Taylor’s law in satellite time series of vegetation productivity

Background: Taylor’s law is considered a semi-natural-law and originally comes from ecology. It describes how the variance of a quantity grows with its mean, following variance = a · mean^b. It was first found for insect and animal counts, but it also shows up in many other data. The exponent b tells us something about stability. If b is below 2, places with high values fluctuate less, relative to their mean, than places with low values. With Landsat data back to 1985, this can be tested for vegetation at every pixel.

Short description: The thesis tests Taylor’s law with long satellite time series of vegetation, for example from Landsat. For each pixel, a robust yearly vegetation measure is derived, and its mean and variance over the years are calculated. The relationship between mean and variance across all pixels then gives the exponent b. Comparing b between land cover types, regions or elevation zones shows where vegetation is more or less stable. Part of the work is to check how far the results depend on the choice of vegetation index and on the number of available observations.

Taylor’s law from satellite time series (schematic). Each Landsat pixel becomes one point, and the slope b shows how variability scales with productivity.

Suggested reading

Taylor, L. R. (1961). Aggregation, variance and the mean. Nature, 189, 732–735. https://www.nature.com/articles/189732a0

Badgley, G., Field, C. B., & Berry, J. A. (2017). Canopy near-infrared reflectance and terrestrial photosynthesis. Science Advances.

Vadrevu, K. (2026): Power law scaling reveals systemic divergence in fire risk across South and Southeast Asian countries. Environmental Research Letters. 10.1088/1748-9326/ae962a

Start: Anytime

Prerequisites/qualification

·        Remote Sensing & GIS

·        Basic programming skills

·        Good basic statistics (regression, power laws)

·        Interest in ecology

Storing Sentinel-2 time series in neural networks: Implicit neural representations

Background: Satellite data are usually stored as a grid with one value per pixel. Implicit neural representations (INRs) instead treat the data as a function of location and time and train a neural network to learn it. You give the network a location and a date, and it returns the values for that place and time. This can save a lot of storage, but the network only stores an approximation. Its quality is usually judged by a single number, the average reconstruction error. What this number hides, which information is lost and where, is largely unknown.

Source: https://gkvoelkl.github.io/python-streamlit-neural-earth-field/en/01-geodaten-als-funktion.html

Short description: The thesis trains an INR on a Sentinel-2 time series of a chosen region and compares the reconstruction with the original data, pixel by pixel and date by date. The aim is to find out what gets lost. Possible candidates are sudden changes (e.g., a meadow mown on a specific date that shows a sharp drop in NDVI), small objects such as single buildings or ponds or extreme values such as the peak of the growing season. The result is a clear picture of which information an INR keeps and which it does not.

Suggested reading

Sitzmann, V., Martel, J., Bergman, A., Lindell, D., & Wetzstein, G. (2020). Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems, 33, 7462–7473.

Madadikhaljan, M., Prexl, J., Wittmann, I., Albrecht, C. M., & Schmitt, M. (2026). Location Is All You Need: Continuous spatiotemporal neural representations of Earth observation data. arXiv:2604.07092.

https://gkvoelkl.github.io/python-streamlit-neural-earth-field/en/01-geodaten-als-funktion.html

Start: Anytime

Prerequisites/qualification: This topic is technically demanding and suits students who enjoy coding and have trained neural networks before.

·        Remote Sensing

·        Very good Python skills and experience with PyTorch

·        Experience training neural networks

Observation density and apparent greening trends in pretrained EO embeddings of alpine ecosystems

Background: Pretrained embeddings such as TESSERA or AlphaEarth Foundations summarise a year of satellite observations in one vector per pixel. Yet the number of clear observations behind each embedding varies between places and years. Bayle et al. (2024) showed that Landsat-based greening trends in the Alps were overestimated because the number of summer observations increased over time. Whether embeddings show the same artefact has not been tested, and in embeddings it could easily go unnoticed.

Short description: The thesis transfers the idea of Bayle et al. to embeddings of alpine vegetation from 2017 onwards. Greening trends are calculated from NDVI and from embeddings for the same pixels, for example along a “greenness axis” in embedding space. A locally runnable model such as TESSERA is then fed with time series that have the same number of observations each year. If trends shrink, part of the greening comes from the data rather than the vegetation. Bare rock serves as a control.

 

Processing chain from satellite remote sensing measuring of Earth's surface reflectance to pixelwise greening trends. What happens if we reconstruct this approach with embeddings? Figure source: Bayle, A. (2026)

Suggested reading:

Bayle, A., Gascoin, S., Berner, L. T., & Choler, P. (2024). Landsat-based greening trends in alpine ecosystems are inflated by multidecadal increases in summer observations. Ecography, 2024(12), e07394. https://doi.org/10.1111/ecog.07394

Bayle, A. (2026): Satellite Remote Sensing of Alpine Vegetation Dynamics: Challenges and Perspectives. Global Change Biology, https://doi.org/10.1111/gcb.70968

Feng, Z., Atzberger, C., Jaffer, S., Knezevic, J., Sormunen, S., Young, R., et al. (2025). TESSERA: Temporal Embeddings of Surface Spectra for Earth Representation and Analysis. arXiv:2506.20380. https://doi.org/10.48550/arXiv.2506.20380

Start: Anytime

Prerequisites/qualification

• Remote Sensing & GIS
• Good Python skills
• Good basic statistics (e.g., trend analysis)

Measuring ecosystem recovery with spectral indices, fractional cover and semantic categories

Background: After a disturbance such as a landslide, a fire, an avalanche or a wind-throw, satellite data are often used to track how quickly vegetation returns. A common measure is vegetation cover, but it can be described in very different ways. A spectral index such as NDVI gives a unitless value, spectral unmixing gives the share of vegetation in each pixel, and semantic approaches assign a knowledge-based vegetation category. These approaches exist side by side but are rarely compared. Depending on which one is used, the same surface may appear recovered after three years or only after ten.

Schematic figure of ecosystem recovery, here with tree cover as a proxy. How do recovery intervals differ for spectral indices, fractional cover and a semantic approach? Figure source: Mandl et al., 2024

Short description: The thesis tracks recovery on disturbed areas in a region of choice with Sentinel-2 or Landsat time series. Vegetation cover is described in three ways, as NDVI, as vegetation fraction from unmixing, and as the corresponding semantic vegetation category. For each area, recovery is measured as the time until vegetation reaches its pre-disturbance level or a fixed threshold. Very high resolution imagery from different years serves as reference. The result shows how strongly recovery estimates depend on the chosen measure and where the approaches disagree, for example with sparse, dry or patchy vegetation.

Suggested reading

Mandl, L., Viana-Soto, A., Stritih, A., Seidl, R., & Senf, C. (2024). Spectral-unmixing based forest recovery indicators for predicting long-term recovery success. Remote Sensing of Environment. https://doi.org/10.1016/j.rse.2024.114194

Okujeni, A., van der Linden, S., Tits, L., Somers, B., & Hostert, P. (2013). Support vector regression and synthetically mixed training data for quantifying urban land cover. Remote Sensing of Environment, 137, 184–197. https://doi.org/10.1016/j.rse.2013.06.007

Sudmanns, M., Augustin, H., van der Meer, L., Baraldi, A., & Tiede, D. (2021). The Austrian semantic EO data cube infrastructure. Remote Sensing, 13(23), 4807. https://doi.org/10.3390/rs13234807

Start: Anytime

Prerequisites/qualification: 

·        Remote Sensing & GIS

·        Basic programming

·        Interest in ecosystem dynamics

Three-dimensional surface information in pretrained EO embeddings

Suggested by: Lisa Mandl

Background: Embeddings are derived from images taken from above, but these images contain indirect information about height, such as shadows, texture or radar backscatter from building walls. It is unclear whether embeddings keep this information or mainly encode the land cover type that usually goes with a certain height. AlphaEarth Foundations combines many data sources, including spaceborne lidar, and uses spatial context. TESSERA works pixel by pixel on Sentinel-1 and Sentinel-2 time series.

Short description: The thesis tests how much height information AlphaEarth Foundations and TESSERA embeddings contain, using airborne laser scanning (ALS) data (where available) as reference. Simple models are trained on the embeddings to predict object heights, with classical Sentinel-1 and Sentinel-2 features as a baseline. The key question is whether embeddings capture height differences within a class (e.g., low vs. high buildings) or only between classes. Study area and objects (e.g. buildings, vegetation) are flexible.

Building height map, here for Berlin. How much of height information is stored in geospatial embeddings? Data source: https://zenodo.org/records/4066295#.X4QRNmgzYmI

Suggested reading

Brown, C. F., Kazmierski, M. R., Pasquarella, V. J., Rucklidge, W. J., Samsikova, M., Zhang, C., et al. (2025). AlphaEarth Foundations: An embedding field model for accurate and efficient global mapping from sparse label data. arXiv:2507.22291.

Feng, Z., Atzberger, C., Jaffer, S., Knezevic, J., Sormunen, S., Young, R., et al. (2025). TESSERA: Temporal Embeddings of Surface Spectra for Earth Representation and Analysis. arXiv:2506.20380. https://doi.org/10.48550/arXiv.2507.22291

Frantz, D., Schug, F., Okujeni, A., Navacchi, C., Wagner, W., van der Linden, S., & Hostert, P. (2021). National-scale mapping of building height using Sentinel-1 and Sentinel-2 time series. Remote Sensing of Environment, 252, 112128. https://doi.org/10.1016/j.rse.2020.112128

Start: As soon as possible

Prerequisites/qualification

• Remote Sensing
• Good Python skills + basic machine learning
• Interest in the topic