modelling Level 4

Hyperspectral Imaging

What minerals compose this rock? What chemicals are in these plants? Hyperspectral imaging captures hundreds of narrow spectral bands, enabling material identification through spectral signatures. This model derives absorption feature analysis, implements spectral unmixing algorithms, and shows how to map minerals, vegetation biochemistry, and water quality from hyperspectral data.

Prerequisites: spectroscopy, spectral unmixing, dimensionality reduction, absorption features

27 Feb 2026 Updated 12 Mar 2026 13 min read

1. The Question

Can we identify minerals from orbit using reflected sunlight?

Hyperspectral imaging captures continuous spectra across hundreds of narrow bands.

Comparison:

Multispectral (Landsat, Sentinel-2):

8-12 broad bands
Band width: 50-200 nm
Coarse spectral sampling

Hyperspectral (AVIRIS, EnMAP, PRISMA):

100-400+ narrow bands
Band width: 5-10 nm
Fine spectral sampling
Continuous spectrum 400-2500 nm

Key advantage:

Narrow bands capture diagnostic absorption features that reveal:

Mineral composition (geology)
Vegetation biochemistry (chlorophyll, water, nitrogen)
Water quality (sediment, algae, dissolved organic matter)
Urban materials (roof types, pavement)

Applications:

Mineral exploration (ore deposits)
Precision agriculture (crop health, nutrient status)
Environmental monitoring (water quality, invasive species)
Geological mapping (lithology discrimination)

2. The Conceptual Model

Spectral Signatures

Reflectance spectrum:

\[\rho(\lambda) = \frac{L_{\uparrow}(\lambda)}{L_{\downarrow}(\lambda)}\]

Where:

$\rho(\lambda)$ = spectral reflectance (0-1)
$\lambda$ = wavelength (nm)
$L_{\uparrow}$ = upwelling radiance
$L_{\downarrow}$ = downwelling irradiance

Absorption features:

Materials absorb specific wavelengths due to:

Electronic transitions (UV-visible, 400-700 nm)
Vibrational overtones (near-infrared, 700-1300 nm)
Molecular vibrations (shortwave infrared, 1300-2500 nm)

Example - vegetation:

680 nm: Chlorophyll absorption (red)
750-900 nm: High reflectance (near-infrared plateau)
1400, 1900 nm: Water absorption
2200 nm: Cellulose/lignin absorption

Example - minerals:

Iron oxides: Absorption 400-700 nm
Hydroxyl minerals (clays): 1400, 2200 nm
Carbonates: 2300-2350 nm

Spectral Libraries

Reference spectra measured in laboratory:

USGS Spectral Library:

1400+ materials
Minerals, rocks, soils
Vegetation, water
Measured at high resolution

Use: Compare image spectra to library for identification

Imaging Spectrometer

Data structure:

Hyperspectral cube:

x dimension: spatial (pixels across)
y dimension: spatial (pixels down)
z dimension: spectral (wavelengths)

Data volume:

1000 × 1000 pixels × 200 bands = 200 million values

Each pixel = complete spectrum (hundreds of measurements)

3. Building the Mathematical Model

Spectral Unmixing

Mixed pixel problem:

Pixel contains multiple materials (vegetation + soil, multiple minerals)

Linear mixing model:

\[\rho(\lambda) = \sum_{i=1}^{n} f_i \rho_i(\lambda) + \varepsilon(\lambda)\]

Where:

$f_i$ = fractional abundance of endmember $i$
$\rho_i(\lambda)$ = reflectance spectrum of pure endmember $i$
$\varepsilon(\lambda)$ = residual error

Constraints:

$\sum_{i=1}^{n} f_i = 1$ (fractions sum to 1)

$f_i \geq 0$ (non-negative fractions)

Matrix formulation:

\[\mathbf{r} = \mathbf{E} \mathbf{f} + \boldsymbol{\varepsilon}\]

Where:

$\mathbf{r}$ = measured spectrum (m × 1 vector, m bands)
$\mathbf{E}$ = endmember matrix (m × n, n endmembers)
$\mathbf{f}$ = abundance vector (n × 1)

Solution (least squares):

\[\mathbf{f} = (\mathbf{E}^T\mathbf{E})^{-1}\mathbf{E}^T\mathbf{r}\]

Subject to non-negativity and sum-to-one constraints.

Absorption Feature Depth

Continuum removal:

Fit continuum (straight line or convex hull) to spectrum
Normalize spectrum by continuum
Measure absorption depth

Continuum-removed reflectance:

\[R_{CR}(\lambda) = \frac{\rho(\lambda)}{\rho_{continuum}(\lambda)}\]

Absorption depth:

\[D = 1 - \min(R_{CR}(\lambda))\]

Example - 2200 nm clay feature:

If minimum $R_{CR} = 0.75$, then $D = 0.25$ (25% absorption)

Spectral Angle Mapper (SAM)

Similarity metric between spectra:

\[\alpha = \arccos\left(\frac{\mathbf{r}_1 \cdot \mathbf{r}_2}{||\mathbf{r}_1|| \cdot ||\mathbf{r}_2||}\right)\]

Where:

$\alpha$ = spectral angle (radians or degrees)
Small angle → similar spectra

Classification:

Compare pixel spectrum to reference library spectra, assign to closest match (smallest angle).

Insensitive to illumination (brightness variations) - measures shape, not magnitude.

Principal Component Analysis (PCA)

Dimensionality reduction:

Transform 200 bands → 5-10 principal components capturing most variance.

Covariance matrix:

\[\mathbf{C} = \frac{1}{n-1}\mathbf{X}^T\mathbf{X}\]

Eigenvalue decomposition:

\[\mathbf{C}\mathbf{v}_i = \lambda_i\mathbf{v}_i\]

Principal components:

\[PC_i = \mathbf{X}\mathbf{v}_i\]

Variance explained:

\[\text{Variance}_i = \frac{\lambda_i}{\sum \lambda_i}\]

First few PCs typically capture >95% variance.

4. Worked Example by Hand

Problem: Unmix pixel containing vegetation and soil.

Measured spectrum (simplified, 5 wavelengths):

λ (nm)	ρ_measured
550	0.08
670	0.06
800	0.35
1600	0.25
2200	0.22

Endmember spectra:

Vegetation:

λ (nm)	ρ_veg
550	0.10
670	0.04
800	0.50
1600	0.35
2200	0.30

Soil:

λ (nm)	ρ_soil
550	0.06
670	0.08
800	0.20
1600	0.15
2200	0.14

Find fractional abundances $f_{\text{veg}}$ and $f_{\text{soil}}$.

Solution

Linear mixing model:

\[\rho(\lambda) = f_{\text{veg}} \rho_{\text{veg}}(\lambda) + f_{\text{soil}} \rho_{\text{soil}}(\lambda)\]

With constraint: $f_{\text{veg}} + f_{\text{soil}} = 1$

Substitute constraint: $f_{\text{soil}} = 1 - f_{\text{veg}}$

\[\rho(\lambda) = f_{\text{veg}} \rho_{\text{veg}}(\lambda) + (1-f_{\text{veg}}) \rho_{\text{soil}}(\lambda)\] \[\rho(\lambda) = f_{\text{veg}} (\rho_{\text{veg}}(\lambda) - \rho_{\text{soil}}(\lambda)) + \rho_{\text{soil}}(\lambda)\]

Solve using one wavelength (800 nm):

$$0.35 = f_{\text{veg}}(0.50 - 0.20) + 0.20$$

$$0.35 = 0.30 f_{\text{veg}} + 0.20$$

$$0.15 = 0.30 f_{\text{veg}}$$

\[f_{\text{veg}} = 0.50\] \[f_{\text{soil}} = 0.50\]

Verify at 670 nm:

$\rho_{pred} = 0.50(0.04) + 0.50(0.08) = 0.02 + 0.04 = 0.06$ ✓

Verify at 2200 nm:

$\rho_{pred} = 0.50(0.30) + 0.50(0.14) = 0.15 + 0.07 = 0.22$ ✓

Result: Pixel is 50% vegetation, 50% soil.

Note: Using all bands simultaneously with least squares gives more robust estimate.

5. Computational Implementation

Below is an interactive hyperspectral analysis tool.

Vegetation fraction: 50% Add noise: Show endmembers: Analysis method:

Estimated vegetation: --%

Estimated soil: --%

RMSE: --

Classification: --

Observations:

Green dashed line shows pure vegetation spectrum with strong NIR reflectance
Brown dashed line shows pure soil spectrum with more uniform response
Blue line with points is measured mixed spectrum
Pink dotted line is reconstructed spectrum from unmixing
Red marker at 670 nm shows chlorophyll absorption feature
Green marker at 850 nm shows NIR plateau characteristic of vegetation
Unmixing algorithm estimates component fractions from spectral shape
Adding noise degrades unmixing accuracy but algorithm remains robust
RMSE quantifies reconstruction quality

Key insights:

Vegetation has distinctive spectral signature with red absorption and NIR reflectance
Mixed pixels can be decomposed into constituent materials
Narrow bands enable precise absorption feature measurement
Spectral libraries provide reference signatures for identification

6. Interpretation

Mineral Mapping

Example - Cuprite, Nevada mining district:

AVIRIS hyperspectral data identified:

Kaolinite (2200 nm absorption)
Alunite (2165 nm absorption)
Buddingtonite (2125 nm absorption)
Iron oxides (500-600 nm absorption)

Economic value:

Mapping alteration minerals indicates hydrothermal systems associated with ore deposits.

Advantage over field mapping:

Cover large areas rapidly
Access remote/dangerous terrain
Objective, repeatable
Subsurface minerals exposed at surface

Precision Agriculture

Crop biochemistry from spectra:

Chlorophyll content:

Red edge position (700-750 nm)
Chlorophyll indices using 550, 670, 750 nm

Water stress:

Water absorption at 1400, 1900 nm
Normalized difference water index

Nitrogen status:

Correlation with chlorophyll
Red edge shifts with N availability

Disease detection:

Early stress signatures before visual symptoms
Spectral changes in stressed tissue

Application:

Variable-rate fertilizer/pesticide application based on hyperspectral maps.

Water Quality

Optically active constituents:

Chlorophyll-a (algae):

Absorption at 440, 675 nm
Fluorescence peak at 685 nm

Suspended sediment:

Scattering increases reflectance across all bands
Particularly strong in red/NIR

Dissolved organic matter (DOM):

Absorption in blue/green
“Yellow substance”

Example - Lake Erie harmful algal blooms:

Hyperspectral imaging from aircraft detects:

Bloom extent
Chlorophyll concentration
Cyanobacteria vs green algae discrimination

Public health application:

Toxic cyanobacteria produce microcystin - early detection enables beach closures.

7. What Could Go Wrong?

Atmospheric Interference

Problem:

Atmosphere absorbs and scatters light.

Water vapor absorption bands:

1400 nm
1900 nm
2500+ nm

In these bands: Surface reflectance unmeasurable from aircraft/satellite.

Solution:

Atmospheric correction (ATREM, FLAASH algorithms)
Use bands outside absorption features
Ground-based measurements for validation

Endmember Variability

Assumption: Pure endmembers have fixed spectra.

Reality:

Vegetation spectra vary with species, health, phenology
Soil spectra vary with moisture, texture, mineralogy
Shadows, topography affect apparent spectra

Problem:

Using library spectra from different location/time introduces error.

Solution:

Image-derived endmembers (select from actual scene)
Multiple endmember spectral mixture analysis (MESMA)
Account for variability in library

Dimensionality Curse

High dimensionality (200+ bands) causes statistical problems:

Hughes phenomenon:

Classification accuracy decreases with too many features
Need exponentially more training samples

Multicollinearity:

Adjacent bands highly correlated
Redundant information

Solution:

Dimensionality reduction (PCA, MNF)
Band selection (choose optimal subset)
Feature extraction (spectral indices)

Non-Linear Mixing

Linear model assumes:

Photons interact with one material before sensor.

Reality - multiple scattering:

Canopy: Photons bounce between leaves, soil
Mineral mixtures: Intimate vs areal mixing behave differently

Non-linear effects especially strong in:

Dense vegetation
Rough surfaces
Multiple scattering media

Solution:

Non-linear unmixing models
Radiative transfer modelling
Acknowledge limitation in interpretation

8. Extension: Derivative Spectroscopy

Spectral derivatives enhance subtle features.

First derivative:

\[\frac{d\rho}{d\lambda} \approx \frac{\rho(\lambda_{i+1}) - \rho(\lambda_{i-1})}{2\Delta\lambda}\]

Second derivative:

\[\frac{d^2\rho}{d\lambda^2} \approx \frac{\rho(\lambda_{i+1}) - 2\rho(\lambda_i) + \rho(\lambda_{i-1})}{\Delta\lambda^2}\]

Advantages:

Remove background (continuum)
Enhance narrow absorption features
Less sensitive to illumination variations

Red edge derivative:

First derivative peak position indicates chlorophyll content and plant stress.

Mineral identification:

Second derivative reveals overlapping absorption features invisible in original spectrum.

Challenge:

Derivatives amplify noise - require smoothing.

9. Math Refresher: Vector Spaces and Similarity

Dot Product

Definition:

\[\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = ||\mathbf{a}|| \cdot ||\mathbf{b}|| \cos\theta\]

Where $\theta$ = angle between vectors.

Cosine Similarity

Normalized dot product:

\[\text{similarity} = \frac{\mathbf{a} \cdot \mathbf{b}}{||\mathbf{a}|| \cdot ||\mathbf{b}||} = \cos\theta\]

Range: -1 to +1

+1: Identical direction
0: Orthogonal
-1: Opposite direction

Spectral Angle Mapper uses this:

Angle $\alpha = \arccos(\text{similarity})$

Small angle → high similarity

Euclidean Distance

\[d = ||\mathbf{a} - \mathbf{b}|| = \sqrt{\sum_{i=1}^{n} (a_i - b_i)^2}\]

Sensitive to magnitude - bright and dark versions of same material have large distance.

Spectral angle insensitive to magnitude - better for reflectance spectra.

Summary

Hyperspectral imaging captures hundreds of narrow spectral bands enabling material identification
Key advantage over multispectral is fine sampling of absorption features diagnostic of composition
Linear spectral unmixing decomposes mixed pixels into fractional abundances of pure materials
Spectral Angle Mapper classifies by comparing spectrum shape to reference library
Applications span mineral exploration, precision agriculture, water quality, and environmental monitoring
Continuum removal and derivative spectroscopy enhance subtle absorption features
Principal Component Analysis reduces dimensionality while preserving most variance
Challenges include atmospheric correction, endmember variability, and non-linear mixing effects
AVIRIS, EnMAP, PRISMA provide operational hyperspectral data from airborne and spaceborne platforms
Critical tool for biochemical and mineralogical mapping from remote platforms