The Gold Standard of Archaeology | Broadband Coherent Sources, Frequency Focusing and Subband Rank Instability

1. Start with the archaeological wavefield and claim

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Define whether the array is locating a controlled source, a reflected boundary, repeating machinery or an ambient emitter. State the frequency band, propagation model, array geometry and time window. A peak in a focused spectrum is not automatically a wall, chamber or activity area.

Broadband processing should be chosen because the signal spans frequencies whose spatial information matters, not because it produces a sharper figure. Preserve the chain from wavefield assumption to archaeological interpretation.

2. Why narrowband subspaces cannot simply be averaged

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At each frequency, sensor phase depends on wavelength and direction. Covariance matrices from different bins therefore live in frequency-dependent array manifolds. Averaging them without alignment can smear peaks or combine incompatible subspaces.

Incoherent methods estimate or score each bin separately and combine spectra. Coherent methods instead transform subspaces or covariance matrices to a reference frequency. The gain in coherence introduces a focusing model that must be checked.

3. Understand coherent sources and lost rank

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Multipath or common forcing can make arrivals highly correlated. The signal covariance then has lower effective rank than the number of physical paths, so standard subspace separation may merge sources or lose them. Spatial smoothing or other decorrelation steps can help, with an aperture cost.

Broadband coherence is not one number. Two paths may be coherent in some bins and separable in others. Diagnose eigenvalue structure across frequency rather than assigning a single source count before looking.

4. Choose a focusing frequency for information, not convenience

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The reference frequency should lie where the array has useful sensitivity, the signal has support and the steering model is reliable. Very low frequencies can have poor directional resolution; very high frequencies can suffer spatial aliasing or attenuation.

Repeat the analysis across plausible reference frequencies. A source that appears only after focusing to one fragile bin should not carry a strong archaeological claim.

5. Construct and test the focusing transformation

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A focusing matrix maps steering or signal subspaces at each bin toward the reference manifold. Some approaches require preliminary direction estimates; errors in those estimates can bias the focused result. Modal or interpolation approaches reduce that dependence but still rely on geometry and field-of-view assumptions.

Measure focusing residuals on a direction grid and with calibration data. Exclude or downweight bins where the mapping error is large rather than assuming every transformed covariance is equally trustworthy.

6. Worked example: two correlated sources beside a buried structure

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A linear array records broadband hammer impacts. Conventional spectra show one broad arrival, while focused processing shows two peaks interpreted as separate reflecting boundaries. The second peak is strongest in three high-frequency bins where the array spacing exceeds half a wavelength and the estimated rank jumps from one to three.

The team removes aliased bins, estimates rank with bootstrap stability and repeats focusing at three references. Only one direction remains stable; the second is reported as a processing artefact rather than a second wall. A follow-up array with closer spacing is designed before excavation.

7. Treat subband rank as an uncertain diagnostic

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Estimate eigenvalues, noise level and source number within each bin and window. Low signal-to-noise ratio can collapse a real component; coloured noise or calibration error can create an extra one. Information criteria provide a rule, not an oracle.

Record rank distributions under resampling and nearby window choices. Consider a frequency-dependent rank model or robust aggregation instead of forcing the maximum rank across all bins.

8. Changed-context transfer check: a compact near-field array

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Move the method from distant plane waves to a compact array close to a test pit. Curved wavefronts now depend on range as well as direction, and a far-field focusing matrix can align the wrong manifold. The transferable question is whether transformed bins represent the same physical parameterisation.

Use a near-field steering model or restrict claims to calibrated locations. Stable focused angles under the wrong geometry do not validate a range interpretation.

9. Weight bins by reliability without hiding the choice

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Possible weights reflect signal power, estimated noise, focusing residual, coherence, calibration quality or effective sample size. Power-only weighting may let a noisy transient dominate; uniform weighting may give empty bins the same vote as informative ones.

Predefine the weighting rule and compare reasonable alternatives. Show a contribution plot so readers can see which frequencies create each peak.

10. Audit the focused spectrum from raw traces to peaks

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Retain sensor coordinates, timing calibration, raw windows, transforms, frequency selection, covariance estimates, smoothing parameters, rank rules, reference frequency, focusing matrices, weights and peak-selection settings. Store unfocused subband spectra beside the combined result.

Use synthetic injection and physical calibration at known locations. An algorithm that recovers simulated sources under its own assumptions but fails the field calibration has not earned an archaeological interpretation.

11. Frequently asked questions

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Does focusing create more data? No; it aligns existing frequency information under a model. Is one stable peak proof of one source? No; coherent paths may remain unresolved. Should every subband use the same rank? Not automatically. Can spatial smoothing always restore rank? No; it reduces effective aperture and depends on array structure.

12. The gold standard keeps broadband sharpness accountable

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A defensible broadband localisation defines the wavefield, diagnoses coherence and rank by subband, chooses a justified reference, measures focusing error, excludes aliased or empty bins and tests sensitivity to weighting. It separates stable physical information from peaks manufactured by transformation choices.

The objective is not the narrowest image. It is an archaeological inference that remains credible when the frequency reference, source count and processing path are changed within reasonable bounds.