Guide: SAR Doppler Tomography

Guide – Exploring the Past with LiDAR

Imagine being able to see the landscape around you in a completely new way—an invisible layer revealing the hidden structures Read more

Guide: Common Features of Iron Age Hillforts

This article attempts to serve as a guide for many of the features of the hillforts found in Britain, in Read more

Guide – Hidden Remains

Identification of features is simplified when the full extent of remains such as Earthworks, can be easily seen. However, once Read more

SAR plotting a volcanic eruption – Scott Manley

SAR Doppler Tomography

Synthetic-Aperture Radar (SAR) already relies on Doppler shifts: echoes from scatterers in a side-looking radar beam have slightly different frequencies as the platform flies past, and focusing those micro-shifts yields a two-dimensional image. Doppler (or Tomo) SAR takes the idea one step further. By collecting a stack of SAR images from slightly different flight tracks or look angles, it treats the Doppler-frequency axis itself as an aperture in elevation.

Unwanted cookie prevention

Clicking on this image will load the YouTube video, and you will get exactly the same cookies as if you were on the YouTube website directly

For more information, see our cookie policy at: YouTube Cookies

The above video, from Scott Manley, explains SAR (prior to doppler), and compliments this article by serving as an introduction to the subject.

Setting the scene

When archaeologists talk about radar they usually mean a device that sends out a burst of radio-waves and times the returning echo. Synthetic-Aperture Radar, or SAR, is a special form that flies on an aircraft or a satellite. Instead of taking a single snapshot, the sensor keeps transmitting while it moves; later, a computer knits all those echoes into a single “synthetic” antenna hundreds or even thousands of metres long. The result is a map whose sharpness rivals optical photographs but that works day or night and through cloud, smoke or forest canopy.

Every echo carries two pieces of information:

• its amplitude (how strong the reflection is)
• its phase (the tiny shift in wavelength that measures how far the wave has travelled).

Because the aircraft is moving, echoes from objects ahead and behind the beam arrive with slightly different frequencies. That frequency shift is called Doppler. By treating the Doppler spectrum as if it were another lens, engineers can focus not just a flat picture but a three-dimensional tomographic slice. The technique is therefore known as SAR Doppler Tomography (or simply TomoSAR).

Three geometric terms matter throughout:

• Wavelength (λ) – the physical length of one radio wave. Longer wavelengths (P-band ≈ 70 cm, L-band ≈ 23 cm) penetrate leaves, light forest or dry sand better than short wavelengths (C-band ≈ 6 cm, X-band ≈ 3 cm).
• Baseline (B⊥) – the sideways separation between two flight tracks. The wider this gap, the finer the vertical resolving power of the tomographic reconstruction.
• Look Angle (θ) – the tilt between the beam and the vertical. It controls how echoes stack up in depth.

After focusing, the computer fills a 3-D grid of tiny boxes called voxels (volume pixels) with Back-scatter strength. One Voxel might represent the forest canopy, another the bare soil beneath, or—if conditions are perfect—the roof, walls and foundations of a buried structure. The clarity of each layer depends on Coherence (how similar the phase of successive echoes remains) and on how far the radio waves can travel through vegetation or soil before they fade.

Wavelengths used

* Penetration depths assume low soil moisture and low salinity; water-logged or clay-rich ground drastically reduces reach.
** Height resolution figures are rule-of-thumb values derived from Δh ≈ λ R ⁄ (2 B⊥ sin θ) for typical ranges and available baselines.

Common terms

Synthetic-Aperture Radar (SAR):  Side-looking radar flown on aircraft or satellites; by combining echoes gathered along the flight path it simulates an antenna hundreds of metres long and produces metre-scale images day or night, through cloud, smoke or vegetation.

Doppler Shift: The tiny change in echo frequency caused by the platform’s motion relative to scatterers ahead or behind the beam; SAR processors exploit these shifts to focus detail along track and, in tomography, to separate echoes coming from different heights.

Tomographic SAR (TomoSAR): A stack of slightly offset SAR passes treated as a vertical aperture so that the Doppler spectrum can be inverted, voxel by voxel, into a three-dimensional back-scatter map.

Voxel: The smallest 3-D element in a tomographic cube; it records echo strength from a specific azimuth-range-height cell, analogous to a pixel in 2-D imagery.

Wavelength (λ): Physical length of one radar wave; long wavelengths (L-band ≈ 23 cm, P-band ≈ 70 cm) penetrate foliage or dry sand far better than short ones (X-band ≈ 3 cm).

Baseline (B⊥): The sideways separation between two flight tracks; greater baselines widen the synthetic vertical aperture and sharpen height resolution.

Look Angle (θ): The tilt between the radar beam and the vertical; together with baseline and wavelength it controls the achievable vertical resolution Δh ≈ λ R / (2 B⊥ sin θ).

Coherence: A measure of how little the radar phase changes between repeated passes; high coherence is essential for sharp tomographic reconstruction and deformation mapping.

Phase: The fractional part of a radar wavelength returned by a target; phase differences underlie SAR focusing, interferometry and tomographic height separation.

Amplitude (Back-scatter): Echo strength returned from a target; variations map surface texture, moisture and roughness, and populate voxel intensities in TomoSAR.

Interferometric SAR (InSAR): Technique that compares phase between two coherent SAR images to measure height or surface motion; forms the 2-D precursor to full 3-D TomoSAR.

D-TomoSAR — Differential Tomographic SAR; combines InSAR phase change with 3-D voxel separation to monitor millimetre-scale deformation on specific height layers (e.g., building façades).

Beam-forming Algorithm (Capon, MUSIC, RIAA): Mathematical filters that convert Doppler-frequency data into height profiles; newer high-resolution methods (e.g., RIAA) halve vertical error compared with classical Fourier focusing.

Penetration Depth: Maximum subsurface reach of a radar pulse; in very dry, quartz-rich soils P-band may penetrate up to \~60 cm, while vegetation penetration can exceed 50 m.

Height Resolution: Smallest vertical separation TomoSAR can distinguish; ranges from \~3 m for airborne P-band with multi-kilometre baselines to \~30 m for single-satellite X-band stacks.

The technology itself, is quite technical, and is likely to have developed further, by the time you read this, but with those ideas: Synthetic aperture, Doppler shift, tomography, wavelength, baseline, look angle, voxel, coherence and penetration, in place, the reader should get a good understanding both of the technology and of it’s application and limitation.

Technical Discussion

This next section shows in detail how engineers turn raw radar echoes into a 3-D model, and the following sections explore what that model has already revealed about standing monuments, jungle-hidden cities and shallow desert ruins.

Making height-focusing feel less like rocket science

If you think of ordinary SAR as shining a very thin torch-beam sideways from an aircraft: while the aircraft moves forward, every ground object briefly sits in the beam and the echo you record contains a tiny frequency shift (its Doppler “note”). Add all those notes together in software and you “focus” a sharp 2-D picture.

Tomographic SAR does exactly the same thing, but with several parallel flight paths, so you hear each object singing at a slightly different pitch every time. If you now stack all those echoes you have, in effect, recorded a little Doppler melody that changes with height. “Playing the melody backwards” (the mathematical term is inverting the spectrum) lets a computer place echo-strength into thousands of 3-D bricks called voxels. The algorithms that perform this reverse process—Fourier beam-forming, or the higher-definition Capon, MUSIC and RIAA filters—differ only in how cleverly they separate the overlapping notes; to the user they all end up filling a 3-D cube of brightness.

Why a radar’s “eyeglass” has limits

A camera’s sharpness depends on lens diameter; for TomoSAR the equivalent “lens” is the sideways gap between the highest and lowest flight lines, called the perpendicular baseline (B⊥). The focusing rule is:

    Δh ≈ λ R ⁄ (2 B⊥ sin θ)

• λ – radar wavelength: longer waves improve penetration but coarsen resolution.
• R – slant range: the straight-line distance from sensor to target (hundreds of kilometres for a satellite, tens for an aircraft).
• B⊥ – total vertical spread of the flight paths; the wider the spread, the finer the vertical “slice”.
• θ – look angle: a steep look (large θ) helps resolution; a very shallow look blurs it.

Two concrete examples

• Satellite X-band stack (TerraSAR-X)
  • λ = 3 cm, R ≈ 700 km, B⊥ ≈ 2 km, θ ≈ 35°.
  • Δh ≈ (0.03 m × 700 000 m) ⁄ (2 × 2 000 m × 0.57) ≈ 18 m.
  • Result: you can separate a city roof from the street, or a canopy from the ground, but you will not pick out individual forest layers.
• Airborne P-band campaign over rainforest
  • λ = 70 cm, R ≈ 10 km, B⊥ ≈ 3 km (dozens of tightly controlled tracks), θ ≈ 45°.
  • Δh ≈ (0.70 m × 10 000 m) ⁄ (2 × 3 000 m × 0.71) ≈ 1.6 m.
  • Result: separate the tree-tops, understory and the bare soil with enough clarity to map ancient causeways hidden under 50 m of jungle.

Accuracy in the real world

Forests: multi-baseline P-band surveys in Gabon and Cambodia quote height RMSE of 2–4 m when checked against airborne LiDAR.

Urban heritage: twelve-scene TerraSAR-X stacks over Milan reproduce façade heights within ±1 m and detect subsidence of <2 mm / year.

Desert archaeology: experimental L-band TomoSAR in Egypt has outlined buried walls under 40 cm of dry sand, though still at a coarse 8-m vertical grid.

So, while the maths hides inside specialist software, the take-away is simple: longer wavelengths plus wider baselines equal sharper and deeper 3-D views, and even the “blurrier” satellite stacks are already accurate enough to monitor buildings or strip the canopy off extensive archaeological landscapes.

Technical summary

A 3-D voxel map of back-scatter is then reconstructed by inverting the Doppler spectrum with a Fourier beam-former or high-resolution spectral estimator such as Capon, MUSIC or the more recent RIAA algorithm (sto.nato.int, mdpi.com). The achievable vertical (height) resolution obeys an optical-style formula Δh ≈ λ R⁄(2 B⊥ sin θ), where λ is wavelength, R slant range, B⊥ the total perpendicular baseline and θ the look angle (earth.esa.int); centimetre wavelengths and kilometre-scale satellite baselines give ~30–50 m, while airborne P-band campaigns with tens of baselines routinely reach 3–5 m.

Workflow in outline

• Acquire a coherent stack (typically 5–50 single-look-complex images) from repeat-pass, bistatic or along-track channels.
• Precisely co-register and compensate for platform motion to a common phase origin.
• Form the Doppler spectral cube and apply beam-forming / spectral inversion to retrieve the elevation profile for every azimuth-range pixel.
• Optionally apply differential processing to isolate deformation or slow motion (D-TomoSAR).

Where it has been used

• Forests and biomass – L- and P-band TomoSAR campaigns by DLR, NASA UAVSAR and ESA’s AfriSAR have mapped canopy layers and ground topography through >50 m tropical forest; recent P-band tests report canopy-height RMSEs of 2–6 m against LiDAR (mdpi.com, sciencedirect.com).
• Urban 3-D mapping – TerraSAR-X stacks over Berlin, Munich and Milan resolve individual building façades and detect millimetre-scale subsidence; height differences compared with airborne laser scans are usually within 1–3 m (engineering.purdue.edu, researchgate.net).
• Cryosphere & topography under vegetation – multilook P-band apertures recover glacier stratigraphy and bare-earth surfaces beneath forest canopy.
• Infrastructure/heritage monitoring – phase-based Doppler tomography of COSMO-SkyMed images has revealed sub-millimetre micro-vibrations inside the Khufu pyramid, producing a 3-D map of hidden chambers (mdpi.com).
• Experimental marine & coastal work – along-track Doppler tomography is being trial-led to image wave fields and coastal subsurface geology (oceanova.nz).

SAR Penetration, note the minimal depth penetration in contrast to SAR Doppler Tomography – Scott Manley

How well does it work?

Vertical resolution is limited by wavelength and total baseline; airborne P-band or low-orbit bistatic constellations can reach 1–3 m, whereas single-platform X-band satellites seldom do better than 30 m.

Height accuracy (bias/RMSE) depends on signal coherence, vegetation density and algorithm choice. In mixed temperate conifer sites an RIAA processor at L-band achieved 2.0 m RMSE in canopy height; a conventional Capon solution on the same data gave 6 m (mdpi.com). Urban point-cloud studies with TerraSAR-X routinely meet ±1 m when control points are available (engineering.purdue.edu).

Limitations include temporal decorrelation in leafy forests, layover in steep terrain, sensitivity to baseline errors, and high computation cost for large stacks; advanced sparse-recovery and machine-learning solvers are an active research front (link.springer.com).

The future

ESA’s BIOMASS P-band mission (launch planned 2025) and the proposed Tandem-L constellation are designed with multi-baseline geometries precisely to exploit Doppler SAR tomography for global 3-D biomass, deformation and surface-change mapping. With those satellite stacks expected to refresh every 7–16 days, D-TomoSAR will shift from experimental campaign tool to routine Earth-observation workhorse over the next decade.

SAR use for Archaeology – Scott Manley

Use in Archaeology

For archaeology, the technique has been applied in two quite different ways.

Monitoring of standing monuments

The first and most mature use is for monitoring standing monuments. With a long stack of very high-resolution satellite scenes, tomographic processing can separate the echoes that come from the façades, the roof and the ground around a building, effectively turning a side-looking radar into a three–dimensional laser scanner.

The TerraSAR-X stacks that were raised over Milan, Berlin and Munich resolve individual stories on masonry towers and track sub-millimetre subsidence month by month; comparisons with laser scans show the height values are usually within about one metre of the true surface and the measured rates of movement match classical levelling to within a millimetre a year. These same data are now used by conservation teams to decide where to grout, underpin or restrict visitor numbers.

Buried or canopy-covered archaeology

The second, newer strand concerns buried or canopy-covered archaeology. By flying a radar that works at a very long wavelength—twenty-three centimetres at L-band or sixty to seventy centimetres at P-band—and by repeating the flight on dozens of slightly shifted tracks, researchers can treat the Doppler frequency axis as a vertical aperture and reconstruct a voxel model of everything that reflects in the first ten to fifteen metres above the true ground.

In tropical forests the method pulls two quite independent surfaces out of the data cube: one is the canopy top; the other, tens of metres lower, is the bare earth. When the DLR and NASA teams applied P-band TomoSAR to the Angkor and Caracol catchments it delivered under-canopy digital-terrain models whose vertical error against ground survey was about three to five metres; the resulting maps revealed long rectilinear embankments, reservoirs and causeways that had never been seen in the jungle before. (caracol.org) Similar experiments on the gravel terraces north of the Nile have detected buried wall lines under half a metre of wind-blown sand, though that work is still in conference proceedings rather than a full peer-reviewed paper.

Penetration into vegetation

Penetration into vegetation is therefore excellent—the radar sees right through a fifty-metre canopy at P-band—but penetration into soil is modest. In very dry, quartz-rich sand a P-band pulse can reach something approaching one metre; in damp loam or clay the depth falls to a few centimetres, and practically no energy survives to be tomographically focused. That is why the clearest sub-surface successes so far come from hyper-arid desert margins or from gravelly terraces where the water table is deep.

A cautious rule of thumb from the experimental campaigns is that you may expect reliable mapping of structures buried up to half the operating wavelength—so perhaps thirty centimetres at L-band and sixty centimetres at P-band—provided the soil is dry, salts are low, and the target surface is fairly extensive. Where those conditions hold, height resolution inside the ground layer is on the order of the vertical resolution in the air, namely three to five metres for airborne P-band baselines of a few hundred metres and perhaps thirty metres for present single-platform X-band satellites. (onlinelibrary.wiley.com, sciencedirect.com)

Proven tool, still being developed

SAR Doppler tomography is already a proven tool for millimetre-scale deformation studies on standing heritage and a promising, though still experimental, way to strip forest canopies from archaeological landscapes and to glimpse shallow masonry in desert terrain. The degree of ground penetration is governed less by the mathematics of tomography than by physics: the longer the wavelength and the drier the soil, the further the pulse can travel and the more archaeological detail it will return.