Hybrid Spherical-Plane Wave Model (HSPWM)

• Hybrid Spherical-Plane Wave Model (HSPWM) is a modeling pattern that dynamically selects either spherical or plane-wave approximations based on geometry, channel gain, and effective rank.
• It employs explicit demarcation and subarray partitioning to optimize modeling accuracy and computational efficiency in extremely large-scale arrays and THz UM-MIMO environments.
• HSPWM underpins diverse applications from XL-array communications and IRS systems to wavefield modeling and galaxy clustering, providing tailored solutions across different physical domains.

Hybrid Spherical-Plane Wave Model (HSPWM) denotes a class of hybrid representations that combine spherical-wave and plane-wave descriptions when neither a purely spherical-wave model (SWM) nor a purely plane-wave model (PWM) is adequate over the full domain of interest. In extremely large-scale array communications, HSPWM is constructed from explicit PWM/SWM demarcations based on channel gain and effective rank, and it selects SWM or PWM according to geometry-dependent thresholds (Li et al., 2023). In Terahertz ultra-massive MIMO, a closely related formulation, often written as HSPM, uses PWM within subarrays and SWM among subarrays (Chen et al., 2021). The same label, or a structurally analogous hybridization, also appears in integrated UM-MIMO–IRS channels, modular XL-array localization, distributed multi-UAV near-field communications, active-source wavefield modeling, Gaussian-basis scattering theory, and hybrid-basis galaxy clustering inference (Chen et al., 2022, Zhang et al., 18 Apr 2025, Huo et al., 23 Aug 2025, Bhaumik et al., 2024, Mahato et al., 18 May 2026, Wang et al., 2020).

1. Terminology and scope

The literature does not use HSPWM as a single standardized construction. Instead, the recurring idea is to combine a spherical description where curvature, range variation, or global geometry matters, with a plane-wave description where local structure or computational tractability makes planarization valid. This suggests that HSPWM is best understood as a modeling pattern rather than a unique formula.

Domain	Hybridization principle	Representative paper
XL-array communications	SWM inside demarcated near-field region, PWM outside	(Li et al., 2023)
THz UM-MIMO	PWM within subarray, SWM among subarrays	(Chen et al., 2021)
UM-MIMO–IRS cascaded channels	Subarray-wise planar steering with spherical inter-subarray phases	(Chen et al., 2022)
Modular XL-array localization	Planar steering within each SA, spherical variation across SAs	(Zhang et al., 18 Apr 2025)
Multi-UAV near-field communications	Kronecker form: spherical across UAVs, planar within each UPA	(Huo et al., 23 Aug 2025)
Layered half-space wavefields	Cylindrical Hankel source field with plane-wave layer solution	(Bhaumik et al., 2024)
Quantum scattering / chemistry	SGTOs combined with plane-wave-modulated SGTOs	(Mahato et al., 18 May 2026)
Galaxy clustering	Spherical Fourier modes at low $k$, plane-wave FFT block at high $k$	(Wang et al., 2020)

Within this broad family, the most explicit decision-theoretic formulation is the XL-array communication model derived from PWM/SWM applicability boundaries. Other papers instead implement hybridization through subarray partitioning, basis design, or asymptotic matching.

2. Demarcation-based HSPWM in extremely large-scale arrays

For XL-array communications, the underlying question is not whether SWM or PWM is universally preferable, but where each is applicable. The baseline PWM assumes that the transmitter–receiver distance is sufficiently large that the wavefront is locally planar across the array, so path gain depends on a common distance $r$ and phase slope is determined by an incident angle. For a ULA, the far-field steering vector is

$$\bar h_n=\frac{\lambda}{4\pi r}\exp\!\Bigl(-j\frac{2\pi}{\lambda}\bigl(r-d(n-\tfrac{N-1}{2})\sin\theta\bigr)\Bigr).$$

The SWM removes the planar approximation and assigns a distinct propagation distance $r_n$ to each element:

$$h_n=\frac{\lambda}{4\pi r_n}\exp\!\bigl(-j\frac{2\pi}{\lambda}r_n\bigr),$$

with

$$r_n=\sqrt{r^2-2rd\,(n-\tfrac{N-1}{2})\sin\theta+\bigl(d(n-\tfrac{N-1}{2})\bigr)^2 }.$$

These definitions are treated for four single-LoS setups: point-to-ULA, point-to-UPA, ULA-to-ULA, and ULA-to-UPA (Li et al., 2023).

For point-to-ULA, the normalized received-power ratio under SWM versus PWM with MRC is

$$\mu(r,\theta)=\frac{G_{\rm near}(r,\theta)}{G_{\rm far}(r,\theta)} =\frac{r^2}{N}\sum_{n=1}^N\frac{1}{r_n^2}.$$

A closed form is

$$\mu(r,\theta) =\frac{r}{N d\cos\theta}\Biggl[ \arctan\!\Bigl(\frac{N d}{2r\cos\theta}+\tan\theta\Bigr) +\arctan\!\Bigl(\frac{N d}{2r\cos\theta}-\tan\theta\Bigr) \Biggr].$$

Analysis of $\partial^2\mu/\partial r^2$ yields an inflection point at $k$0. For $k$1, $k$2 always, so PWM uniformly over-estimates SWM power; for $k$3, $k$4 first exceeds $k$5 and then returns to $k$6 (Li et al., 2023).

For point-to-UPA, the analogous quantity $k$7 produces an equi-power surface. In the uniform circular PA case with $k$8, the dividing curve is

$k$9

When $r$0, PWM always over-estimates, whereas for $r$1, $r$2 first exceeds $r$3 and then converges to $r$4.

For MIMO, the paper adopts the effective-rank metric

$r$5

Under pure PWM LoS MIMO, $r$6; under SWM, $r$7 in the near field. For ULA-to-ULA, the threshold surface is approximated by

$r$8

and for ULA-to-UPA an upper bound is

$r$9

The same study also notes that scatterers cause $$\bar h_n=\frac{\lambda}{4\pi r}\exp\!\Bigl(-j\frac{2\pi}{\lambda}\bigl(r-d(n-\tfrac{N-1}{2})\sin\theta\bigr)\Bigr).$$0 to fluctuate and extend the near-field region when scatterer power rises (Li et al., 2023).

These results lead directly to a regime-switching HSPWM. For point-to-ULA, one uses SWM if $$\bar h_n=\frac{\lambda}{4\pi r}\exp\!\Bigl(-j\frac{2\pi}{\lambda}\bigl(r-d(n-\tfrac{N-1}{2})\sin\theta\bigr)\Bigr).$$1, where $$\bar h_n=\frac{\lambda}{4\pi r}\exp\!\Bigl(-j\frac{2\pi}{\lambda}\bigl(r-d(n-\tfrac{N-1}{2})\sin\theta\bigr)\Bigr).$$2 solves $$\bar h_n=\frac{\lambda}{4\pi r}\exp\!\Bigl(-j\frac{2\pi}{\lambda}\bigl(r-d(n-\tfrac{N-1}{2})\sin\theta\bigr)\Bigr).$$3, and PWM otherwise. In indicator form,

$$\bar h_n=\frac{\lambda}{4\pi r}\exp\!\Bigl(-j\frac{2\pi}{\lambda}\bigl(r-d(n-\tfrac{N-1}{2})\sin\theta\bigr)\Bigr).$$4

For other geometries, $$\bar h_n=\frac{\lambda}{4\pi r}\exp\!\Bigl(-j\frac{2\pi}{\lambda}\bigl(r-d(n-\tfrac{N-1}{2})\sin\theta\bigr)\Bigr).$$5 is replaced by the appropriate elementwise distance.

3. Subarray-based channel modeling in THz UM-MIMO and IRS systems

A second major interpretation of HSPWM appears in THz ultra-massive MIMO, where the hybridization is spatially hierarchical rather than boundary-based. The key assumption is that a subarray aperture is small enough for a plane-wave approximation to hold within the subarray, while the distances and angles among subarray reference antennas require spherical modeling. In this formulation, the $$\bar h_n=\frac{\lambda}{4\pi r}\exp\!\Bigl(-j\frac{2\pi}{\lambda}\bigl(r-d(n-\tfrac{N-1}{2})\sin\theta\bigr)\Bigr).$$6 block of path $$\bar h_n=\frac{\lambda}{4\pi r}\exp\!\Bigl(-j\frac{2\pi}{\lambda}\bigl(r-d(n-\tfrac{N-1}{2})\sin\theta\bigr)\Bigr).$$7 is

$$\bar h_n=\frac{\lambda}{4\pi r}\exp\!\Bigl(-j\frac{2\pi}{\lambda}\bigl(r-d(n-\tfrac{N-1}{2})\sin\theta\bigr)\Bigr).$$8

where $$\bar h_n=\frac{\lambda}{4\pi r}\exp\!\Bigl(-j\frac{2\pi}{\lambda}\bigl(r-d(n-\tfrac{N-1}{2})\sin\theta\bigr)\Bigr).$$9 retains exact distance-dependent phase and amplitude, and $r_n$0 and $r_n$1 are planar steering vectors within the corresponding subarrays (Chen et al., 2021).

This model was evaluated against full SWM by the normalized Frobenius errors

$r_n$2

For $r_n$3 THz, subarray spacing $r_n$4, and $r_n$5, $r_n$6 is up to $r_n$7 dB lower than $r_n$8 at $r_n$9 m. The parameter count is also intermediate: SWM requires $$h_n=\frac{\lambda}{4\pi r_n}\exp\!\bigl(-j\frac{2\pi}{\lambda}r_n\bigr),$$0 real parameters, PWM requires $$h_n=\frac{\lambda}{4\pi r_n}\exp\!\bigl(-j\frac{2\pi}{\lambda}r_n\bigr),$$1, and HSPM requires

$$h_n=\frac{\lambda}{4\pi r_n}\exp\!\bigl(-j\frac{2\pi}{\lambda}r_n\bigr),$$2

The same paper develops a two-phase channel-estimation mechanism using a DCNN for reference subarrays and geometric extrapolation to the remaining subarray pairs (Chen et al., 2021).

The integrated UM-MIMO–IRS case extends the same idea to cascaded channels. The near-/far-field partition is expressed through the Rayleigh distance

$$h_n=\frac{\lambda}{4\pi r_n}\exp\!\bigl(-j\frac{2\pi}{\lambda}r_n\bigr),$$3

For the HSPWM channel matrix,

$$h_n=\frac{\lambda}{4\pi r_n}\exp\!\bigl(-j\frac{2\pi}{\lambda}r_n\bigr),$$4

For the cascaded channel $$h_n=\frac{\lambda}{4\pi r_n}\exp\!\bigl(-j\frac{2\pi}{\lambda}r_n\bigr),$$5, the rank obeys

$$h_n=\frac{\lambda}{4\pi r_n}\exp\!\bigl(-j\frac{2\pi}{\lambda}r_n\bigr),$$6

Numerically, at $$h_n=\frac{\lambda}{4\pi r_n}\exp\!\bigl(-j\frac{2\pi}{\lambda}r_n\bigr),$$7 m and $$h_n=\frac{\lambda}{4\pi r_n}\exp\!\bigl(-j\frac{2\pi}{\lambda}r_n\bigr),$$8, the capacity of HSPWM deviates from full SWM by only $$h_n=\frac{\lambda}{4\pi r_n}\exp\!\bigl(-j\frac{2\pi}{\lambda}r_n\bigr),$$9 bits/s/Hz, whereas PWM is approximately $$r_n=\sqrt{r^2-2rd\,(n-\tfrac{N-1}{2})\sin\theta+\bigl(d(n-\tfrac{N-1}{2})\bigr)^2 }.$$0 bits/s/Hz worse. The same work introduces a subarray-based sparse representation and two compressive-sensing estimation algorithms, separate-side estimation (SSE) and dictionary-shrinkage estimation (DSE); DSE is reported to be about $$r_n=\sqrt{r^2-2rd\,(n-\tfrac{N-1}{2})\sin\theta+\bigl(d(n-\tfrac{N-1}{2})\bigr)^2 }.$$1 dB better than SSE at low SNR (Chen et al., 2022).

4. Localization, compressed sensing, and tensor formulations

In modular XL-array localization, HSPWM is tied to spatial non-stationarity (SNS). A base station with $$r_n=\sqrt{r^2-2rd\,(n-\tfrac{N-1}{2})\sin\theta+\bigl(d(n-\tfrac{N-1}{2})\bigr)^2 }.$$2 subarrays uses planar steering within each subarray and spherical variation of $$r_n=\sqrt{r^2-2rd\,(n-\tfrac{N-1}{2})\sin\theta+\bigl(d(n-\tfrac{N-1}{2})\bigr)^2 }.$$3 across subarrays. Visibility regions are modeled by binary indicators $$r_n=\sqrt{r^2-2rd\,(n-\tfrac{N-1}{2})\sin\theta+\bigl(d(n-\tfrac{N-1}{2})\bigr)^2 }.$$4. The resulting localization pipeline has three stages: visible-SA selection and AoA estimation via SOMP, coarse 3-D position estimation via weighted least squares (WLS), and reduced-dictionary CS followed by a final WLS refinement. The paper reports that SA interval $$r_n=\sqrt{r^2-2rd\,(n-\tfrac{N-1}{2})\sin\theta+\bigl(d(n-\tfrac{N-1}{2})\bigr)^2 }.$$5 must balance angular diversity and link quality, with a simulation optimum of about $$r_n=\sqrt{r^2-2rd\,(n-\tfrac{N-1}{2})\sin\theta+\bigl(d(n-\tfrac{N-1}{2})\bigr)^2 }.$$6 m in a $$r_n=\sqrt{r^2-2rd\,(n-\tfrac{N-1}{2})\sin\theta+\bigl(d(n-\tfrac{N-1}{2})\bigr)^2 }.$$7 grid at THz $$r_n=\sqrt{r^2-2rd\,(n-\tfrac{N-1}{2})\sin\theta+\bigl(d(n-\tfrac{N-1}{2})\bigr)^2 }.$$8 GHz; with total $$r_n=\sqrt{r^2-2rd\,(n-\tfrac{N-1}{2})\sin\theta+\bigl(d(n-\tfrac{N-1}{2})\bigr)^2 }.$$9 elements, the reported optimum is $$\mu(r,\theta)=\frac{G_{\rm near}(r,\theta)}{G_{\rm far}(r,\theta)} =\frac{r^2}{N}\sum_{n=1}^N\frac{1}{r_n^2}.$$0, $$\mu(r,\theta)=\frac{G_{\rm near}(r,\theta)}{G_{\rm far}(r,\theta)} =\frac{r^2}{N}\sum_{n=1}^N\frac{1}{r_n^2}.$$1 (Zhang et al., 18 Apr 2025).

In distributed multi-UAV near-field communications, HSPWM is also called the cross-field model. The array response is approximated by a Kronecker product

$$\mu(r,\theta)=\frac{G_{\rm near}(r,\theta)}{G_{\rm far}(r,\theta)} =\frac{r^2}{N}\sum_{n=1}^N\frac{1}{r_n^2}.$$2

where $$\mu(r,\theta)=\frac{G_{\rm near}(r,\theta)}{G_{\rm far}(r,\theta)} =\frac{r^2}{N}\sum_{n=1}^N\frac{1}{r_n^2}.$$3 captures the planar manifold within each UAV subarray and $$\mu(r,\theta)=\frac{G_{\rm near}(r,\theta)}{G_{\rm far}(r,\theta)} =\frac{r^2}{N}\sum_{n=1}^N\frac{1}{r_n^2}.$$4 captures spherical variation across UAVs. Under MRC, the SNR becomes

$$\mu(r,\theta)=\frac{G_{\rm near}(r,\theta)}{G_{\rm far}(r,\theta)} =\frac{r^2}{N}\sum_{n=1}^N\frac{1}{r_n^2}.$$5

which reduces to a single double sum over UAV indices rather than the four-fold sum of full SWM. The paper states that $$\mu(r,\theta)=\frac{G_{\rm near}(r,\theta)}{G_{\rm far}(r,\theta)} =\frac{r^2}{N}\sum_{n=1}^N\frac{1}{r_n^2}.$$6 nearly coincides with $$\mu(r,\theta)=\frac{G_{\rm near}(r,\theta)}{G_{\rm far}(r,\theta)} =\frac{r^2}{N}\sum_{n=1}^N\frac{1}{r_n^2}.$$7 over all tested angles and spacings, while retaining lower analytical complexity. This structure yields a rank-1 tensor representation for each path and motivates tensor-OMP; simulation results show tensor-OMP achieves NMSE comparable to spherical-domain OMP (SD-OMP), with reduced computational complexity and improved scalability (Huo et al., 23 Aug 2025).

These algorithmic developments show that HSPWM is not only a forward model. It also defines the geometry of the inference problem: support sets in CS, visible-region selection, reduced dictionaries, and low-rank tensor atoms all inherit the specific way spherical and planar components are separated.

5. Extensions beyond wireless communications

The same hybrid logic appears in several other fields, although the mathematical objects differ substantially.

In active-source wavefield modeling for a layered half-space, the model replaces the usual planar-wave assumption of free-vibration forward models with a cylindrically spreading source field described by Hankel functions, while retaining a plane-wave-based layered eigenproblem. The vertical and radial surface responses are written as modal sums of $$\mu(r,\theta)=\frac{G_{\rm near}(r,\theta)}{G_{\rm far}(r,\theta)} =\frac{r^2}{N}\sum_{n=1}^N\frac{1}{r_n^2}.$$8 and $$\mu(r,\theta)=\frac{G_{\rm near}(r,\theta)}{G_{\rm far}(r,\theta)} =\frac{r^2}{N}\sum_{n=1}^N\frac{1}{r_n^2}.$$9 terms, including both propagating and decaying modes. Reported runtimes are at least two orders of magnitude faster than numerical methods; for example, Profile I gives HSPWM $$\mu(r,\theta) =\frac{r}{N d\cos\theta}\Biggl[ \arctan\!\Bigl(\frac{N d}{2r\cos\theta}+\tan\theta\Bigr) +\arctan\!\Bigl(\frac{N d}{2r\cos\theta}-\tan\theta\Bigr) \Biggr].$$0 s versus SGFD $$\mu(r,\theta) =\frac{r}{N d\cos\theta}\Biggl[ \arctan\!\Bigl(\frac{N d}{2r\cos\theta}+\tan\theta\Bigr) +\arctan\!\Bigl(\frac{N d}{2r\cos\theta}-\tan\theta\Bigr) \Biggr].$$1 s, and Profile II gives HSPWM $$\mu(r,\theta) =\frac{r}{N d\cos\theta}\Biggl[ \arctan\!\Bigl(\frac{N d}{2r\cos\theta}+\tan\theta\Bigr) +\arctan\!\Bigl(\frac{N d}{2r\cos\theta}-\tan\theta\Bigr) \Biggr].$$2 s versus SGFD $$\mu(r,\theta) =\frac{r}{N d\cos\theta}\Biggl[ \arctan\!\Bigl(\frac{N d}{2r\cos\theta}+\tan\theta\Bigr) +\arctan\!\Bigl(\frac{N d}{2r\cos\theta}-\tan\theta\Bigr) \Biggr].$$3 s. The method also captures modal osculation and leaky waves (Bhaumik et al., 2024).

In quantum physics and chemistry, HSPWM denotes an analytic framework for free-particle Green’s-function matrix elements over spherical Gaussian-type orbitals (SGTOs) and plane-wave-modulated SGTOs. Plane-wave modulation shifts Gaussian centers into the complex domain, with $$\mu(r,\theta) =\frac{r}{N d\cos\theta}\Biggl[ \arctan\!\Bigl(\frac{N d}{2r\cos\theta}+\tan\theta\Bigr) +\arctan\!\Bigl(\frac{N d}{2r\cos\theta}-\tan\theta\Bigr) \Biggr].$$4 for the $$\mu(r,\theta) =\frac{r}{N d\cos\theta}\Biggl[ \arctan\!\Bigl(\frac{N d}{2r\cos\theta}+\tan\theta\Bigr) +\arctan\!\Bigl(\frac{N d}{2r\cos\theta}-\tan\theta\Bigr) \Biggr].$$5-type case, and the general PW-SGTO matrix elements reduce to SGTO core integrals evaluated at the complex separation vector

$$\mu(r,\theta) =\frac{r}{N d\cos\theta}\Biggl[ \arctan\!\Bigl(\frac{N d}{2r\cos\theta}+\tan\theta\Bigr) +\arctan\!\Bigl(\frac{N d}{2r\cos\theta}-\tan\theta\Bigr) \Biggr].$$6

The framework provides recurrence relations for two-center radial integrals and asymptotic expansions for large $$\mu(r,\theta) =\frac{r}{N d\cos\theta}\Biggl[ \arctan\!\Bigl(\frac{N d}{2r\cos\theta}+\tan\theta\Bigr) +\arctan\!\Bigl(\frac{N d}{2r\cos\theta}-\tan\theta\Bigr) \Biggr].$$7 and large $$\mu(r,\theta) =\frac{r}{N d\cos\theta}\Biggl[ \arctan\!\Bigl(\frac{N d}{2r\cos\theta}+\tan\theta\Bigr) +\arctan\!\Bigl(\frac{N d}{2r\cos\theta}-\tan\theta\Bigr) \Biggr].$$8, with the stated purpose of stable continuum-electron calculations in scattering and autoionization studies (Mahato et al., 18 May 2026).

In galaxy clustering inference, the hybridization occurs in spectral space rather than physical space. The density contrast is decomposed into a spherical Fourier–Bessel block for $$\mu(r,\theta) =\frac{r}{N d\cos\theta}\Biggl[ \arctan\!\Bigl(\frac{N d}{2r\cos\theta}+\tan\theta\Bigr) +\arctan\!\Bigl(\frac{N d}{2r\cos\theta}-\tan\theta\Bigr) \Biggr].$$9 and a Cartesian plane-wave FFT block for $\partial^2\mu/\partial r^2$0. The implementation adopts a sharp cutoff at $\partial^2\mu/\partial r^2$1 with $\partial^2\mu/\partial r^2$2 for $\partial^2\mu/\partial r^2$3, giving $\partial^2\mu/\partial r^2$4 SFB modes. The total likelihood is taken as

$\partial^2\mu/\partial r^2$5

under the approximation that the two blocks are weakly correlated; mock-based cross-correlations are reported to be smaller than $\partial^2\mu/\partial r^2$6 (Wang et al., 2020).

Related mathematical constructions make the same spherical/plane-wave duality explicit. One work shows how exact spherical electromagnetic and Robinson–Trautman gravitational waves approach plane-fronted limits and then constructs a hybrid metric

$\partial^2\mu/\partial r^2$7

with matching in a buffer zone (Hogan et al., 2021). Another proves that any Helmholtz solution in a ball can be represented as a continuous superposition of evanescent plane waves, and numerical tests report machine-precision accuracy up to $\partial^2\mu/\partial r^2$8 with bounded coefficients for EPW approximation sets (Galante, 2023). A further mathematical line expresses reproducing kernels of spherical, complex, and symplectic harmonics as plane-wave integrals over Stiefel manifolds (Bie et al., 2017).

6. Computational profile, limitations, and recurrent misconceptions

In the XL-array formulation derived from PWM/SWM demarcations, the computational contrast is explicit. SWM-based channel generation costs $\partial^2\mu/\partial r^2$9 distance-square-root and exponential evaluations per user, whereas PWM uses a precomputed steering vector and a common factor $k$00. The stated implementation strategy is to precompute equi-power or equi-rank boundary surfaces $k$01 offline, then, at runtime, compute $k$02 or $k$03, look up $k$04, select SWM or PWM, and form the channel vector or matrix accordingly. The threshold parameter is chosen in practice as $k$05–$k$06 (Li et al., 2023).

Across fields, the computational motivation is similar but not identical. In layered half-space modeling, the hybrid model is reported to be hundreds to thousands of times faster than SGFD or DSGFD on representative profiles (Bhaumik et al., 2024). In the multi-UAV setting, tensor-OMP exploits the Kronecker structure induced by HSPWM to reduce projection cost relative to SD-OMP (Huo et al., 23 Aug 2025). In the cosmology hybrid basis, a single evaluation of the compressed spherical likelihood takes about $k$07 min, while the FFT-based plane-wave step is sub-second per likelihood call (Wang et al., 2020).

Several misconceptions recur. First, HSPWM is not a universally fixed acronym with a single canonical formula; the literature uses the label for non-equivalent constructions in communications, wave physics, chemistry, cosmology, and mathematical analysis. Second, HSPWM is not always a literal interpolation between a spherical formula and a planar formula. In wireless channels it is often a regime selector or a subarray decomposition; in galaxy clustering it is a split in $k$08-space; in Gaussian-basis scattering it is a hybrid basis with complex-shifted centers. Third, the hybrid model is not a claim that full SWM is unnecessary. The XL-array results show that applicability depends on angle, array structure, effective rank, and scatterers, while the THz and distributed-array papers tie hybridization to subarray aperture and spacing rather than to a single universal near-/far-field boundary (Li et al., 2023, Chen et al., 2021, Huo et al., 23 Aug 2025).

Taken together, these works define HSPWM as a family of technically distinct but structurally related methods: each preserves spherical behavior where curvature, range dependence, or global geometry is decisive, and each retains plane-wave structure where local approximation or algorithmic efficiency is decisive.