular to an offshore/onshore baseline at regular along-shore intervals (typically $25 - 50\,\text{m}$).

Peer-Reviewed Foundations

  • Thieler, E. R., Himmelstoss, E. A., Zichichi, J. L., & Ergul, A. (2009). Digital Shoreline Analysis System (DSAS) version 4.0—An ArcGIS extension for calculating shoreline change. U.S. Geological Survey Open-File Report 2008-1278, 72 p.
    DOI: 10.3133/ofr20081278 Applied in: analyze_coastal_erosion
  • Vos, K., Splinter, K. D., Harley, M. D., Simmons, J. A., & Turner, I. L. (2019). CoastSat: A Google Earth Engine-enabled Python toolkit to extract shorelines from publicly available satellite imagery. Environmental Modelling & Software, 122, 104528.
    DOI: 10.1016/j.envsoft.2019.104528 Applied in: analyze_coastal_erosion

Copernicus DEM & 8-Neighbor Hydrologic Inundation

Digital elevation model geodetic specifications, local topographic gradients, and hydro-connected coastal inundation percolation modeling.

Tool: simulate_sea_level_rise → Tool: get_elevation_profile →

Copernicus DEM GLO-30 Geodetic Specifications

Topographic analysis in eo-mcp utilizes the European Space Agency's Copernicus DEM GLO-30 product (30 m / 1 arcsecond resolution), derived from TanDEM-X interferometric X-band radar. Independent global validation using airborne LiDAR and ICESat-2 spaceborne altimetry proves that Copernicus DEM achieves a vertical Linear Error at 90% confidence ($\text{LE}90 < 2.0\,\text{m}$), substantially outperforming SRTM and NASADEM across coastal zones (ESA, 2020; Guth & Geoffroy, 2021).

Local Topographic Surface Derivatives (Horn's Formulation) Terrain Analysis
$$\text{Slope } S = \arctan \sqrt{\left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} \qquad\qquad \text{Aspect } \theta = 270^{\circ} - \arctan2\left(\frac{\partial z}{\partial y}, -\frac{\partial z}{\partial x}\right)$$
$$\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}$$
East-West and North-South partial elevation gradients calculated over a 3x3 pixel kernel using distance-weighted finite differences (Horn, 1981)
$$S$$
Local topographic terrain slope (degrees or radians, $[0^{\circ}, 90^{\circ}]$)
$$\theta$$
Downslope azimuth aspect direction clockwise from geographic North ($[0^{\circ}, 360^{\circ}]$)
Hydro-Connected Flood-Fill vs. Planar Bathtub Inundation Hydraulic Modeling
$$\text{Flooded}(p) = \text{True} \iff Z(p) \le h_{\text{tide}} + \Delta_{\text{SLR}} \quad \land \quad \exists \text{ path } \mathcal{P} = (p_0, p_1, \dots, p) \text{ such that } \forall q \in \mathcal{P}, Z(q) \le h_{\text{tide}} + \Delta_{\text{SLR}}$$
$$Z(p)$$
Orthometric ground surface elevation above mean sea level (EGM2008 geoid) at pixel $p$
$$h_{\text{tide}}$$
Astronomical tide height above datum (e.g. Mean Higher High Water, MHHW)
$$\Delta_{\text{SLR}}$$
Scenario-driven sea level rise increment (IPCC AR6 projection)
$$\mathcal{P}$$
Continuous 8-connected grid percolation path originating from oceanic seed boundary cells $p_0$
Poulter & Halpin (2008); Gesch (2009, 2018). Standard naive "bathtub" models assume any pixel with $Z(p) \le h$ will drown, generating severe false-positive flooding in low-lying agricultural polders, coastal basins, and dry interior depressions protected by natural berms, dikes, or elevated roadbeds. eo-mcp requires verified 8-connected topological percolation before flagging inundation.

IPCC AR6 Sea Level Rise Scenario Projections

IPCC AR6 ScenarioSocioeconomic NarrativeMedian SLR by 2050 (m)Median SLR by 2100 (m)Likely Range by 2100 (5th–95th %)
SSP1-2.6Aggressive global decarbonization, net-zero greenhouse gases by 2050+0.19 m+0.44 m0.32 m – 0.61 m
SSP2-4.5Intermediate emissions pathway, current national policy trajectories+0.23 m+0.56 m0.41 m – 0.76 m
SSP5-8.5High-fossil development without additional climate mitigation+0.30 m+0.77 m0.63 m – 1.01 m

Peer-Reviewed Foundations

  • Poulter, B., & Halpin, P. N. (2008). Raster modelling of coastal flooding from sea-level rise. International Journal of Geographical Information Science, 22(2), 167–182.
    DOI: 10.1080/13658810701371858 Applied in: simulate_sea_level_rise
  • Gesch, D. B. (2018). Best practices for elevation-based assessments of sea-level rise and coastal flooding exposure. Frontiers in Earth Science, 6, 230.
    DOI: 10.3389/feart.2018.00230 Applied in: simulate_sea_level_rise
  • Fox-Kemper, B. et al. (2021). Ocean, cryosphere and sea level change. In Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change, pp. 1211–1362. Cambridge University Press.
    DOI: 10.1017/9781009157896.011 Applied in: simulate_sea_level_rise

Multi-Decadal Surface Water Dynamics & Reservoir Drought

Long-term hydrological transition analysis, JRC global water recurrence dynamics, and hypsometric reservoir volume modeling.

Operational Tool Reference: analyze_reservoir_drought →

EC JRC Global Surface Water (GSW) Multi-Decadal Physics

Quantifying hydrological drought and reservoir depletion requires isolating seasonal water fluctuations from multi-decadal climate trends. eo-mcp builds upon the European Commission Joint Research Centre (JRC) Global Surface Water methodology developed by Pekel et al. (2016, Nature), analyzing over 3 million Landsat scenes spanning 38 years (1984–present).

Multi-Decadal Water Occurrence ($WO$) Formulation Hydrological Persistence
$$WO = \frac{\sum_{t=1}^{T} W_{\text{water}}(t)}{\sum_{t=1}^{T} V_{\text{valid}}(t)} \times 100\%$$
$$WO$$
Water Occurrence frequency percentage over the entire observation window ($[0\%, 100\%]$)
$$W_{\text{water}}(t)$$
Binary water detection flag (1 if classified as water via MNDWI and multispectral rules, 0 otherwise) at time $t$
$$V_{\text{valid}}(t)$$
Binary observation validity flag (1 if unobstructed by clouds, snow, or sensor anomalies, 0 otherwise) at time $t$
Pekel et al. (2016). Based on $WO$ thresholds, pixels are partitioned into hydrological recurrence classes: Permanent Water ($WO \ge 80\%$), Seasonal Water ($20\% \le WO < 80\%$), and Ephemeral / Occasional Water ($0\% < WO < 20\%$). Loss of permanent water over consecutive years serves as an unambiguous remote sensing indicator of severe regional meteorological drought.
Hypsometric Reservoir Volume Estimation Volumetric Integration
$$V(h) = \int_{z_{\min}}^{h} A(z) \, dz \approx \sum_{k=1}^{M-1} \frac{A(z_k) + A(z_{k+1})}{2} \cdot (z_{k+1} - z_k)$$
$$V(h)$$
Estimated water volume ($\text{m}^3$) stored in the reservoir at surface water elevation $h$
$$A(z)$$
Reservoir surface area ($\text{m}^2$) evaluated at contour elevation $z$ using the Copernicus DEM
$$z_{\min}$$
Lowest bathymetric ground elevation in the reservoir basin
Trapezoidal hypsometric integration relating satellite-observed surface area expansion/contraction to volumetric water storage capacity.

Peer-Reviewed Foundations

  • Pekel, J.-F., Cottam, A., Gorelick, N., & Belward, A. S. (2016). High-resolution mapping of global surface water and its long-term changes. Nature, 540(7633), 418–422.
    DOI: 10.1038/nature20584 Applied in: analyze_reservoir_drought

Radiative Transfer LST & Urban Heat Island

Single-channel atmospheric radiative transfer inversion, Planck blackbody derivation, and Fractional Vegetation Cover emissivity estimation.

Operational Tool Reference: analyze_urban_heat_island →

Thermal Infrared Radiative Transfer Physics

Retrieving kinetic Land Surface Temperature ($T_s$) from satellite thermal sensors (such as Landsat 8/9 TIRS Band 10, $10.60 - 11.19\,\mu\text{m}$) requires solving the thermal infrared Radiative Transfer Equation (RTE). At thermal infrared wavelengths, sensor-observed radiance consists of three components:

  1. Surface thermal radiance attenuated by atmospheric transmittance $\tau$.
  2. Reflected downwelling atmospheric thermal irradiance $(1 - \epsilon) L^{\downarrow}$ transmitted to sensor.
  3. Thermal upwelling atmospheric path radiance $L^{\uparrow}$ emitted directly toward the sensor aperture.
Thermal Radiative Transfer Equation (RTE) & Radiance Inversion Single-Channel RTE
$$L_{\text{sensor}} = \left[ \epsilon \cdot B(T_s) + (1 - \epsilon) L^{\downarrow} \right] \tau + L^{\uparrow} \quad \Longrightarrow \quad B(T_s) = \frac{L_{\text{sensor}} - L^{\uparrow} - \tau (1 - \epsilon) L^{\downarrow}}{\tau \cdot \epsilon}$$
$$L_{\text{sensor}}$$
At-sensor calibrated Top-of-Atmosphere spectral radiance ($\text{W}\cdot\text{m}^{-2}\cdot\text{sr}^{-1}\cdot\mu\text{m}^{-1}$)
$$B(T_s)$$
Planck blackbody spectral radiance emitted by the surface at kinetic temperature $T_s$
$$\epsilon$$
Land Surface Emissivity (dimensionless, $[0.95, 0.99]$)
$$\tau$$
Atmospheric spectral transmittance in the thermal infrared band
$$L^{\uparrow}, L^{\downarrow}$$
Upwelling atmospheric path radiance and downwelling hemispherical sky irradiance
Sobrino, Jiménez-Muñoz, & Paolini (2004); Jiménez-Muñoz et al. (2009). Atmospheric parameters ($\tau, L^{\uparrow}, L^{\downarrow}$) are derived from atmospheric profile soundings or MODTRAN simulations.
Planck Function Inversion for Land Surface Temperature ($T_s$) Thermodynamic Inversion
$$T_s = \frac{K_2}{\ln\left( \frac{K_1}{B(T_s)} + 1 \right)} \qquad\qquad T_{\text{Celsius}} = T_s - 273.15$$
$$T_s$$
Kinetic Land Surface Temperature in Kelvin
$$K_1, K_2$$
Thermal calibration constants for Landsat 8/9 TIRS Band 10: $K_1 = 774.8853\,\text{W}/(\text{m}^2\cdot\text{sr}\cdot\mu\text{m})$, $K_2 = 1321.0789\,\text{K}$
Fractional Vegetation Cover (FVC) & Sobrino Emissivity Estimation Surface Physics
$$\text{FVC} = \left( \frac{\text{NDVI} - \text{NDVI}_{\text{soil}}}{\text{NDVI}_{\text{veg}} - \text{NDVI}_{\text{soil}}} \right)^2 \qquad\qquad \epsilon = \begin{cases} \epsilon_{\text{soil}} & \text{if } \text{NDVI} < 0.2 \\ \epsilon_{\text{veg}} \cdot \text{FVC} + \epsilon_{\text{soil}} \cdot (1 - \text{FVC}) + d\epsilon & \text{if } 0.2 \le \text{NDVI} \le 0.5 \\ \epsilon_{\text{veg}} + d\epsilon & \text{if } \text{NDVI} > 0.5 \end{cases}$$
$$\text{FVC}$$
Fractional Vegetation Cover (Valor & Caselles, 1996). Standard thresholds: $\text{NDVI}_{\text{soil}} = 0.2$, $\text{NDVI}_{\text{veg}} = 0.5$
$$\epsilon_{\text{soil}}, \epsilon_{\text{veg}}$$
Soil emissivity ($\approx 0.960$) and healthy dense canopy emissivity ($\approx 0.985$)
$$d\epsilon$$
Internal cavity geometric term: $d\epsilon \approx (1 - \epsilon_{\text{soil}}) \epsilon_{\text{veg}} F'(1 - \text{FVC})$ where $F' \approx 0.55$
Sobrino et al. (2004, 2008). Accurately captures high thermal emissivity over vegetative green spaces and low emissivity over urban impervious concrete, enabling accurate Urban Heat Island (UHI) temperature anomaly quantification.

Peer-Reviewed Foundations

  • Sobrino, J. A., Jiménez-Muñoz, J. C., & Paolini, L. (2004). Land surface temperature retrieval from LANDSAT TM 5. Remote Sensing of Environment, 90(4), 434–440.
    DOI: 10.1016/j.rse.2004.02.003 Applied in: analyze_urban_heat_island
  • Valor, E., & Caselles, V. (1996). Mapping land surface emissivity from NDVI: Application to European, African, and South American areas. Remote Sensing of Environment, 57(3), 167–184.
    DOI: 10.1016/0034-4257(96)00039-9 Applied in: analyze_urban_heat_island
  • Jiménez-Muñoz, J. C., Cristóbal, J., Sobrino, J. A., Sòria, G., Ninyerola, M., & Pons, X. (2009). Revision of the single-channel algorithm for land surface temperature retrieval from Landsat thermal-infrared data. IEEE Transactions on Geoscience and Remote Sensing, 47(1), 339–349.
    DOI: 10.1109/TGRS.2008.2007125 Applied in: analyze_urban_heat_island

Active Wildfires (FRP) & Post-Fire Burn Severity

Mid-infrared sub-pixel thermal anomaly physics, Fire Radiative Power biomass combustion rates, and relativized burn severity metrics.

Tool: detect_active_wildfires → Tool: calculate_burn_severity →

Active Fire Physics: Wien's Displacement Law & Mid-Infrared Sensitivity

According to Wien's Displacement Law ($\lambda_{\max} \cdot T = 2898\,\mu\text{m}\cdot\text{K}$), the peak emission of ambient Earth surfaces ($\approx 300\,\text{K}$) occurs in the Thermal Infrared (TIR) at $\approx 9.7\,\mu\text{m}$. In contrast, actively flaming biomass combustion ($800 - 1200\,\text{K}$) shifts the blackbody emission curve dramatically toward the Mid-Infrared (MIR, $3.7 - 4.0\,\mu\text{m}$).

By observing in the $3.9\,\mu\text{m}$ atmospheric window (VIIRS Band I4, $375\,\text{m}$ GSD), a sub-pixel fire occupying as little as $0.01\%$ of a pixel increases observed MIR radiance by several hundred percent, while barely affecting the $11\,\mu\text{m}$ TIR channel (VIIRS Band I5).

Contextual Brightness Temperature Anomaly ($\Delta T$) Thermal Anomaly
$$\Delta T = T_{3.9} - T_{11} \qquad \Longrightarrow \qquad \text{Active Fire} \iff T_{3.9} > T_{3.9,\text{bg}} + n \cdot \delta_{3.9} \quad \land \quad \Delta T > \Delta T_{\text{bg}} + n \cdot \delta_{\Delta T}$$
$$T_{3.9}, T_{11}$$
Brightness temperatures (Kelvin) inverted from calibrated radiances in Mid-Infrared ($3.9\,\mu\text{m}$) and Thermal-Infrared ($11\,\mu\text{m}$)
$$T_{3.9,\text{bg}}, \Delta T_{\text{bg}}$$
Mean background brightness temperature and temperature difference calculated over surrounding non-fire cloud-free pixels
$$\delta$$
Standard deviation of background context; $n$ is threshold multiplier ($n \ge 3.0$)
VIIRS 375 m active fire detection algorithm (Schroeder et al., 2014; Giglio et al., 2016). Contextual adaptive statistics suppress false alarms over sunlit bare rock and cloud edges.
Fire Radiative Power (FRP) & Biomass Combustion Rate Energetic Formulation
$$\text{FRP} = \frac{A_{\text{pixel}} \cdot \sigma}{a \cdot \tau_{\text{MIR}}} \left( L_{\text{MIR}} - L_{\text{MIR,bg}} \right) \qquad\qquad \frac{dM}{dt} = C_{\text{biomass}} \cdot \text{FRP}$$
$$\text{FRP}$$
Fire Radiative Power ($\text{MW}$), instantaneous rate of radiative heat energy released by burning biomass
$$A_{\text{pixel}}$$
Ground pixel footprint area ($375 \times 375\,\text{m}^2 \approx 140,625\,\text{m}^2$ for VIIRS at nadir)
$$\sigma = 5.6704 \times 10^{-8}$$
Stefan-Boltzmann constant ($\text{W}\cdot\text{m}^{-2}\cdot\text{K}^{-4}$)
$$a$$
Sensor-specific empirical MIR radiance fitting coefficient ($a \approx 3.0 \times 10^{-9}\,\text{W}\cdot\text{m}^{-2}\cdot\text{sr}^{-1}\cdot\mu\text{m}^{-1}\cdot\text{K}^{-4}$)
$$\frac{dM}{dt}$$
Fuel biomass consumption rate ($\text{kg/s}$)
$$C_{\text{biomass}}$$
Combustion factor ($C_{\text{biomass}} = 0.368 \pm 0.015\,\text{kg/MJ}$)
Wooster et al. (2003, 2005). Proves direct physical proportionality between Fire Radiative Energy (FRE) and total combusted fuel mass, providing smoke and carbon emission inputs for climate models.
Differenced NBR ($dNBR$) & Relativized Burn Ratio ($RBR$) Burn Severity
$$dNBR = (NBR_{\text{pre}} - NBR_{\text{post}}) \times 1000 \qquad\qquad RBR = \frac{dNBR}{NBR_{\text{pre}} + 1.001}$$
$$NBR_{\text{pre}}, NBR_{\text{post}}$$
Pre-fire and post-fire Normalized Burn Ratio arrays ($(\rho_{\text{NIR}} - \rho_{\text{SWIR2}}) / (\rho_{\text{NIR}} + \rho_{\text{SWIR2}})$)
$$dNBR$$
Difference Normalized Burn Ratio (Key & Benson, 2006)
$$RBR$$
Relativized Burn Ratio (Parks et al., 2014), adding $1.001$ to denominator to avoid division by zero while normalizing for low pre-fire vegetative cover

USGS & European Forest Fire Information System (EFFIS) Thresholds

Severity LeveldNBR Index RangeRBR Index RangeEcological Landscape Interpretation
Enhanced Regrowth$< -100$$< -50$Vigorous post-fire vegetative sprouting or herbaceous flush
Unburned$-100 \text{ to } +99$$-50 \text{ to } +99$Intact green canopy, unburned duff and surface litter
Low Severity$+100 \text{ to } +269$$+100 \text{ to } +174$Surface fire, scorched understory, minor tree crown needle scorch
Moderate-Low$+270 \text{ to } +439$$+175 \text{ to } +269$Understory consumed, 25–50% canopy foliage scorched
Moderate-High$+440 \text{ to } +659$$+270 \text{ to } +399$Deep charring, 50–80% canopy mortality, soil organic layer consumed
High Severity$\ge +660$$\ge +400$Complete overstory canopy mortality ($>80\%$), deep white ash, mineral soil alteration

Peer-Reviewed Foundations

  • Schroeder, W., Oliva, P., Giglio, L., & Csiszar, I. A. (2014). The New VIIRS 375 m active fire detection data product: Algorithm description and initial assessment. Remote Sensing of Environment, 143, 85–96.
    DOI: 10.1016/j.rse.2013.12.008 Applied in: detect_active_wildfires
  • Wooster, M. J., Roberts, G., Perry, G. L. W., & Kaufman, Y. J. (2005). Retrieval of biomass combustion rates and totals from fire radiative power observations: FRP derivation and calibration relationships. Journal of Geophysical Research: Atmospheres, 110(D24), D24311.
    DOI: 10.1029/2005JD006318 Applied in: detect_active_wildfires (FRP biomass rate)
  • Parks, S. A., Dillon, G. K., & Miller, C. (2014). A new metric for quantifying burn severity: The Relativized Burn Ratio. Remote Sensing, 6(3), 1827–1844.
    DOI: 10.3390/rs6031827 Applied in: calculate_burn_severity (RBR metric)

Atmospheric Chemistry & TROPOMI DOAS Spectroscopy

Pushbroom grating spectrometer physics, Differential Optical Absorption Spectroscopy, Air Mass Factor conversions, and trace gas retrievals.

Operational Tool Reference: monitor_atmospheric_emissions →

Differential Optical Absorption Spectroscopy (DOAS) Physics

The TROPOspheric Monitoring Instrument (TROPOMI) aboard Copernicus Sentinel-5 Precursor (S5P) is an advanced pushbroom grating imaging spectrometer measuring backscattered solar radiation across Ultraviolet (UV, $270 - 495\,\text{nm}$), Visible (VIS, $405 - 500\,\text{nm}$), Near-Infrared (NIR, $675 - 775\,\text{nm}$), and Shortwave-Infrared (SWIR, $2305 - 2385\,\text{nm}$) at $3.5 \times 5.5\,\text{km}^2$ nadir resolution (Veefkind et al., 2012).

Retrieval of trace gas atmospheric columns relies on the DOAS principle, separating narrow molecular absorption bands from broadband Rayleigh and aerosol scattering:

DOAS Spectral Fitting Equation (Beer-Lambert Inversion) Spectroscopic Law
$$\ln\left( \frac{I_0(\lambda)}{I(\lambda)} \right) = \sum_{i} \sigma_i(\lambda) \cdot S_i + P(\lambda) + \alpha_{\text{Rayleigh}}(\lambda) + \alpha_{\text{Mie}}(\lambda)$$
$$I_0(\lambda), I(\lambda)$$
Solar extraterrestrial reference irradiance and earthshine radiance measured at top of atmosphere
$$\sigma_i(\lambda)$$
Laboratory-measured molecular absorption cross-section of trace gas species $i$ ($\text{cm}^2/\text{molecule}$)
$$S_i$$
Slant Column Density (SCD) integrated along the effective atmospheric optical photon path ($\text{mol}/\text{m}^2$ or $\text{molec}/\text{cm}^2$)
$$P(\lambda)$$
Low-order polynomial accounting for smooth broadband surface reflectance and aerosol extinction
DOAS spectral fitting separates high-frequency ro-vibrational absorption fingerprints from continuum scattering.
Air Mass Factor (AMF) & Stratosphere-Troposphere Separation Vertical Column Derivation
$$\text{VCD}_{\text{total}} = \frac{\text{SCD}}{\text{AMF}} \qquad\qquad \text{VCD}_{\text{trop}} = \text{VCD}_{\text{total}} - \text{VCD}_{\text{strat}}$$
$$\text{VCD}$$
Vertical Column Density ($\text{mol}/\text{m}^2$ or $\mu\text{mol}/\text{m}^2$), representing the vertical integral of gas concentration from surface to top of atmosphere
$$\text{AMF}$$
Air Mass Factor computed from radiative transfer models (e.g. DAK), integrating viewing geometry, cloud fraction, surface albedo, and a priori vertical gas profile shapes
$$\text{VCD}_{\text{trop}}$$
Tropospheric Vertical Column Density, isolating boundary-layer anthropogenic pollution from natural stratospheric background reservoirs
van Geffen et al. (2020). Chemical transport models (such as TM5-MP) assimilate meteorological fields and high-altitude soundings to subtract the stratospheric ozone and $\text{NO}_2$ column, delivering boundary layer urban emissions.

Target Atmospheric Species & Retrieval Windows

Trace Gas MoleculeFitting Spectral BandTypical Tropospheric ColumnAtmospheric LifetimeAnthropogenic Emission Sources
Nitrogen Dioxide ($\text{NO}_2$)VIS ($405 - 465\,\text{nm}$)$10 - 250\,\mu\text{mol}/\text{m}^2$2 – 8 hours (boundary layer)Thermal power generation, vehicle internal combustion engines, industrial boilers
Carbon Monoxide ($\text{CO}$)SWIR ($2305 - 2385\,\text{nm}$)$1.5 - 4.5\,\text{mmol}/\text{m}^2$1 – 2 monthsIncomplete fossil fuel combustion, large-scale tropical biomass wildfires
Methane ($\text{CH}_4$)SWIR ($2305 - 2385\,\text{nm}$)$1800 - 1950\,\text{ppb}$ mixing ratio9 – 12 yearsOil & gas pipeline fugitive leaks, agricultural ruminants, landfill outgassing
Sulfur Dioxide ($\text{SO}_2$)UV ($312 - 390\,\text{nm}$)$< 1.0\,\text{DU}$ (background) to $> 50\,\text{DU}$1 – 2 daysCoal-fired power plants, metal smelters, volcanic explosive eruptions

Peer-Reviewed Foundations

  • Veefkind, J. P., Aben, I., McMullan, K., Förster, H., de Vries, J., Otter, G., Claas, J., Eskes, H. J., de Haan, J. F., Kleipool, Q., van Weele, M., Hasekamp, O., Hoogeveen, R., Landgraf, J., Snel, R., Tol, P., Ingmann, P., Voors, R., Kruizinga, B., Vink, R., Visser, H., & Levelt, P. F. (2012). TROPOMI on the ESA Sentinel-5 Precursor: A GMES mission for global observations of the atmospheric composition for climate, air quality and ozone layer applications. Remote Sensing of Environment, 120, 70–83.
    DOI: 10.1016/j.rse.2011.09.027 Applied in: monitor_atmospheric_emissions
  • van Geffen, J., Boersma, K. F., Eskes, H., Sneep, M., ter Linden, M., Zara, M., & Veefkind, J. P. (2020). S5P TROPOMI NO2 slant column retrieval: Method, stability, uncertainties and comparisons with OMI. Atmospheric Measurement Techniques, 13(3), 1315–1335.
    DOI: 10.5194/amt-13-1315-2020 Applied in: monitor_atmospheric_emissions

Verified Peer-Reviewed Scientific Bibliography

Authoritative foundational literature, geodetic standards, and algorithmic citations underpinning the eo-mcp planetary protocol.

  • Alpers, W., & Hühnerfuss, H. (1988). Radar signatures of oil films floating on the sea surface and the Marangoni effect. Journal of Geophysical Research: Oceans, 93(C4), 3642–3648.
    DOI: 10.1029/JC093iC04p03642
  • Bioresita, F., Puissant, A., Stumpf, A., & Malet, J.-P. (2018). A method for automatic and rapid mapping of water surfaces from Sentinel-1 imagery. Remote Sensing, 10(2), 217.
    DOI: 10.3390/rs10020217
  • Crisp, D. J. (2004). The state-of-the-art in ship detection in synthetic aperture radar imagery. Defence Science and Technology Organisation (DSTO), Research Report DSTO-RR-0272.
    DTIC: ADA426451
  • European Space Agency. (2020). Copernicus Complex Digital Elevation Model (COP-DEM) Validation Report. Issue 4.0, Airbus Defence and Space & ESA.
    Official ESA Report
  • Finn, H. M., & Johnson, R. S. (1968). Adaptive detection mode with threshold control as a function of spatially sampled clutter-level estimates. RCA Review, 29(3), 414–464.
    Foundational CA-CFAR Radar Derivation
  • Fox-Kemper, B. et al. (2021). Ocean, cryosphere and sea level change. In Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change, pp. 1211–1362. Cambridge University Press.
    DOI: 10.1017/9781009157896.011
  • Gao, B.-C. (1996). NDWI—A normalized difference water index for remote sensing of vegetation liquid water from space. Remote Sensing of Environment, 58(3), 257–266.
    DOI: 10.1016/S0034-4257(96)00067-3
  • Gesch, D. B. (2018). Best practices for elevation-based assessments of sea-level rise and coastal flooding exposure. Frontiers in Earth Science, 6, 230.
    DOI: 10.3389/feart.2018.00230
  • Giglio, L., Schroeder, W., & Justice, C. O. (2016). The collection 6 MODIS active fire detection algorithm and fire products. Remote Sensing of Environment, 178, 31–41.
    DOI: 10.1016/j.rse.2016.02.054
  • Guth, P. L., & Geoffroy, T. M. (2021). LiDAR point cloud and ICESat-2 evaluation of 1 second global digital elevation models: Copernicus wins. Transactions in GIS, 25(5), 2245–2261.
    DOI: 10.1111/tgis.12825
  • Himmelstoss, E. A., Henderson, R. E., Kratzmann, M. G., & Farris, A. S. (2018). Digital Shoreline Analysis System (DSAS) version 5.0 user guide. U.S. Geological Survey Open-File Report 2018-1179, 110 p.
    DOI: 10.3133/ofr20181179
  • Horn, B. K. P. (1981). Hill shading and the reflectance map. Proceedings of the IEEE, 69(1), 14–47.
    DOI: 10.1109/PROC.1981.11918
  • Huete, A. et al. (2002). Overview of the radiometric and biophysical performance of the MODIS vegetation indices. Remote Sensing of Environment, 83(1–2), 195–213.
    DOI: 10.1016/S0034-4257(02)00096-2
  • Jiménez-Muñoz, J. C., Cristóbal, J., Sobrino, J. A., Sòria, G., Ninyerola, M., & Pons, X. (2009). Revision of the single-channel algorithm for land surface temperature retrieval from Landsat thermal-infrared data. IEEE Transactions on Geoscience and Remote Sensing, 47(1), 339–349.
    DOI: 10.1109/TGRS.2008.2007125
  • Jönsson, P., & Eklundh, L. (2004). TIMESAT—A program for analyzing time-series of satellite sensor data. Computers & Geosciences, 30(8), 833–845.
    DOI: 10.1016/j.cageo.2004.05.006
  • Key, C. H., & Benson, N. C. (2006). Landscape Assessment (LA): Sampling and analysis methods. FIREMON: Fire Effects Monitoring and Inventory System, USDA Forest Service GTR-RMRS-164-CD, pp. LA 1–55.
    USDA Treesearch: 24503
  • McFeeters, S. K. (1996). The use of the Normalized Difference Water Index (NDWI) in the delineation of open water features. International Journal of Remote Sensing, 17(7), 1425–1432.
    DOI: 10.1080/01431169608948714
  • Otsu, N. (1979). A threshold selection method from gray-level histograms. IEEE Transactions on Systems, Man, and Cybernetics, 9(1), 62–66.
    DOI: 10.1109/TSMC.1979.4310076
  • Parks, S. A., Dillon, G. K., & Miller, C. (2014). A new metric for quantifying burn severity: The Relativized Burn Ratio. Remote Sensing, 6(3), 1827–1844.
    DOI: 10.3390/rs6031827
  • Pekel, J.-F., Cottam, A., Gorelick, N., & Belward, A. S. (2016). High-resolution mapping of global surface water and its long-term changes. Nature, 540(7633), 418–422.
    DOI: 10.1038/nature20584
  • Pelich, R. et al. (2019). Large-scale automatic vessel monitoring based on dual-polarization Sentinel-1 and AIS data. Remote Sensing, 11(9), 1078.
    DOI: 10.3390/rs11091078
  • Poulter, B., & Halpin, P. N. (2008). Raster modelling of coastal flooding from sea-level rise. International Journal of Geographical Information Science, 22(2), 167–182.
    DOI: 10.1080/13658810701371858
  • Rouse, J. W., Haas, R. H., Schell, J. A., & Deering, D. W. (1974). Monitoring vegetation systems in the Great Plains with ERTS. Third Earth Resources Technology Satellite-1 Symposium, NASA SP-351, 1, 309–317.
    NASA Accession No. N74-30724 (Foundational NDVI paper)
  • Schroeder, W., Oliva, P., Giglio, L., & Csiszar, I. A. (2014). The New VIIRS 375 m active fire detection data product: Algorithm description and initial assessment. Remote Sensing of Environment, 143, 85–96.
    DOI: 10.1016/j.rse.2013.12.008
  • Sobrino, J. A., Jiménez-Muñoz, J. C., & Paolini, L. (2004). Land surface temperature retrieval from LANDSAT TM 5. Remote Sensing of Environment, 90(4), 434–440.
    DOI: 10.1016/j.rse.2004.02.003
  • Thieler, E. R., Himmelstoss, E. A., Zichichi, J. L., & Ergul, A. (2009). Digital Shoreline Analysis System (DSAS) version 4.0—An ArcGIS extension for calculating shoreline change. U.S. Geological Survey Open-File Report 2008-1278, 72 p.
    DOI: 10.3133/ofr20081278
  • Tucker, C. J. (1979). Red and photographic infrared linear combinations for monitoring vegetation. Remote Sensing of Environment, 8(2), 127–150.
    DOI: 10.1016/0034-4257(79)90013-0
  • Twele, A., Cao, W., Plank, S., & Martinis, S. (2016). Sentinel-1-based flood mapping: A fully automated processing chain. International Journal of Remote Sensing, 37(13), 2990–3004.
    DOI: 10.1080/01431161.2016.1192304
  • Valor, E., & Caselles, V. (1996). Mapping land surface emissivity from NDVI: Application to European, African, and South American areas. Remote Sensing of Environment, 57(3), 167–184.
    DOI: 10.1016/0034-4257(96)00039-9
  • van Geffen, J. et al. (2020). S5P TROPOMI NO2 slant column retrieval: Method, stability, uncertainties and comparisons with OMI. Atmospheric Measurement Techniques, 13(3), 1315–1335.
    DOI: 10.5194/amt-13-1315-2020
  • Veefkind, J. P. et al. (2012). TROPOMI on the ESA Sentinel-5 Precursor: A GMES mission for global observations of the atmospheric composition for climate, air quality and ozone layer applications. Remote Sensing of Environment, 120, 70–83.
    DOI: 10.1016/j.rse.2011.09.027
  • Vos, K., Splinter, K. D., Harley, M. D., Simmons, J. A., & Turner, I. L. (2019). CoastSat: A Google Earth Engine-enabled Python toolkit to extract shorelines from publicly available satellite imagery. Environmental Modelling & Software, 122, 104528.
    DOI: 10.1016/j.envsoft.2019.104528
  • Wooster, M. J., Roberts, G., Perry, G. L. W., & Kaufman, Y. J. (2005). Retrieval of biomass combustion rates and totals from fire radiative power observations: FRP derivation and calibration relationships. Journal of Geophysical Research: Atmospheres, 110(D24), D24311.
    DOI: 10.1029/2005JD006318
  • Xu, H. (2006). Modification of normalised difference water index (NDWI) to enhance open water features in remotely sensed imagery. International Journal of Remote Sensing, 27(14), 3025–3033.
    DOI: 10.1080/01431160600589179
  • Zhang, X. et al. (2003). Monitoring vegetation phenology using MODIS. Remote Sensing of Environment, 84(3), 471–475.
    DOI: 10.1016/S0034-4257(02)00135-9
BibTeX Reference Database (docs/methodology/references.bib) 
@article{tucker1979red,
  author  = {Tucker, Compton J.},
  title   = {Red and photographic infrared linear combinations for monitoring vegetation},
  journal = {Remote Sensing of Environment},
  volume  = {8},
  number  = {2},
  pages   = {127--150},
  year    = {1979},
  doi     = {10.1016/0034-4257(79)90013-0}
}

@article{pekel2016high,
  author  = {Pekel, Jean-Fran{\c{c}}ois and Cottam, Andrew and Gorelick, Noel and Belward, Alan S.},
  title   = {High-resolution mapping of global surface water and its long-term changes},
  journal = {Nature},
  volume  = {540},
  number  = {7633},
  pages   = {418--422},
  year    = {2016},
  doi     = {10.1038/nature20584}
}

@article{sobrino2004land,
  author  = {Sobrino, Jos{\'e} A. and Jim{\'e}nez-Mu{\~n}oz, Juan Carlos and Paolini, Leonardo},
  title   = {Land surface temperature retrieval from LANDSAT TM 5},
  journal = {Remote Sensing of Environment},
  volume  = {90},
  number  = {4},
  pages   = {434--440},
  year    = {2004},
  doi     = {10.1016/j.rse.2004.02.003}
}

@article{pelich2019large,
  author  = {Pelich, Ramona and Chini, Marco and Hostache, Renaud and Matgen, Patrick and L{\'o}pez-Mart{\'i}nez, Carlos and Nuevo, Miguel and Ries, Philippe and Eiden, Gaston},
  title   = {Large-Scale Automatic Vessel Monitoring Based on Dual-Polarization Sentinel-1 and AIS Data},
  journal = {Remote Sensing},
  volume  = {11},
  number  = {9},
  pages   = {1078},
  year    = {2019},
  doi     = {10.3390/rs11091078}
}

@techreport{crisp2004state,
  author      = {Crisp, David J.},
  title       = {The State-of-the-Art in Ship Detection in Synthetic Aperture Radar Imagery},
  institution = {Defence Science and Technology Organisation (DSTO)},
  number      = {DSTO-RR-0272},
  year        = {2004}
}