Geometrical Acoustics Focusing Algorithm (GOAT)

• The paper demonstrates that GOAT corrects ultrasound beamforming errors by substituting homogeneous delay models with physically accurate, ray-based travel times.
• GOAT employs a layered-ray formulation using Snell’s law and Fermat’s principle to compute unique refracted ray trajectories across smooth, stratified interfaces.
• GOAT integrates with standard delay-and-sum beamforming using Newton-type solvers to deliver precise focusing corrections, as validated through numerical and phantom experiments.

Geometrical Acoustics based Focusing Algorithm (GOAT) denotes a deterministic, model-based focusing method for medical ultrasound in known layered media with continuously differentiable medium boundaries, introduced to correct aberrations caused by using homogeneous beamforming delays in heterogeneous propagation paths (Hackl et al., 7 Oct 2025). Its central operation is the replacement of straight-line, constant-speed times of flight by ray-based times of flight computed in a layered medium: rays are straight inside each layer, refract at interfaces according to Snell’s law, and are constrained to connect a transducer element to a prescribed focus point. In this formulation, GOAT is not a new beamforming architecture; it preserves the standard delay-and-sum structure and substitutes physically corrected transmit and receive delays for the homogeneous-model delays (Hackl et al., 7 Oct 2025).

1. Physical setting and meaning of “focusing”

The problem addressed by GOAT arises in pulse-echo ultrasound with a linear or planar array of transducer elements at the surface of a patient or phantom, where each element both transmits and receives (Hackl et al., 7 Oct 2025). Conventional delay-and-sum beamforming assumes a homogeneous medium with constant sound speed $c=1540\,\mathrm{m/s}$. Under that assumption, transmit focusing is implemented by delaying element excitations so that all wavefronts arrive in phase at a chosen transmit focus, and receive focusing is implemented by delaying received signals so that echoes from a chosen image point add coherently. In layered media, however, the actual propagation path is refracted and the segmentwise propagation speeds differ from the homogeneous model, so the computed times of flight are wrong; the resulting focusing delays are wrong as well, and the focal spot becomes shifted, broadened, and distorted (Hackl et al., 7 Oct 2025).

The medium model used by GOAT consists of a finite number of layers $n=1,\ldots,N$, each with constant sound speed $c_n$, separated by smooth interfaces

$$B_n=\{(x,b_n(x))\mid x\in[0,1]\},\qquad b_n\in C^1([0,1]),\qquad n=1,\ldots,N-1.$$

This graph-based representation excludes overhangs and encodes what the paper calls a layered medium with continuously differentiable medium boundaries (Hackl et al., 7 Oct 2025).

In a different branch of acoustics, “focusing” is formalized in linear discrete inverse problems through the pseudoinverse decomposition

$$\mathbf{G}^+ = \mathbf{G}^H(\mathbf{G}\mathbf{G}^H)^{-1},$$

where $\mathbf{G}^H$ is interpreted as a matched-filter focusing stage and $(\mathbf{G}\mathbf{G}^H)^{-1}$ as an inversion stage cancelling crosstalk (Hamdan et al., 2020). GOAT addresses a different problem: physical travel-time correction in refracting media rather than array inversion in a discrete transfer-matrix model.

2. Layered-ray formulation and the GOAT system

For a source element $P_{0,m}$ and a target focus $P_N$ in the final layer, GOAT represents the physical ray as a broken line

$$P_{0,m}\to P_{1,m}\in B_1 \to P_{2,m}\in B_2 \to \cdots \to P_{N-1,m}\in B_{N-1}\to P_N,$$

where the unknowns are the interface intersection points $n=1,\ldots,N$0 (Hackl et al., 7 Oct 2025). Within each layer the ray segment is straight because the sound speed is constant within that layer. Once these intersection points are known, the total time of flight is

$n=1,\ldots,N$1

For a fixed source, the notation is simplified to $n=1,\ldots,N$2, with $n=1,\ldots,N$3 and $n=1,\ldots,N$4 fixed and

$n=1,\ldots,N$5

At each interface, the local tangent angle $n=1,\ldots,N$6 is defined by

$n=1,\ldots,N$7

The incident and transmitted angles with respect to the normal, $n=1,\ldots,N$8 and $n=1,\ldots,N$9, satisfy the layerwise Snell relation

$c_n$0

A key feature of GOAT is that the angular quantities are written explicitly in terms of the unknown interface coordinates. The paper derives

$c_n$1

and

$c_n$2

The GOAT system of equations is therefore the coupled nonlinear system, for $c_n$3, consisting of: Snell’s law, interface membership, tangent slope, and the two coordinate-based angle formulas. In the form given in the paper, this is a system of $c_n$4 equations in $c_n$5 unknowns $c_n$6 (Hackl et al., 7 Oct 2025). Solving this system yields the unique refracted ray, when it exists and is unique, that intersects each interface once and terminates at the prescribed focus.

3. Fermat principle, existence, and uniqueness

The theoretical core of GOAT is the equivalence between Snell-consistent ray construction and stationarity of the travel-time functional. Writing

$c_n$7

the time-of-flight functional is

$c_n$8

The paper proves that, for each interface index $c_n$9, the following are equivalent: $$B_n=\{(x,b_n(x))\mid x\in[0,1]\},\qquad b_n\in C^1([0,1]),\qquad n=1,\ldots,N-1.$$0 is a stationary point of $$B_n=\{(x,b_n(x))\mid x\in[0,1]\},\qquad b_n\in C^1([0,1]),\qquad n=1,\ldots,N-1.$$1, Snell’s law is satisfied at each interface point $$B_n=\{(x,b_n(x))\mid x\in[0,1]\},\qquad b_n\in C^1([0,1]),\qquad n=1,\ldots,N-1.$$2, and the interface slope obeys the explicit relation

$$B_n=\{(x,b_n(x))\mid x\in[0,1]\},\qquad b_n\in C^1([0,1]),\qquad n=1,\ldots,N-1.$$3

This is the paper’s Fermat-principle formulation of the GOAT ray and gives a stationary-time characterization of the refraction problem (Hackl et al., 7 Oct 2025).

The existence theorem is phrased in geometric ray terms. Let $$B_n=\{(x,b_n(x))\mid x\in[0,1]\},\qquad b_n\in C^1([0,1]),\qquad n=1,\ldots,N-1.$$4 and $$B_n=\{(x,b_n(x))\mid x\in[0,1]\},\qquad b_n\in C^1([0,1]),\qquad n=1,\ldots,N-1.$$5 be two rays originating at $$B_n=\{(x,b_n(x))\mid x\in[0,1]\},\qquad b_n\in C^1([0,1]),\qquad n=1,\ldots,N-1.$$6 that pass the focus $$B_n=\{(x,b_n(x))\mid x\in[0,1]\},\qquad b_n\in C^1([0,1]),\qquad n=1,\ldots,N-1.$$7 on opposite sides, one above and one below. If, for all rays between $$B_n=\{(x,b_n(x))\mid x\in[0,1]\},\qquad b_n\in C^1([0,1]),\qquad n=1,\ldots,N-1.$$8 and $$B_n=\{(x,b_n(x))\mid x\in[0,1]\},\qquad b_n\in C^1([0,1]),\qquad n=1,\ldots,N-1.$$9, no total internal reflection occurs and there is exactly one intersection with each interface and with the line $$\mathbf{G}^+ = \mathbf{G}^H(\mathbf{G}\mathbf{G}^H)^{-1},$$0, then at least one solution to the GOAT system exists (Hackl et al., 7 Oct 2025). The proof uses a continuous lateral-miss function $$\mathbf{G}^+ = \mathbf{G}^H(\mathbf{G}\mathbf{G}^H)^{-1},$$1 and the intermediate value theorem.

The corresponding uniqueness statement is equally strong: there exists a unique solution to the GOAT system if and only if on every interface $$\mathbf{G}^+ = \mathbf{G}^H(\mathbf{G}\mathbf{G}^H)^{-1},$$2 there exists a unique point $$\mathbf{G}^+ = \mathbf{G}^H(\mathbf{G}\mathbf{G}^H)^{-1},$$3 fulfilling the slope condition above (Hackl et al., 7 Oct 2025). Operationally, this means that the stationary optical path from the source element to the focus is unique. The paper further notes that no-total-reflection and unique-intersection conditions can be written as explicit inequalities involving ray directions and interface slopes.

4. Algorithmic pipeline and embedding in beamforming

The computational problem solved by GOAT is termed the Multilayered Focusing Problem. Its inputs are the layered medium $$\mathbf{G}^+ = \mathbf{G}^H(\mathbf{G}\mathbf{G}^H)^{-1},$$4, the set of source positions $$\mathbf{G}^+ = \mathbf{G}^H(\mathbf{G}\mathbf{G}^H)^{-1},$$5, and a desired focus point $$\mathbf{G}^+ = \mathbf{G}^H(\mathbf{G}\mathbf{G}^H)^{-1},$$6; its outputs are the focusing delays $$\mathbf{G}^+ = \mathbf{G}^H(\mathbf{G}\mathbf{G}^H)^{-1},$$7 and $$\mathbf{G}^+ = \mathbf{G}^H(\mathbf{G}\mathbf{G}^H)^{-1},$$8 for all sources (Hackl et al., 7 Oct 2025).

Algorithmically, GOAT proceeds element by element. For each source position $$\mathbf{G}^+ = \mathbf{G}^H(\mathbf{G}\mathbf{G}^H)^{-1},$$9, the nonlinear GOAT system is solved for the unknown interface points and angles. The corresponding time of flight $\mathbf{G}^H$0 is then evaluated from the segmentwise distance-over-speed sum. After all source-element times have been computed, the transmit delays are formed as

$\mathbf{G}^H$1

and the receive delays as

$\mathbf{G}^H$2

These corrected delays are then inserted into an otherwise standard delay-and-sum beamforming chain (Hackl et al., 7 Oct 2025).

The paper describes the nonlinear solve as a Newton-type procedure and explicitly mentions a trust-region dogleg method, such as MATLAB’s fsolve, for the joint solve over all interface variables (Hackl et al., 7 Oct 2025). The proposed initialization is geometrically simple: ignore refraction, intersect the straight line from $\mathbf{G}^H$3 to $\mathbf{G}^H$4 with each interface, set $\mathbf{G}^H$5 to those intersection abscissae, then set $\mathbf{G}^H$6 and initialize the angles accordingly. Because typical medical aberrations are moderate, this straight-ray initialization is reported to work well; the authors also note that a small number of bisection or line-search iterations on the first interface position can be used to improve robustness.

Transmit and receive are treated symmetrically at the level of travel times. By reciprocity, the travel time from an element to a focal point is the same as the travel time from that point back to the element, so the same ray-theoretic solver supports both transmit focusing corrections and receive focusing corrections (Hackl et al., 7 Oct 2025). In the experimental part of the paper, the receive side is corrected explicitly, while transmit focusing remains standard.

5. Numerical validation and phantom experiments

The numerical study evaluates GOAT against full-wave simulations performed in 2D with k-Wave (Hackl et al., 7 Oct 2025). The reported setup uses grid spacing $\mathbf{G}^H$7 in both $\mathbf{G}^H$8 and $\mathbf{G}^H$9, a $(\mathbf{G}\mathbf{G}^H)^{-1}$0 grid, time step $(\mathbf{G}\mathbf{G}^H)^{-1}$1, and total simulation time $(\mathbf{G}\mathbf{G}^H)^{-1}$2. The sound speeds used in the layer models are

$(\mathbf{G}\mathbf{G}^H)^{-1}$3

representing, respectively, fat-like, soft-tissue-like, and faster cover-material conditions. Three geometries are examined: a straight horizontal interface with fat over tissue, a curved elliptic interface with fat over tissue, and a “transducer cover” configuration consisting of an elliptical inner region and a $(\mathbf{G}\mathbf{G}^H)^{-1}$4-thick elliptical annulus with $(\mathbf{G}\mathbf{G}^H)^{-1}$5 above tissue (Hackl et al., 7 Oct 2025). Sources are point sources in the bottom layer emitting Gaussian pulses at $(\mathbf{G}\mathbf{G}^H)^{-1}$6, and sensors form a line array in the top layer.

The principal simulation result is that GOAT times of flight and focusing delays match k-Wave almost exactly in all three geometries, with errors at numerical precision, whereas the homogeneous model shows significant travel-time and delay errors (Hackl et al., 7 Oct 2025). In the fat-over-tissue settings, homogeneous-model times are systematically biased because the true speed differs from $(\mathbf{G}\mathbf{G}^H)^{-1}$7. The curved-interface case has a similar time-of-flight error magnitude to the flat-interface case, indicating that the speed mismatch dominates the total error more strongly than ray bending. In the transducer-cover setting, the layer is thin, so absolute time errors are moderated, but the lateral variation of error is steeper and therefore the delay errors are larger. A small residual GOAT error in one right-shifted-source case is attributed in the paper to multiple reflections or wave effects not captured by ray theory.

The experimental validation uses a GE Healthcare 9L-D linear array, a $(\mathbf{G}\mathbf{G}^H)^{-1}$8-thick cuboid PVC-based Proxon aberrator, salt water as reference coupling medium, and a wire phantom with nominal background speed $(\mathbf{G}\mathbf{G}^H)^{-1}$9 (Hackl et al., 7 Oct 2025). Two acquisitions are compared: one with the Proxon layer and one with the Proxon removed and replaced by salt water. The sound speed of Proxon is estimated from the depth shift of wire targets and reported as lying in $P_{0,m}$0, with median $P_{0,m}$1 and mean $P_{0,m}$2; the reconstruction then adopts $P_{0,m}$3.

For the Proxon acquisition, receive focusing is recomputed with GOAT using a two-layer model consisting of a $P_{0,m}$4 planar Proxon layer above a $P_{0,m}$5 phantom medium (Hackl et al., 7 Oct 2025). The observed image changes are specific: homogeneous beamforming with Proxon produces broadened wire targets and degraded near-field lateral resolution, while GOAT produces substantially sharper wire targets and corrects the depth shifts, yielding images that closely resemble the no-aberrator reference. The lateral beam-profile analysis reports narrower main lobes and restored peak-to-background contrast for GOAT, with the strongest improvements in the near field. The paper also notes a bright horizontal reverberation line at approximately $P_{0,m}$6 in all images, attributed to multiple reflections between interface and transducer.

6. Relation to broader geometrical acoustics, caustics, and extensions

GOAT belongs to a broader class of high-frequency ray methods based on the WKB ansatz

$P_{0,m}$7

with eikonal equation

$P_{0,m}$8

and amplitude transport equation

$P_{0,m}$9

for time-harmonic acoustics (Potter et al., 2022). In that general framework, rays are characteristics of the eikonal, and amplitude growth is governed by geometric spreading. Related nonlinear geometrical-acoustics work shows that caustics occur when the ray-tube Jacobian vanishes, $P_N$0, equivalently $P_N$1, so linear WKB predicts $P_N$2 at focal singularities (Afanasyev et al., 2012). GOAT, as formulated for layered medical ultrasound, avoids explicit caustic analysis because it uses refraction-corrected travel times rather than full amplitude-field reconstruction.

This delimitation is important for interpreting the method’s scope. GOAT assumes known medium composition, validity of geometrical acoustics, interfaces representable as graphs $P_N$3, and propagation paths with no total reflection and a single intersection with each interface (Hackl et al., 7 Oct 2025). Arbitrary 2D or 3D sound-speed maps are not directly supported; strong diffraction, scattering, interference, and mode conversion are neglected by construction. The computational load is also explicit in the paper: for each element–point pair, one solves a nonlinear system of size $P_N$4 (Hackl et al., 7 Oct 2025).

At the same time, several adjacent arXiv lines indicate natural technical extensions. Numerical geometric-acoustics solvers on tetrahedral meshes already propagate phase and amplitude through piecewise-linear 3D environments with reflections, UTD edge diffraction, dynamic-programming ray plans, and an approximate origin function that localizes shadow and reflection boundaries (Potter et al., 2022). This suggests a route from GOAT’s layered setting to more general 3D geometries, provided that the travel-time correction remains the primary objective. A different extension concerns caustics: metaplectic geometrical optics replaces standard configuration-space GO by a mixed representation

$P_N$5

chosen so that GO remains valid in $P_N$6-space without caustic singularities, and then maps the solution back by metaplectic transforms (Lopez et al., 2021, Lopez, 2022). This suggests a mathematically natural way to augment GOAT if the target application moves from layered aberration correction toward full field synthesis in strongly focusing media.

Within the literature surveyed by the ultrasound paper, GOAT is positioned against full-wave simulation, straight-ray travel-time correction, phase-screen methods, and earlier geometrical-acoustics approaches in nondestructive evaluation. Its stated novelty is the combination of a medical-ultrasound-oriented layered refraction model, a complete focusing-delay pipeline for transmit and receive, and an existence-and-uniqueness analysis tied directly to Fermat stationarity and Snell refraction (Hackl et al., 7 Oct 2025). In that sense, GOAT is best understood not as a general synonym for geometrical-acoustics focusing, but as a specific layered-media aberration-correction algorithm whose broader significance lies in showing how rigorous ray theory can be inserted directly into practical ultrasound beamforming.