Spatial autocorrelation, abbreviated SPAC, is a passive seismic-array method that uses relationships among ground-vibration recordings at different sensor positions to estimate surface-wave dispersion. If you are trying to interpret SPAC microtremor curves, interstation coherence, the Bessel J0 function, array radius or an inverted shear-wave velocity profile, begin with the pair spacing and the frequency axis. An oscillating correlation coefficient is not a directly measured soil stiffness.
The method compares signals recorded at separated sensors and, under suitable assumptions about the ambient wavefield, relates their spatial coherence to the phase velocity of surface waves. The inferred dispersion curve can then constrain a layered shear-wave velocity model. Both steps have assumptions: the noise sources need sufficient directional coverage for conventional averaging, the selected frequency band must be informative, and different subsurface velocity models can predict similar data.
This article explains the SPAC method from ambient-noise acquisition through spatial averaging, coherence, Bessel J0 curves, zero crossings, phase-velocity estimates and Vs inversion. It is written for readers interpreting a real figure or report rather than operating field equipment from a recipe. Worked values are explicitly illustrative, and the distinctions among correlation, velocity, wavelength, depth and model uncertainty remain visible throughout.
Quick answer: what does a SPAC curve measure?
A SPAC curve describes a frequency-dependent relationship between ground motions recorded by separated sensors, commonly after normalisation and averaging over selected station pairs. Under an idealised isotropic surface-wave model, a particular spatial correlation coefficient is related to the Bessel function J0 of a dimensionless product involving frequency, separation and phase velocity.
Because phase velocity changes with frequency in layered materials, the observed correlation pattern can be used to estimate a dispersion curve. That curve is then compared with predictions from potential shear-wave velocity structures. The interpretation is an inverse problem, not direct observation of the depth and thickness of every buried layer.
Why ambient vibrations can be useful seismic data
The ground is continually excited by natural and human sources. Ocean-generated waves, weather, vehicles and industry contribute to a changing background wavefield. Passive seismic methods exploit that motion instead of relying solely on controlled impacts or explosive sources. This can make array surveys practical where active shots are inconvenient, but it also makes source direction and temporal variability important.
A passive measurement is not automatically free from source bias. A strong road source on one side of an array may dominate certain frequencies, violating the directional idealisation used by conventional SPAC analysis. The method’s power comes from linking observable spatial correlations to a stated wavefield model and checking where the observations support that model.
What an array adds to a single-station recording
A single three-component station can record local motion and support methods such as HVSR. An array measures relationships between signals at distinct positions. Each separation samples how coherent the wavefield remains across a particular distance at a given frequency. Short and long station spacings often probe different wavelength regimes, subject to noise and processing limitations.
Array geometry is therefore part of the measurement, not a decorative map. Record the sensor coordinates and station pairs used in each averaged estimate. A circle, triangle or irregular layout may provide different distributions of directions and distances. When two studies use different arrays, the range of defensible phase velocities may differ even if their sites share similar geology.
Correlation is not the same as equal ground motion
Two sensors can record signals with different absolute amplitudes but still show a strong normalised relationship in phase or spectral behaviour. A spatial correlation coefficient does not tell you that the ground displacement at those sensors is identical. Instead, it quantifies a particular relationship under the analysis convention and frequency band.
The distinction matters when an SPAC ordinate is mistakenly labelled as vibration amplitude. Depending on the implementation, coefficients can change sign and oscillate with frequency. A negative value does not mean a physically negative seismic velocity or negative soil strength. It reflects the phase relationships captured by the correlation function.
Why the separation r appears in the model
For a selected pair of receivers, r describes their spatial separation, typically measured in metres. The same seismic wavelength produces different relative phases at different spacings. If stations are close together relative to wavelength, their recordings may remain strongly coherent. At larger separations, the averaged relationship can weaken or change sign under the surface-wave model.
A frequency alone cannot determine this behaviour because distance matters. The dimensionless argument of the conventional J0 relationship contains the product of frequency and pair spacing divided by phase velocity, multiplied by two pi. Forgetting r destroys the geometric meaning of the method and makes correlation peaks look like unexplained soil properties.
What the Bessel J0 function is doing here
J0 is a special mathematical function that appears when oscillating wave contributions are averaged over azimuth under an idealised isotropic distribution. In a simplified SPAC surface-wave relation, correlation at frequency f and separation r is modelled by J0 of 2πfr divided by phase velocity c. The argument has no physical units because hertz times metres divided by metres per second is dimensionless.
The oscillations are not evidence that soil stiffness alternates between positive and negative values. They arise from coherent wave phases and spatial averaging. The formula is valuable because, if its assumptions are adequate, the observed coefficient pattern can constrain c at given frequencies. It must not be used without considering the wavefield and array geometry.
A simple units check for the Bessel argument
Suppose a sensor pair is separated by 20 metres, the analysed frequency is 4 hertz and a trial Rayleigh-wave phase velocity is 200 metres per second. The dimensionless Bessel argument is 2π times 4 times 20 divided by 200, approximately 2.513. All seconds and metres cancel.
If someone inserts 200 kilometres per second without converting units, the argument changes dramatically. If someone treats frequency in milliseconds rather than inverse seconds, the calculation becomes meaningless. Dimensional analysis is the fastest way to catch such mistakes. SPAC works with a particular relationship among frequency, distance and propagation speed; every term must use compatible units.
The first J0 zero is a useful landmark
The first positive zero of J0 occurs at approximately 2.4048. Under the idealised single-mode relation, an observed zero of the spatial-correlation curve at a known pair spacing and frequency can help estimate phase velocity. For frequency 4 Hz and separation 20 metres, a trial velocity near 2π times 4 times 20 divided by 2.4048 is about 209 metres per second.
That estimate is illustrative and conditional on correctly identifying the first zero and the relevant wave mode. Noise, limited frequency resolution, source directionality and multiple modes can shift apparent crossings. It is not appropriate to label every zero crossing a particular shear-wave velocity layer without testing the full dispersion evidence.
Phase velocity and shear-wave velocity differ
The quantity inferred from Rayleigh-wave dispersion is phase velocity: the speed at which a phase of the surface wave propagates under the selected frequency and mode. Shear-wave velocity Vs is a subsurface material property that influences wave behaviour. They are related by wave mechanics but are not universally identical.
In layered soil, Rayleigh-wave phase velocity can depend on frequency because waves interact with material at different depths. An inversion uses this frequency-dependent behaviour to estimate a Vs profile. Treating every phase-velocity pick as a direct in-situ Vs reading at a particular depth skips the actual model and can create unrealistic subsurface maps.
Why dispersion is the bridge to a depth profile
Different frequencies tend to be influenced by different parts of a layered near-surface structure, although sensitivity is broad and model-dependent. An observed phase velocity changing with frequency is called dispersion. The shape of that curve can constrain layer velocities, thicknesses and contrasts when compared with theoretical forward models.
One common error is equating each frequency to exactly one depth using a fixed multiplier. Wavelength can provide a rough investigation-scale clue, but sensitivity kernels spread over depth and depend on modes and contrasts. A reliable SPAC interpretation distinguishes the measured correlation, derived phase-velocity curve and modelled Vs layers rather than claiming the original sensor recorded a soil layer directly.
Wavelength is a derived scale, not depth
For a phase velocity c and frequency f, wavelength λ is c divided by f. If an illustrative picked c equals 240 metres per second at 6 Hz, λ equals 40 metres. That number represents the wavelength of the selected wave, not a measured boundary depth. Changing frequency changes λ even when the same array geometry is used.
A common beginner mistake is writing that a 40-metre wavelength proves the sensor sampled precisely 40 metres down. Actual sensitivity varies with depth and model structure, and an array may lack resolution at particular scales. Use wavelength to reason about sampling and likely sensitivity, then inspect formal inversion sensitivity and independent geological constraints before claiming depth.
Why multiple radii can resolve more information
A survey with several station-pair separations can produce correlation estimates over different ranges of the dimensionless product kr. Short pairs may remain coherent at frequencies where long pairs already show oscillation or noise. Together, multiple separations can provide more evidence about phase velocity across a broader frequency range.
However, adding many sensors is not automatically equivalent to adding independent information at every frequency. Pair geometry, sensor noise, angular coverage and correlated source disturbances can limit useful data. A publication should show which radius contributed to which part of the dispersion interpretation. Array design sets a real boundary on what the inversion can claim.
Circular symmetry and azimuthal averaging
Traditional SPAC approaches often use approximately symmetric sensor layouts so correlations among station pairs at similar separation can be averaged across different directions. This averaging helps approximate an isotropic or azimuthally balanced wavefield. If nearly all pairs face the same direction, a strong directional source can bias the result.
Irregular arrays and modified formulations can still be useful, but their mathematical treatment must match the geometry. The USGS work on wavenumber-normalised SPAC discusses approaches for asymmetric arrays. Do not assume that a simple circular-array formula automatically applies unchanged to every arbitrary collection of sensor coordinates.
Spatial aliasing creates false apparent waves
If sensors are spaced too far apart for the relevant wavelengths, distinct wave patterns can become difficult to distinguish from sampled phases. This spatial aliasing can produce misleading correlation or apparent dispersion behaviour. The problem is analogous to sampling a rapidly changing signal too sparsely in time, but here the missing information lies in space.
A survey report should discuss receiver spacings, investigated frequencies and quality criteria. A mathematically neat inversion can still fit an aliased dispersion branch. When apparent velocities jump implausibly at high frequencies or disagree with shorter-spacing estimates, examine spatial sampling before inventing a sudden physical layer. Aliasing is a measurement limitation, not a hidden rock type.
Frequency resolution also depends on recording time
A long recording contains more cycles of low-frequency motion, offering the possibility of better spectral estimates than a very short window. But ambient sources can change throughout the observation, which affects assumptions of stationarity. Analysts divide data into windows and may compare the results to assess stability.
At a frequency of 0.2 Hz, one cycle lasts five seconds. A 10-second window contains only two cycles, providing a fragile basis for a low-frequency correlation claim. A longer window improves cycle coverage but must still be inspected for changing sources and equipment problems. Report accepted duration as well as the total recording time.
Coherence curves deserve uncertainty bands
SPAC coefficients can vary among time windows, station pairs and repeated acquisitions. Averaging produces a central estimate, but the spread is scientifically meaningful. If a model curve fits the mean while falling outside the variation observed across reliable windows, the fit may be less convincing than its graphical appearance suggests.
Ask how uncertainty was calculated and whether samples are independent. Many overlapping spectral windows from one recording are not equivalent to many independent field sites. A model can be overconfident if it treats correlated measurements as independent. Good plots show uncertainty in the observed correlation before they display a finely segmented Vs profile that depends on it.
Rayleigh-wave modes can overlap
A layered ground model can support fundamental and higher-order Rayleigh-wave modes. The conventional simple J0 interpretation is clearest when a suitable single-mode approximation is justified. When multiple modes contribute strongly in the same frequency band, the observed spatial correlation can be a mixture rather than the response of one branch.
An analyst who forces all data onto a fundamental-mode curve may generate a plausible but incorrect phase-velocity sequence. Look for modal evidence, processing choices and comparison with forward models. This is a reason to use multiple scales and complementary methods rather than turning a noisy spectral wiggle into a highly detailed velocity profile.
The source field is not guaranteed to be isotropic
Traffic from a nearby highway, machinery on one side of the array or ocean noise arriving predominantly from one direction can violate the ideal of waves travelling equally from all azimuths. This may distort spatial correlations and cause the standard J0 model to fit poorly or yield biased velocities.
A robust interpretation checks different time windows, directional analyses where applicable and alternative pair groupings. If the results change markedly between daytime traffic and quieter conditions, report that source sensitivity. The method is not invalid merely because real-world noise is directional; the conclusion must reflect which approximations the data can support.
The need for accurate timing and coordinates
SPAC relies on comparing waveforms across sensors. Clock offsets and timing drift can distort spectral phase relationships, while incorrect coordinates change the assumed pair separations. These errors can produce apparent correlations that no subsurface model can explain correctly. Equipment synchronisation and station surveying therefore belong in the quality-control record.
An otherwise elegant J0 fit cannot compensate for a pair distance entered in the wrong unit. If one station was moved during recording, the separation is no longer constant. Good field metadata includes coordinates, timing references, sampling rate, calibration and any station changes. Measurement interpretation begins with accurate geometry.
Why an inversion is not a direct sounding
Once phase-velocity information has been extracted, an inverse model estimates layer velocities and thicknesses that could explain it. Several models can yield similar predicted curves because sensitivity is finite and data may cover only a restricted frequency band. Regularisation or parameter constraints often guide selection among those models.
The final profile is a reasoned model conditional on assumptions, not a photograph of underground rock. A report should include observed dispersion data, fitted model response, uncertainty and the search or regularisation procedure. A single smooth coloured profile without these elements hides where the evidence ends and the analyst’s choices begin.
Identifying the correct Bessel lobe matters
The Bessel J0 function crosses zero more than once and oscillates with decreasing amplitude at increasing argument. If an observed spatial correlation reaches zero, a reader must establish which theoretical lobe it belongs to before converting that event into a phase velocity. Mistaking a later zero for the first changes the inferred argument and thus the velocity.
A plausible phase-velocity trend across neighbouring frequencies and several array radii helps resolve this ambiguity. Noise, imperfect source isotropy and modal mixing can also create apparent crossings. The correct method examines a family of measurements rather than plugging every isolated zero into one formula. A zero is a useful feature, but its interpretation is not automatic.
Why a single radius cannot determine a unique Vs model
A fixed station spacing produces one spatial-correlation relationship per frequency under the processing definition. A layered ground model may affect different frequencies in related ways, but the data do not directly provide one independent depth measurement for each sample. Limited frequency coverage and correlation uncertainty permit multiple combinations of layer thickness and velocity.
Including several radii can improve the constraint, though additional radii do not eliminate all model trade-offs. A report should show what range of Vs models remains consistent with the observations. An attractive colour-coded depth column is only one possible representation of that uncertainty. Where the study needs an engineering parameter, report the sensitivity of that parameter to plausible model choices.
How the depth of investigation is decided
SPAC surveys do not have a fixed penetration depth determined solely by the longest pair separation. Investigation depth depends on observed wavelengths, modal sensitivity, source energy, site velocity structure, noise and inversion constraints. A large array can in principle support long wavelengths, but only if the recorded correlations are useful at the relevant low frequencies.
A defensible report discusses the frequency band and model sensitivity rather than claiming an exact maximum depth from array diameter alone. Deep layers may be poorly resolved even where the best-fit model includes them. Sometimes a broad constraint on average deep velocity is all the data justify. A useful scientific statement names which depth intervals are strongly or weakly informed.
A worked phase-velocity illustration with two radii
Suppose a model with phase velocity c equal to 240 metres per second at 3 Hz is examined using station spacings of 10 and 30 metres. The corresponding dimensionless arguments are 2π times 3 times 10 divided by 240, about 0.785, and 2π times 3 times 30 divided by 240, about 2.356.
Those values occupy very different regions of the J0 curve, even though the frequency and hypothetical soil are unchanged. The longer pair approaches the first zero, while the shorter pair remains in the first positive lobe. This is why multiple pair distances help distinguish wave speeds and why a ‘correlation at 3 Hz’ cannot be interpreted without geometry.
Why the horizontal and vertical components need labels
Many conventional SPAC derivations focus on particular wave components, often the vertical component of Rayleigh-wave motion. Three-component sensors can supply additional information, but the expected correlation function and modal interpretation may differ with component choice. A plot labelled only ‘spatial coherence’ can therefore hide an important modelling assumption.
Read the channel definitions, polarisation processing and whether the method applied a vertical-only, radial or transverse component. Do not use the same J0 formula mechanically for every possible combination. An instrument recording three components does not imply the analyst correctly identified each wave mode. The physical observable must match the theoretical relationship invoked.
Distinguish coherence from phase velocity
An observed coherence curve is a function of correlation or cross-spectral relationships; a derived phase-velocity curve is the propagation-speed interpretation under a model. Both may be shown against frequency, making them easy to confuse. The former is dimensionless or defined by the normalisation, while the latter is expressed in metres per second.
If a journal figure displays a coefficient near zero at 5 Hz, it is incorrect to write that the seismic velocity is zero at 5 Hz. A Bessel crossing corresponds to a phase relationship at a particular separation, not an immobile wave. A properly annotated figure labels both the measured data and the model-derived velocity if both are presented.
The inverse step adds a second model layer
There are two important inference stages: converting spatial-correlation information to dispersion evidence, then converting dispersion evidence to a Vs profile. Each stage has uncertainties. An analyst may fit observed SPAC spectra directly with layered-earth forward predictions instead of first selecting a single dispersion point at every frequency, but the chain remains model-based.
Readers should locate the exact forward model and what it predicts: correlations, phase velocities or another transformed quantity. A good fit to a derived curve may not reflect the original measurement uncertainty if the transformation discarded covariance. Whenever possible, show the original observed coefficients next to the predicted ones. A model should be evaluated where data were measured.
Regularisation can make an uncertain model look smooth
Inversion often requires constraints to prevent unrealistic fluctuations in layer velocity or thickness. Smoothness penalties, limited layer counts, bounds on Vs or prior geological information can guide the model. These choices can be sensible, but they also shape the visual output. A smooth Vs curve is not direct evidence that velocity increases smoothly through the earth.
Compare models built with reasonable alternative regularisation choices. If a sharp contact appears only under one particular parameter setting, it should be qualified. The appropriate geological interpretation combines fit to data with independent observations and stated assumptions. Regularisation is a decision rule for an ill-posed problem, not a hidden additional sensor.
Multiple models may fit the same dispersion curve
Imagine two candidate layered-earth structures producing similar Rayleigh-wave phase velocities across the measured band: one with a thin slow surface layer over moderate material, another with a thicker but slightly faster layer. The dispersion curves might be indistinguishable within observational uncertainty. The underlying profiles can nevertheless yield different predictions outside the measured band.
This is model non-uniqueness. It cannot be solved by printing more decimal places in the final Vs table. Analysts may use ensembles, posterior distributions or multiple starting models to explore plausible solutions. When the output is used for design, sensitivity analysis and comparison with independent measurements are more valuable than a single best-fit line presented as geological certainty.
Vs30 is a summary, not a complete profile
Vs30 is the travel-time-based average shear-wave velocity in the upper 30 metres, commonly defined as 30 divided by the sum of layer thickness divided by layer Vs through that depth. It is a particular scalar derived from a velocity model or suitable measurements. A high value does not automatically establish a simple geological structure, and different profiles can share the same Vs30.
A SPAC inversion may estimate Vs30 if its data adequately constrain the upper 30 metres, but some passive arrays have limited shallow resolution. Where short wavelengths are absent, that derived number may depend strongly on assumptions. Readers should check model sensitivity near the surface before treating a reported Vs30 as equivalent to a direct downhole measurement.
Worked Vs30 calculation with correct travel times
Suppose an illustrative model has 10 metres with Vs equal to 200 metres per second, followed by 20 metres with Vs equal to 400 metres per second. The shear-wave travel times are 10 divided by 200, or 0.05 seconds, and 20 divided by 400, also 0.05 seconds. Total time across 30 metres is 0.10 seconds.
Vs30 is therefore 30 metres divided by 0.10 seconds, or 300 metres per second. The arithmetic average of the two velocities, 300, happens to match here, but that is a coincidence of the chosen thicknesses and speeds; the correct operation is the harmonic travel-time formula. Do not average layer velocities without the thickness and travel-time basis.
Why short-wavelength data are essential to shallow claims
High-frequency surface-wave behaviour often contributes information about shallower portions of the velocity structure, while lower frequencies and longer wavelengths can provide broader constraints on deeper material. The exact sensitivity is model-dependent. An array designed only with long pair spacings may have limited support for the shallowest few metres.
When a report includes a very detailed near-surface velocity profile despite little useful high-frequency information, ask what constrained it. Perhaps an independent borehole or MASW dataset was used; perhaps it was a regularisation assumption. Both are possible, but readers deserve to know. An unobserved shallow layer should not be presented as a uniquely resolved SPAC discovery.
How SPAC differs from ReMi
Both SPAC and ReMi can use ambient vibrations to infer surface-wave dispersion, but they process spatial information differently. ReMi commonly uses a linear array and slowness–frequency representation, where wave arrival direction can make apparent phase velocity exceed true phase velocity. SPAC focuses on spatial correlations and their relationship to a wavefield and array model.
The ReMi guide discusses how choosing a lower-velocity envelope depends on directional illumination. SPAC has its own vulnerability when isotropic averaging assumptions are not met. Neither method can simply substitute another’s graph axes or processing formula. Together they may provide complementary checks, not automatic agreement.
How SPAC differs from MASW
MASW uses multichannel recordings to estimate surface-wave dispersion, often with controlled active sources but also in passive configurations. SPAC uses interstation correlation structure of ambient noise and often exploits approximately circular or otherwise designed receiver arrays. Both may ultimately constrain Vs through dispersion inversion, but their acquisition and processing are different.
The MASW guide emphasises modal picking and a non-unique inversion. If SPAC and MASW yield different velocities, inspect their frequency bands, source conditions, mode assumptions and lateral sampling areas. Disagreement can reveal methodological limitations or real spatial variation, not necessarily incompetent analysis.
How SPAC complements HVSR
HVSR compares horizontal-to-vertical spectral ratios at a station, sometimes showing a predominant site-frequency peak. SPAC uses correlations among spatially separated recordings to infer phase-velocity behaviour. The two methods can constrain different properties of a site model: a possible resonance frequency and a surface-wave dispersion response.
The HVSR guide explains why an H/V peak alone does not uniquely give sediment thickness or absolute amplification. Comparing its frequency with candidate Vs structures from SPAC may improve a geological interpretation, but the result remains conditional on wavefield and modelling assumptions. Distinct data reduce ambiguity only when their physical meanings are kept separate.
What the 2022 microtremor-array research established
A USGS-indexed article, Microtremor Array Method Using Spatial Autocorrelation Analysis of Rayleigh-Wave Data, discusses field planning, data processing and limitations. It demonstrates that SPAC requires informed choices about array configuration, correlation computation and interpretation rather than one universal button marked ‘Vs’.
The original research explains a practical method in the context of real seismic waves. It does not guarantee that every curve from every array can be inverted uniquely. A responsible article cites the actual methodological source, clarifies which idealisations its examples use and avoids claiming that a Bessel fit itself measured a buried soil layer.
What irregular arrays change
A perfectly circular array offers a convenient geometry for azimuthal averaging at selected radii. Real sites may prevent placing sensors on that geometry. Modified processing can accommodate unequal separations or asymmetry under justified models; one approach normalises coherence using the dimensionless product of wavenumber and spacing.
The USGS-indexed KRSPAC work illustrates why irregular geometry is a methodological issue rather than a reason every survey must be discarded. But a reader should not assume a modified technique preserves all properties of conventional SPAC without conditions. Check the processing method and effective pair distribution.
A systematic guide to reading a SPAC figure
First identify the plotted quantity: a correlation coefficient, a dispersion estimate or an inverted velocity profile. Next read frequency units, station spacing or array radius and whether the curves correspond to different pairs. Determine how component signals were combined, whether time windows were screened and what uncertainty bands represent.
Then inspect whether the theoretical model is J0 of a dimensionless frequency–radius–velocity argument, and whether directional-source assumptions are plausible. If the result moves from correlation to phase velocity or Vs, trace each inversion stage. Finally ask which parts of the profile are independently checked by borehole, HVSR or other data. This order keeps measurement separate from interpretation.
An error in reading a negative coefficient
Suppose a coefficient plotted near 8 Hz is minus 0.25. A student writes, ‘The soil has a negative shear-wave velocity at 8 Hz.’ That is wrong. Spatial correlations can be negative because signals have phase relationships under the wavefield model. Propagation speed remains a positive physical magnitude in this setting.
The correct description is that the observed, normalised spatial correlation is negative at the stated frequency and separation. A forward model may predict such a value through the oscillatory Bessel response. The next step is to compare the coefficient with model predictions and uncertainties, not to assign a negative material property.
A second error: confusing dispersion with a geological log
A phase-velocity point of 250 metres per second at a given frequency is not the same as a borehole shear-wave velocity of 250 metres per second at a specific depth. Rayleigh waves sample distributed properties and the phase-velocity relationship must be inverted. Depth information is inferred from model sensitivity, not directly attached to each frequency label.
An accurate result says, ‘The correlation-derived dispersion supports candidate Vs structures with these constraints.’ If a particular layer is important, show what additional observations locate its boundary. Readers who preserve this distinction can understand why two valid studies may report similar dispersion curves but different detailed layering.
Practice: identify which quantity is measured
Imagine a report containing three panels. Panel A shows SPAC coefficients against frequency at several radii. Panel B shows selected phase velocities against frequency. Panel C shows a layered Vs profile with depth. Which quantity is closest to the instrument output? Panel A, after the specified signal processing, represents the observational correlation quantity. Panel B is interpreted through the wavefield model, and Panel C is inferred through the layered-earth inversion.
The separation matters when evaluating uncertainty. If the model fits Panel B but Panel A is poorly resolved, Panel C may be overconfident. A reliable report should carry the limitations from one panel to the next. It should never present Panel C as more directly observed than its underlying evidence.
Common mistakes and repairs
A frequent mistake is treating the correlation coefficient as a velocity or amplitude. Repair it by checking axes and units. Another is assigning a Bessel zero without identifying the correct lobe. Repair it by examining nearby frequencies and multiple radii. A third is treating the wavelength as a directly measured exploration depth. Repair it with model sensitivity analysis.
Further mistakes include ignoring directional noise, treating repeated time windows as independent field sites, overlooking higher surface-wave modes and presenting one inversion model as unique geology. Each has a repair: inspect source conditions, define independent replication, test modal alternatives and compare multiple plausible models. Better interpretations arise from admitting which assumptions make the calculation possible.
Frequently asked questions
What does SPAC stand for? Spatial autocorrelation. Why use ambient noise? It supplies wave energy without controlled shots, subject to source and geometry limits. What does a negative coefficient mean? A correlation phase relationship, not negative velocity. Why does J0 appear? Idealised azimuthal averaging of surface-wave phases produces a Bessel relationship. Is a J0 zero a soil boundary? No.
Does SPAC directly measure Vs? No; it constrains dispersion and a modelled Vs profile. Are circular arrays mandatory? No, but geometry and processing must be compatible. Can one radius solve the whole profile? Generally not uniquely. Does a long baseline always prove deep resolution? No. Is SPAC the same as HVSR or ReMi? No. Can a model fit be accurate yet non-unique? Yes.
Original sources and further learning
The USGS-hosted 2022 overview of microtremor-array SPAC discusses theory, practical array work and method limitations. The USGS research on KRSPAC for asymmetric arrays documents why pair geometry matters. The USGS Bayesian SPAC inversion data release provides an example of treating a velocity profile as an inference with uncertainty.
Readers can compare HVSR resonance curves, ReMi slowness plots, MASW dispersion and cross-linguistic transfer of technical terms. Each method contributes different evidence.
Final lesson: the correlation is measured, the profile is inferred
SPAC turns spatial relationships in ambient ground motion into useful constraints on near-surface wave propagation. Its strength lies in analysing the geometry and phase relationships of wavefields rather than requiring a controlled seismic source. Its limitations lie in source distribution, array sampling, mixed modes and the non-uniqueness of the velocity inversion.
A trustworthy explanation always names the measured coefficient, the Bessel-based wavefield assumption, the estimated dispersion curve and the inferred Vs model as separate stages. When those links are visible, the reader can use the result without mistaking a mathematical fit for a directly observed map of the earth.