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Mathematical Foundations

This document establishes the mathematical foundations for Earth observation machine learning.

Coordinate Reference Systems

Geodetic Datum

The World Geodetic System 1984 (WGS84) defines the reference ellipsoid:

$$\frac{x^2 + y^2}{a^2} + \frac{z^2}{b^2} = 1$$

where a = 6378137 m (semi-major axis) and b = 6356752.314 m (semi-minor axis).

Reference: National Imagery and Mapping Agency (2000). Department of Defense World Geodetic System 1984. NIMA TR8350.2.

Universal Transverse Mercator (UTM)

UTM divides Earth into 60 zones, each 6 degrees wide:

$$\text{zone} = \left\lfloor \frac{\lambda + 180}{6} \right\rfloor + 1$$

Reference: Snyder, J.P. (1987). Map Projections: A Working Manual. USGS Professional Paper 1395. DOI: 10.3133/pp1395

Spectral Indices

NDVI (Normalized Difference Vegetation Index)

$$\text{NDVI} = \frac{\rho_{NIR} - \rho_{Red}}{\rho_{NIR} + \rho_{Red}}$$

NDVI ranges from -1 to +1, with healthy vegetation typically 0.2-0.9.

Reference: 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

EVI (Enhanced Vegetation Index)

$$\text{EVI} = G \cdot \frac{\rho_{NIR} - \rho_{Red}}{\rho_{NIR} + C_1 \cdot \rho_{Red} - C_2 \cdot \rho_{Blue} + L}$$

with G = 2.5, C₁ = 6, C₂ = 7.5, and L = 1.

Reference: 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

NDWI (Normalized Difference Water Index)

$$\text{NDWI} = \frac{\rho_{Green} - \rho_{NIR}}{\rho_{Green} + \rho_{NIR}}$$

Reference: McFeeters, S.K. (1996). The use of the Normalized Difference Water Index (NDWI). International Journal of Remote Sensing, 17(7), 1425-1432. DOI: 10.1080/01431169608948714

Spatial Statistics

Moran's I

Global measure of spatial autocorrelation:

$$I = \frac{n}{\sum_i \sum_j w_{ij}} \cdot \frac{\sum_i \sum_j w_{ij}(x_i - \bar{x})(x_j - \bar{x})}{\sum_i (x_i - \bar{x})^2}$$
Value Interpretation
I > 0 Positive autocorrelation (clustering)
I ≈ 0 Random pattern
I < 0 Negative autocorrelation (dispersion)

Reference: Moran, P.A.P. (1950). Notes on Continuous Stochastic Phenomena. Biometrika, 37(1/2), 17-23. DOI: 10.2307/2332142

Semivariogram

$$\gamma(h) = \frac{1}{2|N(h)|} \sum_{(i,j) \in N(h)} (z_i - z_j)^2$$

Reference: Matheron, G. (1963). Principles of geostatistics. Economic Geology, 58(8), 1246-1266. DOI: 10.2113/gsecongeo.58.8.1246

Deep Learning Foundations

Convolution

$$(I * K)[i,j] = \sum_m \sum_n I[i+m, j+n] \cdot K[m, n]$$

Reference: LeCun, Y., Bengio, Y., & Hinton, G. (2015). Deep learning. Nature, 521(7553), 436-444. DOI: 10.1038/nature14539

Attention Mechanism

$$\text{Attention}(Q, K, V) = \text{softmax}\left(\frac{QK^T}{\sqrt{d_k}}\right)V$$

Reference: Vaswani, A., et al. (2017). Attention Is All You Need. NeurIPS, 30. arXiv:1706.03762

Loss Functions

Cross-Entropy Loss

$$\mathcal{L}_{CE} = -\sum_{c=1}^{C} y_c \log(\hat{y}_c)$$

Dice Loss

$$\mathcal{L}_{Dice} = 1 - \frac{2 \sum_i p_i g_i}{\sum_i p_i + \sum_i g_i}$$

Reference: Milletari, F., et al. (2016). V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation. 3DV. DOI: 10.1109/3DV.2016.79

See Also