Based on the three-dimensional wave equation and the boundary conditions of continuous displacement and stress at the interface of the medium, Zoeppritz (1919) deduced the expression of the reflection and transmission coefficient when the wave propagates to the interface, which is described as the Zoeppritz equation. Assuming a solid–solid interface between two homogeneous isotropic elastic half-spaces, P-wave velocity, S-wave velocity, and density of the upper half-space are denoted by V P 1 , V S 1 , ρ 1 , respectively; P-wave velocity, S-wave velocity, and density of the lower half-space are denoted by V P 2 , V S 2 , ρ 2 , respectively; i 1 and i 2 are the incident angle and transmitted angle of P-wave; j 1 and j 2 are the incident angle and transmitted angle, respectively, of S-wave; the ray parameter is constant and described as p = sin ⁡ i 1 / V P 1 = sin ⁡ i 2 / V P 2 = sin ⁡ j 1 / V S 1 = sin ⁡ j 2 / V S 2 . Aki and Richards (2002) derived the accurate solutions of the Zoeppritz equation as R P P = b cos ⁡ i 1 V P 1 − c cos ⁡ i 2 V P 2 F − a + d cos ⁡ i 1 V P 1 cos ⁡ j 2 V S 2 H p 2 D , where a = ρ 2 1 − 2 V S 2 2 p 2 − ρ 1 1 − 2 V S 1 2 p 2 b = ρ 2 1 − 2 V S 2 2 p 2 + 2 ρ 1 V S 1 2 p 2 c = ρ 1 1 − 2 V S 1 2 p 2 + 2 ρ 2 V S 2 2 p 2 d = 2 ρ 2 V S 2 2 − ρ 1 V S 1 2 and E = b cos ⁡ i 1 V P 1 + c cos ⁡ i 2 V P 2 F = b cos ⁡ j 1 V S 1 + c cos ⁡ j 2 V S 2 G = a − d cos ⁡ i 1 V P 1 cos ⁡ j 2 V S 2 H = a − d cos ⁡ i 2 V P 2 cos ⁡ j 1 V S 1 D = E F + G H p 2 .

As shown above, although the Zoeppritz equation is a highly nonlinear function with respect to these properties, the formulations are explicitly valid in the isotropic media, and it uses few numbers of approximations, which allows the prediction of the reflection coefficients to be accurate from near to far incident angles. Then, global optimization schemes are suggested to be involved to treat AVO inversion. Combining the exact Zoeppritz equation with the global optimization methods can help inversion converge to global solutions.

Nonlinear inversions are usually done by minimizing an objective function. The L2 norm and cross-correlation coefficient are commonly objective functions to define an error, misfit, or similarity in the optimization problem. Then, we can obtain the objective function of the form as follows: m I N V = arg min m r − G m T r − G m , where r is the data vector, denoted as r = r 1 , r 2 , … , r n T , and model parameters m = m 1 , m 2 , … , m p T are all positive; and n and p are the numbers of observed data points and model parameters, respectively. Arg min returns the value of m, which minimizes the given function. It should be noted that these objective functions are the simplest form and are affected by the energy of noise.

Simple ACO functions in two operating modes: forward (from the nest toward the food) and backward (from food back to the nest). Forward ants build a solution by probabilistically choosing the next node to move to among those in the adjacent positions with respect to the current node. This probabilistic choice is biased by pheromone trails previously deposited on the paths by other ants. Forward ants do not deposit any pheromone, which, when associated with deterministic backward moves, helps eliminate loop formation.

First, an ant k locating on node i moves to j as the next node using the following constraint for the transition function: m j = arg max τ i j t α ∙ η i j β , r ≤ r 0 j ′ , r > r 0 , where τ i j is the pheromone trail at time t, η i j is the problem-specific heuristic information, α and β are impacts of pheromone trail and heuristic information, respectively, r is the random number with uniform distribution in [0, 1], r0 is the pre-specified parameter ranging from 0 to 1, and j ′ is the target point selected according to the following probability distribution: P i j k t = τ i j t α ∙ η i j β ∑ j ∈ N i k τ i j t α ∙ η i j β , where N i k is the neighborhood of ant k when located on node i. Ant k traversing in the backward mode through the arc (i; j) will update the pheromone value as follows: τ i j * = 1 − ρ τ i j + ∑ k = 1 K ∆ τ i j k , where ∆ τ i j k indicates the amount of pheromone deposition to all arcs, ρ ∈ ( 0 , 1 is the pheromone evaporation parameter, and K indicates the amount of ants.

In a quantum computer, a two-level quantum system called qubit acts as the physical media to store the information units. A qubit is the basic unit of information in quantum computation and is described by a superposition of the basis states φ ⟩ = α   0 ⟩ + β   1 ⟩ , where 0 ⟩ and 1 ⟩ are denoted as two basic states (Han and Kim, 2000); α and β are complex numbers and satisfy α 2 + β 2 = 1 . α 2 and β 2 are called the probability amplitude of the corresponding states 0 ⟩ and 1 ⟩ of qubit, respectively. We can note that a qubit is not a value of 0 or 1, but there is a possibility that it can be represented with values. A qubit can contain the information of both state |0⟩ and |1⟩, and a superposition state φ ⟩ also can be represented by a unit vector of a two-dimensional Hilbert space as φ ⟩ = α β (Williams, 2010).

The quantum rotating gate is the core operator in the evolution operation, and it directly affects the performance of the algorithm. As model parameters are in superposition states, the genetic qubits in the population should be updated by the quantum rotating gates to adjust the probability amplitude and constitute new individuals. Quantum rotating gates can be designed according to the practical problems and can be usually defined as G = cos ⁡ θ − sin ⁡ θ sin ⁡ θ cos ⁡ θ , where θ is the rotating angle. The rotation strategy adopted is given by the following equation: φ ′ ⟩ = G θ × φ ⟩ = cos ⁡ θ − sin ⁡ θ sin ⁡ θ cos ⁡ θ α β = α ′ β ′ , where φ ′ ⟩ is the updated quantum superposition state and α ‘and β' are the probability amplitudes of the quantum state after rotation. Generally, the rotation angle and rotation direction of the quantum gates are empirically determined in advance (Layeb and Saidouni, 2007; Han and Kim, 2000), and the adjustment strategy of the rotation angle and rotation direction for the quantum gate is updated in accordance with the self-adaptive rotating strategy proposed by Cheng et al. (2022b). A dynamic rotating method to adjust the rotation angle of the quantum rotating gate is proposed as follows: ∆ θ = − s g n A × θ max − θ min × F C u r − F min C ,

where A = α B α i β B β i , α B , β B are the probability amplitudes of a qubit in the currently optimal solution, and α i , β i are the probability amplitudes of the corresponding qubits in the current search solution. Then, the direction of the rotation angle is taken as follows: when A ≠ 0 , the direction is − s g n A ; when A = 0 , the direction can be positive or negative. θ max represents the maximum value in the interval, θ min represents the minimum value in the interval, F min represents the fitness of the best optimal individual, F C u r represents the fitness of the current individual, and C is a positive integer constant. The proposed method can associate the rotating angle with the fitness value and dynamically adjust the rotating angle, thereby improving the total convergence rate without losing the accuracy. Adjusting the corresponding qubit based on the fitness can make the individual update toward the direction of the optimal model parameter, and the rotating angle gradually reduces as the fitness converges to the optimal solution. Results listed in Table 1 represent the average evaluation number requested to the global minimum, which reduces from 321 to 217 after applying the self-adaptive rotating strategy, and the optimal solution is closer to the global minimum.

Quantum gate, in practical applications, is an assistant operation with the purpose to avoid losing important information of the population and enhance the local searching ability, as the diversity maintained by the superposition state is not enough to reflect the superiority of the quantum algorithm and premature convergence may occur due to the effect of noise. Therefore, quantum gates will be added to improve the performance of the algorithm.

The quantum NOT gate (Pauli-X gate) is commonly adopted to implement quantum gates. Through exchanging the probability amplitude of each gene 0 ⟩ and 1 ⟩ , the chance of the obtained result after measurement is exchanged. Moreover, the quantum NOT gate, as the quantum gate operator, rotates the angle of the gene by π 2 − 2 θ . As the rotation angle is slightly large, the convergence speed of the algorithm can be accelerated in the early stage of the quantum genetic algorithm, and at the same time, the diversity of the population can be guaranteed. However, the quantum NOT gate may cause premature phenomenon in the late stage. Consequently, we consider using Hadamard gates for quantum gates when the convergence is in the late stage. The matrix representation of the Hadamard gate is H = 1 2 1 1 1 − 1 , and the mutation of genes can be represented as H ψ ⟩ = 1 2 1 1 1 − 1 cos ⁡ θ sin ⁡ θ = cos π 4 − θ sin π 4 − θ = cos θ + π 4 − 2 θ sin θ + π 4 − 2 θ .

From the formula, we can find that the angle of the qubit is rotated by π 4 − 2 θ . Therefore, the Hadamard gate is more suitable as a mutation operator when the individual is close to the optimal value, to prevent the information of the current optimal individual from disappearing and to ensure the stability of the population. Table 1 shows that using quantum gates leads to higher efficiency, the average calculation number decreases to 151, and the higher accuracy is when optimal solution closes to 0.