Fourth-Order Aperture Statistics

• Fourth-order aperture statistics are scale-dependent observables that compress complex four-point data via finite-aperture filtering.
• They are applied in weak lensing, optical imaging, and turbulence to probe non-Gaussian features and extract higher-order correlations.
• Practical challenges include distinguishing raw moments from connected cumulants, filter design sensitivity, and computational estimation methods.

Fourth-order aperture statistics are fourth-order observables constructed after finite-aperture filtering, finite-aperture integration, or finite detector/reference selection. In weak gravitational lensing, they include the aperture-mass fourth moment $\langle M_{\rm ap}^4\rangle$, its connected part $\langle M_{\rm ap}^4\rangle_{\rm c}$, and mixed moments such as $\langle \mathcal N^3 M_{\rm ap}\rangle$; in optical imaging, they include equal-time fourth-order intensity correlations $G^{(4)}$ measured for finite detector configurations; in wave propagation and turbulence, they include aperture-integrated variances whose evaluation requires fourth-order coherence functions. Across these uses, the common feature is that an aperture or aperture-like window compresses a high-dimensional four-point structure into a scale-dependent statistic, but the precise meaning of “fourth order” depends on whether the observable is a raw fourth moment, a connected cumulant, a correlation function, or a variance of an aperture-integrated bilinear field quantity (Peel et al., 2018, Silvestre-Rosello et al., 9 Sep 2025, Pearce et al., 2015, Charnotskii, 2018).

1. Domain and conceptual scope

The literature does not use a single universal definition of fourth-order aperture statistics. Instead, several technically distinct constructions recur.

Domain	Representative fourth-order statistic	Aperture object
Weak lensing	$\langle M_{\rm ap}^4\rangle$, $\langle M_{\rm ap}^4\rangle_{\rm c}$	Compensated aperture-mass filter
Higher-order intensity imaging	$G^{(4)}$, $\mathfrak G^{(4)}$	Detector/reference geometry for a source aperture
OAM through turbulence	$\sigma_{\rm OAM}^2$	Finite receiving aperture $A(\mathbf r)$
Wave propagation in random media	Fourth-order field and intensity moments	Finite-window smoothing or aperture integration
Galaxy-galaxy lensing	$\langle M_{\rm ap}^4\rangle_{\rm c}$0	Joint galaxy-count and shear apertures

In weak lensing, the aperture is usually a compensated circular filter acting on convergence or shear. In higher-order intensity imaging, the parameter of interest can be the physical radius of a circular thermal-light aperture, inferred from spatial equal-time intensity correlations. In aperture-averaged orbital angular momentum (OAM), the aperture is the receiver transmission function $\langle M_{\rm ap}^4\rangle_{\rm c}$1, and the fourth-order object enters because the OAM variance depends on a fourth-order coherence function rather than only on ordinary second-order coherence. In random-media wave propagation, finite-window or smoothed measurements reduce to integrals of fourth-order field or intensity moments. In galaxy-galaxy lensing, the fourth-order aperture statistic correlates three filtered foreground galaxy-density factors with one filtered lensing factor (Pearce et al., 2015, Charnotskii, 2018, Garnier et al., 2014, Oel et al., 20 Apr 2026).

A recurring implication is that “fourth-order aperture statistics” are best understood as a family of aperture-compressed four-point observables rather than a single canonical estimator. This suggests that comparisons across subfields require careful attention to whether the statistic is connected or disconnected, normalized or unnormalized, and map-space or correlation-function based.

2. Weak-lensing aperture mass and the fourth-order moment

In weak lensing, the basic aperture statistic is the aperture mass

$\langle M_{\rm ap}^4\rangle_{\rm c}$2

with compensation condition

$\langle M_{\rm ap}^4\rangle_{\rm c}$3

This compensation removes sensitivity to the mass-sheet degeneracy. In the modified-gravity study of higher weak-lensing moments, all measurements were performed directly on simulated pixellated convergence maps $\langle M_{\rm ap}^4\rangle_{\rm c}$4, not on shear catalogues. The implementation used the starlet transform, i.e. an isotropic undecimated wavelet transform, whose aperture filter is compensated, compactly supported, non-oscillatory, and localized in both map and Fourier space. A single transform produced aperture-mass maps at dyadic scales $\langle M_{\rm ap}^4\rangle_{\rm c}$5 pixels; for $\langle M_{\rm ap}^4\rangle_{\rm c}$6 maps with pixel scale $\langle M_{\rm ap}^4\rangle_{\rm c}$7 arcsec, these were $\langle M_{\rm ap}^4\rangle_{\rm c}$8 (Peel et al., 2018).

The fourth-order aperture statistic in that work was not a standardized kurtosis, not an excess kurtosis, and not a connected fourth-order cumulant. It was the non-standardized central fourth moment of the filtered map,

$\langle M_{\rm ap}^4\rangle_{\rm c}$9

The same map-space construction was used for the second- and third-order moments. In the paper’s own interpretive language, the fourth-order moment was treated as “kurtosis,” with positive kurtosis implying a higher peak and larger wings than a Gaussian with the same mean and variance, and negative kurtosis implying a wider peak and shorter wings; however, in modern statistical language the implemented estimator is more precisely a raw central fourth moment (Peel et al., 2018).

A separate computational line established that the aperture mass statistic is formally identical to a wavelet transform of the convergence map at the corresponding scale, provided one chooses an appropriate compensated wavelet. This identification is especially important for fourth-order work because same-scale and cross-scale fourth moments can then be regarded as fourth-order statistics of wavelet bands. The starlet or isotropic undecimated wavelet transform was emphasized as advantageous because it is localized in real and Fourier space and admits very fast algorithms. On a $\langle \mathcal N^3 M_{\rm ap}\rangle$0 image, reported speed-up factors were $\langle \mathcal N^3 M_{\rm ap}\rangle$1 for aperture radii from $\langle \mathcal N^3 M_{\rm ap}\rangle$2 to $\langle \mathcal N^3 M_{\rm ap}\rangle$3 pixels, relative to usual aperture-mass algorithms (Leonard et al., 2012).

3. Connected fourth-order aperture statistics, shear 4PCFs, and trispectra

A distinct fourth-order weak-lensing formalism treats $\langle \mathcal N^3 M_{\rm ap}\rangle$4 as a filtered version of the shear four-point correlation function (4PCF) or of the convergence trispectrum. Using the complex aperture measure

$\langle \mathcal N^3 M_{\rm ap}\rangle$5

with the Gaussian/Crittenden filter

$\langle \mathcal N^3 M_{\rm ap}\rangle$6

the general $\langle \mathcal N^3 M_{\rm ap}\rangle$7-th order aperture moment can be written in real space in terms of shear $\langle \mathcal N^3 M_{\rm ap}\rangle$8-point functions and in Fourier space in terms of convergence polyspectra. For $\langle \mathcal N^3 M_{\rm ap}\rangle$9,

$G^{(4)}$0

The relevant real-space object is the shear 4PCF, decomposed into eight natural components $G^{(4)}$1, $G^{(4)}$2. After integrating out quadrilateral center of mass and global orientation, the fourth-order aperture measures become 5D integrals over three side lengths and two angles, with explicitly derived kernels $G^{(4)}$3. These complex measures are then linearly combined to obtain the pure $G^{(4)}$4-mode statistic and the corresponding mixed $G^{(4)}$5-mode quantities (Silvestre-Rosello et al., 9 Sep 2025).

The crucial conceptual distinction in this formalism is between the full fourth-order moment and its connected part. Because $G^{(4)}$6 by isotropy, the disconnected contributions at $G^{(4)}$7 come from second-order pairings. For a Gaussian random field,

$G^{(4)}$8

so the connected part vanishes. In cosmological applications the target statistic is therefore

$G^{(4)}$9

for the pure $\langle M_{\rm ap}^4\rangle$0-mode equal-scale non-tomographic case. The connected statistic isolates genuinely non-Gaussian information and was the quantity used in the Fisher analysis. That analysis found only a minimal improvement when $\langle M_{\rm ap}^4\rangle$1 was added to a $\langle M_{\rm ap}^4\rangle$2 joint analysis for a DES-Y3-like, non-tomographic, equal-scale setup (Silvestre-Rosello et al., 9 Sep 2025).

4. Estimation strategies and empirical performance

Fourth-order aperture statistics are computationally difficult if they are estimated through brute-force four-point correlation counting. One direct-from-catalogue route rewrites aperture-moment estimators of arbitrary order into linear sums over galaxies within each aperture. For equal radii, the fourth-order estimator takes the explicit form

$\langle M_{\rm ap}^4\rangle$3

where $\langle M_{\rm ap}^4\rangle$4 and $\langle M_{\rm ap}^4\rangle$5 are power sums built from weighted tangential ellipticities and aperture filters. This is the fourth-order specialization of a Bell-polynomial construction and scales linearly in the number of galaxies inside an aperture. Applied to Gaussian mocks, the method accurately recovered equal-scale and multiscale fourth-order statistics; applied to SLICS $\langle M_{\rm ap}^4\rangle$6-body mocks, it indicated that connected fourth-order aperture statistics should be detectable in a KiDS-1000-like survey, with signal-to-noise peaking around $\langle M_{\rm ap}^4\rangle$7 and with higher-order estimation requiring aperture oversampling $\langle M_{\rm ap}^4\rangle$8 (Porth et al., 2021).

A separate weak-lensing programme developed an efficient estimator for the shear 4PCF itself by using a multipole decomposition of the natural 4PCF components. The estimator was reported to scale quadratically with galaxy number, and the associated aperture-mass pipeline was validated on Gaussian random fields and on realistic $\langle M_{\rm ap}^4\rangle$9-body simulations. In DES Y3, the method produced a significant detection of the connected pure $\langle M_{\rm ap}^4\rangle_{\rm c}$0-mode fourth-order aperture mass, $\langle M_{\rm ap}^4\rangle_{\rm c}$1. The null hypothesis of no connected $\langle M_{\rm ap}^4\rangle_{\rm c}$2-mode signal was rejected with

$\langle M_{\rm ap}^4\rangle_{\rm c}$3

while the nominally null systematics-sensitive modes were not highly significant. The same study also found that the sampling distribution of the fourth-order aperture mass is significantly skewed, with skewness increasing toward larger aperture radii (Porth et al., 9 Sep 2025).

Fourth-order aperture statistics also appear in mixed galaxy-galaxy-lensing form. The fourth-order mixed aperture moment

$\langle M_{\rm ap}^4\rangle_{\rm c}$4

correlates three aperture-filtered foreground galaxy-density factors with one aperture-filtered lensing factor and probes the projected galaxy-galaxy-galaxy-mass trispectrum. The 4PCF-to-aperture transformation and an FFT-based direct estimator were derived explicitly. In a mock stage-IV setup on $\langle M_{\rm ap}^4\rangle_{\rm c}$5, the connected part of $\langle M_{\rm ap}^4\rangle_{\rm c}$6 was detected with a signal-to-noise ratio of roughly nine on small aperture scales, and the numerical integration pipeline achieved sub-percent accuracy on relevant scales (Oel et al., 20 Apr 2026).

5. Statistical optics, coherence theory, and turbulence-driven variants

In far-field imaging with incoherent thermal light, the fourth-order aperture statistic is a spatial equal-time intensity correlation. The measured sample observable is

$\langle M_{\rm ap}^4\rangle_{\rm c}$7

and its expectation is the unnormalized fourth-order intensity correlation $\langle M_{\rm ap}^4\rangle_{\rm c}$8. For a uniform circular source of angular diameter $\langle M_{\rm ap}^4\rangle_{\rm c}$9, the degree of coherence is

$G^{(4)}$0

In the repeated-reference geometry $G^{(4)}$1,

$G^{(4)}$2

When used with a multivariate-normal likelihood, a full covariance model, and maximum-likelihood estimation of the source radius, fourth-order processing could outperform second-order HBT-type processing for suitable reference-pixel geometry, although in the circular-aperture demonstrator it did not outperform the third-order correlation (Pearce et al., 2015).

In turbulence-induced OAM fluctuations, the central aperture statistic is the normalized variance

$G^{(4)}$3

where $G^{(4)}$4 is the aperture-integrated intrinsic OAM and $G^{(4)}$5 is the mean power through the aperture. This is a fourth-order quantity because the local OAM density is bilinear in the field, so $G^{(4)}$6 involves four field factors. The variance is governed by the fourth-order coherence function

$G^{(4)}$7

For circularly symmetric apertures, the first-order perturbative contribution vanishes, $G^{(4)}$8, and the leading weak-fluctuation term is second order. The paper further showed that aperture averaging of OAM does not obey the standard square-root law and that the weak/strong fluctuation transition is controlled by aperture-dependent criteria involving the coherence-radius Fresnel number $G^{(4)}$9 and aperture Fresnel number $\mathfrak G^{(4)}$0, not by the scintillation index alone (Charnotskii, 2018).

In the white-noise paraxial regime for wave propagation in random media, the central fourth-order object is the field moment

$\mathfrak G^{(4)}$1

Within the scintillation regime, the centered fourth-order moments satisfy the Gaussian summation rule asymptotically, and explicit formulas were derived for the intensity covariance and for the variance of the smoothed Wigner transform. The smoothed Wigner transform is an aperture-like observable because it averages the random field over finite spatial and angular windows. A practical implication is that many fourth-order finite-window statistics in this regime reduce asymptotically to combinations of second-order quantities (Garnier et al., 2014).

6. Interpretation, limitations, and recurrent distinctions

The main terminological ambiguity is between raw fourth moments and connected fourth-order cumulants. In the modified-gravity weak-lensing study, $\mathfrak G^{(4)}$2 was a non-standardized central fourth moment measured directly on filtered maps. In the later 4PCF-based formalism, the cosmologically relevant statistic was the connected quantity $\mathfrak G^{(4)}$3, obtained by subtracting disconnected Gaussian terms. These are not interchangeable objects, even though both are described informally as fourth-order aperture statistics (Peel et al., 2018, Silvestre-Rosello et al., 9 Sep 2025).

A second recurring distinction is between mean response and discriminatory or inferential utility. In the modified-gravity simulations, the aperture-mass kurtosis often showed larger mean departures from $\mathfrak G^{(4)}$4CDM than the variance and decreased systematically as neutrino mass increased in the $\mathfrak G^{(4)}$5 family. Yet, once the full realization-by-realization distributions were examined, the fourth-order moment proved to be a poor discriminator: for $\mathfrak G^{(4)}$6 with $\mathfrak G^{(4)}$7, it did not exceed $\mathfrak G^{(4)}$8 discrimination efficiency at any filter scale or redshift; for the more degenerate $\mathfrak G^{(4)}$9 with $\sigma_{\rm OAM}^2$0 eV, no aperture-mass moment up to fourth order distinguished the model from $\sigma_{\rm OAM}^2$1CDM, while peak counts reached $\sigma_{\rm OAM}^2$2 efficiency at $\sigma_{\rm OAM}^2$3 and $\sigma_{\rm OAM}^2$4 (Peel et al., 2018).

A third distinction concerns detectability versus cosmological leverage. The DES Y3 4PCF analysis reported a significant detection of the connected fourth-order aperture mass and found no compelling evidence for $\sigma_{\rm OAM}^2$5-mode or parity-violating contamination at fourth order. By contrast, the Fisher forecast based on equal-scale, non-tomographic aperture statistics found only minimal improvement when $\sigma_{\rm OAM}^2$6 was added to $\sigma_{\rm OAM}^2$7. These results are not inconsistent: the forecast itself emphasized its restricted setup, its Gaussian-likelihood approximation, and the likelihood that unequal-scale combinations or tomography could alter the information content, while the DES application emphasized that the sampling distribution of $\sigma_{\rm OAM}^2$8 is significantly skewed and that Gaussian likelihood assumptions are questionable (Silvestre-Rosello et al., 9 Sep 2025, Porth et al., 9 Sep 2025).

A final recurring theme is that fourth-order aperture statistics are highly sensitive to filter design, binning, and covariance modeling. In weak lensing, finite radial integration ranges induce $\sigma_{\rm OAM}^2$9 mixing, sharply peaked quadrilateral configurations make 4PCF-to-aperture transforms numerically delicate, and direct aperture estimators are sensitive to coverage fraction, oversampling, and edge handling. In noisy map-based analyses, fourth-order moments can remain modestly informative but may respond non-monotonically to added Gaussian noise. In optical imaging, covariance modeling for $A(\mathbf r)$0-based inference requires eighth-order moments. These recurring technical constraints explain why fourth-order aperture statistics are often most powerful when paired with carefully chosen geometries, explicit connected/disconnected separation, and numerical pipelines designed around the specific four-point structure of the filtered observable (Peel et al., 2018, Pearce et al., 2015, Silvestre-Rosello et al., 9 Sep 2025).