The Gold Standard of Archaeology | Coherent Sources, Spatial Smoothing and False Source Separation

1. Define coherence in the observed window

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Two physical sources can be coherent when their signals have a stable linear relationship in the chosen frequency band and time window. One source can also create coherent direct and reflected paths. State whether coherence is expected from source timing, multipath, periodic machinery or the analysis transform.

Coherence is not a permanent label. It can change across windows and frequencies, so the diagnostic must use the same samples that support the archaeological interpretation.

2. See why covariance rank can collapse

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MUSIC and related subspace methods rely on a covariance matrix whose signal rank reflects separable components. Fully coherent arrivals can contribute a lower-rank structure than their physical count, leaving the method without the independent dimensions it expects.

Finite samples, coloured noise and calibration error also move eigenvalues. A low rank may reflect coherence, too little data or weak sources; a high rank may reflect noise rather than extra arrivals.

3. Understand what spatial smoothing changes

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Spatial smoothing divides an array into overlapping subarrays and averages their covariance matrices. Phase shifts between subarrays can decorrelate coherent components and restore rank. The operation is a designed transformation, not a direct observation of independent sources.

The effective subarray is shorter than the full array. Resolution and identifiability depend on the number and length of subarrays, the geometry and the assumed steering model.

4. Balance rank restoration against aperture loss

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More smoothing averages can improve rank restoration and covariance stability, but shorter subarrays widen beams and can make close directions harder to resolve. Forward-backward smoothing adds conjugate-reversed information under array-symmetry assumptions and may reduce the sensor requirement.

Plot results across admissible subarray lengths. A pair that appears only at one aggressive setting is not yet a robust separation.

5. Choose source count without reading one eigenvalue elbow

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Compare information criteria, noise resampling, neighbouring windows and physically plausible source scenarios. Apply rank rules before and after smoothing and explain why their assumptions are reasonable. Coherent sources can make an unsmoothed eigenvalue count too small; smoothing and noisy subarrays can make it too large.

Carry several plausible ranks through the analysis. If the inferred structure changes, report model-order sensitivity rather than selecting the most exciting image.

6. Worked example: a pump, a reflection and a supposed wall

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A linear array beside a buried settlement records a narrow 22 Hz source from an irrigation pump. A retaining bank creates a coherent reflection. Unsmoothed MUSIC shows one peak; strong smoothing with rank two shows a second peak aligned across the trench and it is initially described as a wall-related scatterer.

Geometry-matched simulation reproduces the second peak using the pump and bank alone. The peak weakens when longer subarrays are used and vanishes after excluding sensors nearest the bank. The report concludes that two coherent paths are present but that the second direction does not identify a buried wall.

7. Recognise false separation patterns

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Warning signs include peaks locked to subarray boundaries, mirror-symmetric directions, unstable peak order, separation below simulated resolution and components that disappear after a small frequency or sensor change. Spatial smoothing can make a spectrum look clean even when the model is wrong.

Compare residual phase, reconstruction error and held-out sensors. A sharp pseudospectrum is not a probability distribution and its narrowness is not a complete uncertainty measure.

8. Changed-context transfer check: repeated active impacts

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Move from a continuous pump to repeated hammer impacts at one known point. Reflections remain coherent, but repeated trials now allow acquisition-level tests. Randomise source timing, change the impact position and analyse single trials before stacking.

The transferable principle is to vary coherence and geometry deliberately. The changed setting makes experimental separation possible, so processing should not carry the entire burden.

9. Distinguish decorrelation from physical source discovery

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Spatial smoothing creates a full-rank statistical representation under conditions; it does not prove that each recovered component is a separate emitter. A direct path, a reflection and a mode conversion may originate from one event.

Use timing, polarisation, dispersion, site geometry and controlled sources to name components. Retain terms such as arrival direction or coherent path until independent evidence supports a physical source.

10. Audit subarrays, calibration and rejected alternatives

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Retain full-array geometry, subarray memberships, smoothing direction, rank rules, covariance windows, diagonal loading, excluded sensors, timing calibration and every stability sweep. Reproduce the result with unsmoothed beamforming and matched simulations.

Document alternatives that were tested and rejected. A final two-peak plot without the one-source reflection model hides the most important part of the inference.

11. Frequently asked questions

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Does spatial smoothing always resolve coherent sources? No; identifiability, aperture, signal strength and model fit still limit it. Is a restored rank the physical source count? Not necessarily. Should the setting with the sharpest peaks be preferred? No; stability and validation matter more. Can a reflection count as a source? It can be a separable arrival component, but should not automatically be labelled a separate emitter or feature.

12. The gold standard separates components without inventing objects

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Coherent wavefields require joint reasoning about covariance rank, subarray geometry, aperture and physical propagation. Diagnose coherence, sweep valid smoothing choices, carry rank uncertainty, simulate alternatives and name only what the evidence distinguishes.

The gold standard treats spatial smoothing as a useful model operation with visible trade-offs. It rejects both premature merging and false separation, leaving archaeological claims proportional to the parts of the wavefield that remain stable under challenge.