Optimized Nonlinear PRI Variation Strategy Using Knowledge-Guided Genetic Algorithm for Staggered SAR Imaging

Staggered synthetic aperture radar (SAR), which operates with variable pulse repetition interval (PRI), staggers blind areas to solve the blind range problem caused by constant PRI in conventional high-resolution wide-swath SAR imaging. The PRI variation strategy determines the blind area distribution, and thus has a significant influence on the imaging performance in staggered mode. Generally, the existing strategies based on linear PRI variation can control the blind areas in a straightforward way, which has achieved impressive results. However, the linearity of the PRI variation imposes regularity or even periodicity on the locations of the blind areas, which limits the distribution of the blind areas. The imaging performance has the potential to be further improved by introducing much more irregularity into the PRI sequences. To this end, this article proposes an optimized nonlinear PRI variation strategy for staggered SAR mode. First, a novel objective function is defined that quantitatively measures the uniformity of the blind area distribution along the slant range and the discontinuity of the blind area distribution along the azimuth. Subsequently, the optimum nonlinear PRI variation strategy is found using an optimization problem and the proposed objective function. A knowledge-guided genetic algorithm is proposed to solve the optimization problem. Comparisons with the existing linear variation strategies show that the proposed strategy can provide a superior imaging performance after reconstruction with a lower objective function value. Simulations and experiments on raw data generated in staggered SAR mode are performed to verify the effectiveness of the optimized nonlinear PRI variation strategy.

capability has become an acknowledged powerful electronic equipment for Earth observation [1], [2], [3]. High spatial resolution images with ultra-swath coverage are essential to the identification, confirmation, and description of objects, which can be widely applied in the field of environmental monitoring, disaster management, sea and land surface imaging, and so on [4], [5]. Due to the minimum SAR antenna area constraint, there is an inherent tradeoff between the achievable resolution and the swath width [6], [7]. This tradeoff can be overcome by multiple-aperture receiver technology at the cost of increasing antenna length and hardware complexity [8], [9], [10], [11]. With the development of digital beamforming technology, scan-on-receive (SCORE) without the necessity to lengthen the antenna provides a potential for mapping an ultrawide swath with high resolution [12], [13], [14], [15]. Traditional SCORE technology uses constant pulse repetition interval (PRI), which, however, determines regular and periodic blind ranges along the transmitted pulses dimension, since the radar cannot receive while it is transmitting [16], [17].
To obtain gapless wide-swath imaging, the staggered SAR concept proposed in [18] combines the SCORE technology with variable PRI. Further investigations and analysis of the system operation are detailed in [19], [20], [21], [22], [23], [24], and [25]. The variable PRI sequence staggers the blind ranges at different transmitted pulses, i.e., the constant blind ranges caused by traditional HRWS SAR can be shifted and distributed over the whole swath in staggered mode. The PRI variation strategy determines the distribution of the blind areas [26]. The blind area distribution impacts the imaging performance after reconstruction significantly [24], [25]. Therefore, one research area on staggered SAR is on the design of PRI variation to distribute the blind areas so as to minimally affect the quality of the reconstructed images.
The existing PRI sequence strategies mainly focus on linear variation, including the slow linear PRI variation, the fast linear PRI variation, and the elaborated PRI variation. A limited PRI span can ensure a proper sampling in azimuth without aggravating the range and azimuth ambiguities. The slow linear PRI variation proposed in [27] has the minimum PRI span such that blind ranges are almost uniformly distributed over the slant ranges of interest, and the positions of the blind ranges are derived detailedly in [28]. However, the main drawback of the slow linear PRI variation strategy is that a large gap (i.e., blind This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ area) due to several consecutive lost pulses appears in the raw azimuth signal. This gap generates high sidelobes near the main lobe in the azimuth impulse response. Villano et al. [29] designed a short sequence to avoid consecutive lost pulses along the azimuth, following the criterion that no more than one pulse of the sequence is lost in the desired swath. Subsequently, the fast linear PRI variation was proposed in [24] and [25], following the criterion that two consecutive samples are never lost for all slant ranges in the desired swath. These two sequences can effectively avoid high sidelobes near the main lobe, but generate high sidelobes far from the main lobe due to the periodicity of the gaps and a high percentage of lost pluses. Based on the fast linear variation, Zhouet et al. [22] proposed stepwise PRI strategy containing multiple constant PRIs in a PRI staggering cycle, which ensures that part of the data is sampled uniformly to reduce the error propagation during reconstruction. By splicing a few sequences of fast linear PRI variation into a long sequence, a more elaborated PRI variation strategy is proposed in [24] and [25], which can break through the periodicity of the blind areas. These linear PRI variation strategies can control the blind areas in a straightforward way. However, the linearity of the PRI variation imposes regularity or even periodicity on the locations of the blind areas, which limits the blind area distribution to some extent. The reconstructed imaging performance has the potential to be further improved, since the energy of such sidelobes can be spread along azimuth by introducing much more irregularity in PRI variation strategies.
In fact, PRIs can be chosen arbitrarily or even randomly on the condition that the PRI span is determined. The sequences with random PRI variation are widely used in the field of radar antijamming, and have the potential to achieve low sidelobes and super-resolution processing [30], [31], [32]. However, when applied to staggered SAR, random PRI variation generates numerous blind areas made up of lost pulses at some certain slant ranges, and very few blind areas at other slant ranges. The distribution of blind areas is extremely nonuniform along the slant range. This phenomenon will result in a significant discrepancy in the reconstructed imaging performance for different slant ranges in the whole desired swath.
In order to overcome the limitations of linear PRI variation and eliminate the discrepancy caused by random PRI variation, this article proposes an optimized nonlinear PRI variation strategy. We transform the design of the PRI variation strategy into a combinatorial optimization problem. An objective function is proposed to measure the performance of the blind area distribution. Then, we use the genetic algorithm (GA) to solve the optimization problem. A guided-knowledge mutation operator is proposed to efficiently provide a superior nonlinear PRI sequence with a lower objective function value, which can bring good imaging performance after reconstruction.
The advantages can be summarized as follows. First, the blind area distribution of the proposed nonlinear PRI variation strategy can meet the requirement of discontinuity, i.e., two continuous pulses are never lost along the azimuth in the whole swath. Second, the distribution of blind areas can meet the requirement of uniformity in the whole swath, i.e., the blind areas are uniformly distributed along the slant range in the whole swath. Moreover, it is possible to demonstrate that the percentage of the lost pulses is small enough. Third, compared with the linear PRI variation strategy, the optimized nonlinear PRI variation strategy introduces more irregularity into the blind area distribution, which can spread the energy of high sidelobes along the azimuth. Fourth, since the sequence of PRIs repeats within the synthetic aperture time, a short sequence of PRIs determines more repetition cycles than a long sequence. Compared with the fast linear PRI variation strategy (short sequence with small number of changing PRI), the proposed nonlinear PRI variation strategy provides a long sequence with large number of changing PRI, which reduces the periodicity of the gaps, so as to further suppress the high sidelobes. Therefore, owing to these advantages, the optimized nonlinear PRI variation strategy can effectively suppress the sidelobes in the azimuth impulse response, and obtain focused images without discrepancy after reconstruction.
Overall, this article proposes an optimized nonlinear PRI variation strategy for staggered SAR imaging. The main contributions are presented as follows.
1) A novel objective function is proposed to quantitatively measure the uniformity and the discontinuity of the blind area distribution. 2) Based on the proposed objective function, the design of nonlinear PRI variation is transformed into an optimization problem. A knowledge-guided GA is proposed to efficiently solve the optimization problem.

3) Simulations and experiments generated in staggered SAR
mode are performed to verify the effectiveness and robustness of the proposed nonlinear PRI strategy. Compared with the sequence of linear PRI variation, the proposed nonlinear PRI variation strategy improves the imaging performance significantly after reconstruction using the best linear unbiased (BLU) interpolation. The rest of this article is organized as follows. The review of staggered SAR mode is first described in Section II. Section III provides the details of the optimized objective function. A knowledge-guided GA is proposed for the nonlinear PRI variation strategy in Section IV. In Sections V and VI, simulations and experiments are used to prove the effectiveness of the nonlinear PRI strategy. A small discussion is addressed in Section VII. Finally, Section VIII concludes this article.

A. Blind Area
Traditional HRWS SAR using multiple elevation beams (MEBs) technology is affected by blind areas located at fixed slant ranges between adjacent swaths, since the echo cannot be received when the radar is transmitting pulses [33]. The width of each blind area between adjacent subswaths can be calculated as where c is the speed of light, T p represents the pulse duration, T dc denotes the duty cycle, and PRF represents the pulse repetition frequency. The blind range problem can be successfully addressed by the staggered SAR system, which combines the MEBs technology with the strategy of PRI variation, as shown in Fig. 1(a). The staggered mode transmits a sequence of M pulses with different PRIs, which can be expressed as PRI m , m = 0, 1, . . ., M − 1. The M pulses repeat periodically. PRI max and PRI min are the maximum and minimum of the M PRIs, respectively. The mth pulse is transmitted at the slow time t m , which is obtained by the sum of PRIs, as When the mth transmitted pulse is located in its nth blind area, the corresponding range time should satisfy where t r = 2R 0 /c denotes the time delay at slant range R 0 . The locations of blind areas change with the PRI sequence. As it can be noted in Fig. 1(c)-(g), using staggered SAR, the blind areas are no longer strips parallel to the axis of the transmitted pulses. The PRI variation tilts the blind areas to some extent. A limited PRI span can guarantee the performance of range and azimuth ambiguities. However, the more limited the PRI span, the less tilted the blind areas can be. These factors should be considered in the values of PRI span. Consequently, the discretely fixed blind areas abovementioned disappear, and are distributed over the whole swath with the capacity for gapless observation.

B. PRI Variation Strategy
The existing PRI strategies mainly focus on linear variation. The PRIs sequence has a linear trend as where δ denotes the difference between two adjacent pulses. Different values of δ lead to different linear strategies. The slow linear PRI variation strategy has the advantages of a large value of PRI min and a small value of PRI max . The main disadvantage is that a large gap made up of several consecutive missing pulses (blind areas) occurs at each slant range, as shown in Fig. 1(c). Consequently, high sidelobes are present in the vicinity of the main lobe in the azimuth impulse response [24]. The fast linear PRI variation strategy adopts shorter sequences with much faster PRI variation. In the blind area distribution, two consecutive pulses are never missed for all slant ranges of interest, as apparent from Fig. 1(d). In this case, the high sidelobes near the main lobe in the azimuth impulse response can be eliminated. The cost is that there are more distant sidelobes in the azimuth impulse response due to the periodicity of gaps. In order to suppress these high sidelobes, the elaborated PRI sequence is designed by splicing several sequences of fast PRI variation, which has achieved impressive results. The periodicity of gaps is eliminated effectively in the blind area distribution, as shown in Fig. 1(f). However, the linearity of the PRI variation could impose regularity on the locations of the blind areas to some extent. The energy of such sidelobes has the potential to be further spread along azimuth by introducing much more irregularity in PRI variation strategies. Moreover, the performance in terms of range ambiguity-signal-ratio (RASR) is limited due to a smaller PRI min . As mentioned in [24], the PRIs can be arbitrarily or even randomly chosen as the PRI span [PRI min , PRI max ] is determined. In the random PRI variation strategy, the mth PRI is selected by where the random variable γ ∼ U (0, 1) follows a uniform distribution. Using the sequences with random PRIs, more irregularity can be introduced into the blind area distribution. However, we found that there would be numerous consecutive lost pulses at some certain slant ranges, and very few lost pulses at other slant ranges, as shown in Fig. 1(e). After azimuth focusing, it will lead to a significant discrepancy in the imaging performance for different slant ranges included in the desired swath. In order to overcome the regularity of linear PRI variation strategy and the discrepancy of random PRI variation strategy, an efficient method proposed in this article is to keep PRI variation nonlinear. A nonlinear PRI variation strategy introduces irregularity to the blind area distribution, which has the potential to spread the energy of high sidelobes in azimuth impulse response. Therefore, we design a nonlinear PRI variation strategy so that the corresponding blind area distribution can meet two requirements: 1) blind areas uniformly distributed along the slant range, and 2) blind areas discontinuously distributed along the azimuth. The former is meant to eliminate the discrepancy caused by the random PRI variation strategy. The latter guarantees that the no consecutive pulses is lost along the azimuth in the whole swath and this, in turn, contributes to mitigate the high sidelobes and achieve superior imaging performance. To this end, it is essential to measure the uniformity and the discontinuity quantitatively, which is defined as an objective function in Section III.

C. BLU Reconstruction Algorithm
The variable PRI solves the blind range problem with the cost of the lost pulses and nonuniform sampling in azimuth. The nonuniform echo signal needs to be reconstructed to a uniform grid, and then the traditional SAR processing can be further performed. In order to verify the effectiveness of the proposed PRI variation strategy in imaging performance, we use BLU to reconstruct the images in Section VI. Thus, a review of BLU is given as follows.
The BLU algorithm operated with the knowledge of the power spectral density (PSD) provides the best interpolation in the least-squares sense [21]. Let us assume the raw signal in azimuth, u(t), is a zero-mean complex random process. The PSD of u(t) can be expressed as a proportion of the antenna power pattern, as where L a denotes the length of synthetic aperture, and v s is the velocity of radar platform. U (f ) represents the spectrum of u(t).
The autocorrelation function R u (ξ) of u(t) can be obtained by the inverse Fourier transform of P u (f ), as The autocorrelation function R u (ξ) should require that It is worth noting that the specific closed-form expressions of the PSD P u (f ) in (6) and the autocorrelation function R u (ξ) in (7) hold for a planar antenna without weighting. When the aperture is no longer planar, or nonuniformly illuminated, the PSD and the autocorrelation function are different, requiring even numerical evaluation if they cannot be represented in a closed form [24]. The BLU interpolation can be carried out for arbitrary autocorrelation functions with sufficient accuracy.
Considering the existence of Gaussian white noise (GWN), the signal-to-noise ratio is denoted as SNR. The correlation function can be represented by where δ(·) denotes the Kronecker delta. The BLU resamples uniform azimuth signal u(t ) using the nonuniformly sampled azimuth signal u(t q ), q = 1, . . . , Q. Assume that u = [u(t 1 ), u(t 2 ), · · · u(t q )] T . The qth element of the column vector r can be expressed as r q = R un (t − t q ), q = 1 . . . Q. The elements of the matrix G are given by Therefore, the azimuth uniformly sampled signal is reconstructed by the optimal linear unbiased estimation interpolation, asû

III. OPTIMIZED OBJECTIVE FUNCTION
This section proposes the optimized objective function used to quantitatively measure the uniformity and the discontinuity of the blind area distribution. The main idea is that we calculate the number of consecutive blind areas along the azimuth at each slant range. The numbers are encouraged to be zero to ensure discontinuity and are uniform along the slant range to ensure uniformity.
The corresponding flowchart is shown in Fig. 2, and the implement procedures are listed in the following. The optimized objective function mainly consists of two parts: 1) measuring the discontinuity of the blind area distribution (Steps 1-3) and 2) measuring the uniformity of the blind area distribution (Steps 4-6).

1) Measuring the Discontinuity:
Concretely, we transform the image of the blind area distribution into a matrix for the convenience of subsequent calculation in Step 1. In Steps 2 and 3, spatial filtering is performed. The number of consecutive blind areas along azimuth is calculated at each slant range.
Step 1: The image of the blind area distribution is converted into a matrix as I(i, j), where i = 1, 2, . . ., M and j = 1, 2. . ., N. The row i represents the transmitted pulses, and M is the number of PRIs in one period. The column j denotes the slant ranges. N denotes the number of slant range units. We assume that I(i, j) = 1 if the blind area occurs at (i, j), whereas I(i, j) = 0 in the case of no blind area, as The gray matrix elements denote the blind areas, as shown in Fig. 2. It should be noted that the choice of the values of M is vital to the design of the sequence, which affects the imaging performance after reconstruction. Gebert and Krieger [27] analyzed the effects of different M values on imaging results, and draws a conclusion that the period time of M PRIs is required to be set as This can decrease the strong dependence of the imaging performance on the target azimuth position. T a (R min ) denotes the shortest illumination time corresponding to the minimum slant range. Thus, equation (13) is also applied in the proposed nonlinear PRI variation strategy.
Step 2: Spatial filtering is performed to measure the discontinuity of the blind area distribution. The spatial filter can be defined as follows: The filter size is N a × 1, where N a = 2a + 1, and a ∈ Z + , −a ≤ i ≤ a. Applying the spatial filter to the matrix I(i, j), the output of the filtering can be expressed as follows: The indicates the number of the blind areas in the neighborhood of I(i, j) along the azimuth. For example, in the top row of Fig. 2, the size of the spatial filter N a = 5. The spatial filtering is denoted as the symbol of "⊗." In the second row and second column of the blind area matrix I(i, j), we have I(2, 2) = 1 as marked by the purple lines. There are no consecutive lost pulses in the neighborhood of I(2, 2) along the azimuth, i.e., I(1, 2) = 0, I(3.2) = 0, I(4, 2) = 0. Thus, the filter response equals to g(2, 2) = 0. Furthermore, in the third row and fourth column of the blind area matrix I(i, j), we have I(3, 4) = 1. Since two blind areas appear in the neighborhood (i.e., I(1, 4) = 1 and I(4, 4) = 1) and the other elements have no blind areas (i.e., I(2, 4) = 0 and I(5, 4) = 0), the filter response is equal to g(3, 4) = 2, which denotes that there are two blind areas in the neighborhood of I(3, 4).
Step 3: Measuring the statistical characteristics of the filter response g(i, j), we count the numbers of matrix elements in each column satisfying both g(i, j) = k − 1 and I(i, j) = 1, which can be denotes as H(k, j), k = 1, 2, . . ., N a − 1. H(k, j) can be written as where the operator N i (·) is used to count the numbers of matrix elements along the rows. The elements at (i, j) without blind area (i.e., I(i, j) = 0) are not considered into the matrix H(k, j), as the bold lines "\" marked in the matrix g(i, j) in Fig. 2. We only count the numbers of the matrix elements (i, j) where the blind areas occur (I(i, j) = 1). Because g(i, j) has N a distinct values, the number of rows in the matrix H(k, j) is equal to N a , i.e., k = 1, 2, . . . , N a . H(k, j) indicates the case that k − 1 blind areas occur in the neighborhood of I(i, j) along the azimuth at the jth slant range unit, i.e., I(i ± 1, j), . . ., I(i ± a, j). Namely, besides I(i, j) = 1, there are k − 1 elements whose values equal to 1 among the elements of I(i ± 1, j), . . ., I(i ± a, j). As shown in Fig. 2, in the second column of g(i, j), there is one element equal to 0, and four elements equal to 1. Thus, we can obtain H(1, 1) = 1 and H(1, 2) = 4.
It should be noted that the parameter of N a will influence the blind area distribution of the optimized nonlinear PRI sequence. In practice, the value of N a can be chosen depending on different requirements. If a = 1 (N a = 3), the objective function based on this case is defined to avoid the consecutive blind areas along the azimuth. If a ≥ 2, the blind areas are not only distributed discontinuously, but also would be more spread out along the azimuth. In the meantime, the value of N a should be limited. The PRI variation strategy has little effect on the total number of blind areas. It would be difficult to significantly reduce the number of blind areas by the design of PRI variation strategy. Thus, when the value of N a is too large, it may be difficult for the genetic operator to generate superior offspring, which would reduce the efficiency of algorithm iteration.
2) Measuring the Uniformity: Concretely, Steps 4-6 define the obtained probability distribution and the expected probability distribution. The concept of relative entropy is introduced to measure the difference between these two distributions, so as to measure the uniformity of the blind area distribution along the slant range.
Step 4: The probability distribution of the matrix H(k, j) is computed as P (k, j). The probability of the elements in matrix where ε denotes a sufficiently small positive value. The number of columns N in the matrix H(k, j) is equal to the numbers of the slant range units. Since H(1, j) represents the case that no consecutive blind areas along the azimuth occur at the jth slant range unit, a larger value of P (1, j) implies that the probability of the appearance of consecutive blind areas is lower, which is expected.
Step 5: The expected probability distribution is computed as Q(k, j). The expected probability distribution should be designed to meet two requirements. The first one is that for all columns (slant ranges), the values of Q(k, j) when k = 1 are expected to be much larger than the values of Q(k, j) when k = 2, 3, . . . , N a . The second requirement is that the percentage of the blind areas is expected to be small enough at each slant range and distributed uniformly along the slant ranges. As seen in Fig. 2, the expected matrix H(1, j) = L in the first row, and H(K, j) = 0, k = 2, 3, . . . , N a . Therefore, the expected probability Q(k, j) can be designed as In the example of Fig. 2, L = 5. Since the concept of entropy is introduced in the next step, the probabilities P (k, j) and Q(k, j), k = 2, 3, . . ., N a cannot equal 0 in the denominators. ε is used to represent a sufficiently small value, as shown in the expected probability Q(k, j) in Fig. 2.
Step 6: Inspired by the concept of relative entropy [35], [36], in order to measure the difference between the obtained probability distribution P (k, j) and the expected probability distribution Q(k, j), the proposed objective function is defined as A small value of the objective function (Q P ) illustrates that the obtained probability distribution P (k, j) approximates to the expected probability distribution Q(k, j). In this case, the above two requirements (i.e., discontinuity and uniformity) of blind area distribution can be satisfied. The design of the nonlinear PRI variation strategy can be transformed into an optimization problem.
This width impacts the design of the PRIs sequence for staggered SAR system. The width of blind area is cT p /2 in the raw data. However, the width in the range compressed data is twice because the echoes not fully received should be discarded [21], [40]. Furthermore, the reconstruction algorithms should also be considered depending on the two cases: (1) resampling the raw data on a uniformly-spaced grid and (2) resampling the range-compressed data on a uniformly spaced grid. In this article, the PRIs sequence is designed for the blind area width in the raw data, and the BLU reconstruction algorithm is also processed by the former case. If the latter reconstruction algorithm is used, the PRIs sequence needs to be designed in a different way. Typically, the proposed objective function can also be adopted. The difference is that for each blind area, the columns satisfying in range-compressed data will be twice in raw data.

IV. KNOWLEDGE-GUIDED GA OPTIMIZATION FOR NONLINEAR PRI VARIATION STRATEGY
Based on the proposed objective function, the optimization of nonlinear PRI variation strategy can be considered as an optimization problem. This section proposes a knowledge-guided GA to obtain superior nonlinear PRI sequence with lower value of the objective function efficiently. Guided by specific knowledge, the mutation operator limits the ranges of selected PRI parameters to explore promising solution areas in Section IV-A. Finally, the framework and the processing of the knowledgeguided GA algorithm are detailed in Section IV-B.

A. Proposed Knowledge-Guided Mutation Operator
In order to solve the optimization problem, the GA is based on mechanisms of the natural evolution and genetics, which transforms the optimization problem into a genetic evolution behavior [37]. The evolutionary processes can be achieved by the mathematical operations, including selection, crossover, and mutation. Ideally, the performance of the sequences of PRIs gets better as the iteration goes on. However, the classic GA easily gets trapped in a locally optimal solution. The sequences would not be satisfactory in practice, requiring a large number of iterations with an increase in computing time. It is because the search space is huge where GA needs to select the ith transmitted pulse randomly and mutate the PRI to an arbitrary value between PRI min and PRI max . Mutating randomly tends to generate worse individuals.
One proposal to overcome these difficulties is to guide the optimization process toward promising solution areas using specific knowledge. Nonrandom mutation operator is of great benefit to reach a faster convergence of GA with fitter solutions, which has been proved in different fields [38], [39]. Inspired by this, we propose a knowledge-guided mutation operator to direct the mutation process using the heuristic rules, which are specifically suitable for nonlinear PRI variation.
The knowledge, i.e., the map of the blind area distribution, is calculated by the PRI sequence. In order to mutate the genes, which get the most rewards to separate the consecutive blind areas, we need to know which are the worse genes causing large numbers of consecutive blind areas. Concretely, the knowledge is used to evaluate each gene (the PRI of the transmitted pulse) by calculating the number of consecutive blind areas. Instead of mutating genes randomly, we select the worse genes with large numbers of the consecutive blind areas in a greedy strategy. Mutating these genes could get the most rewards to eliminate the consecutive blind areas. Moreover, to improve the diversity of the mutating, we mutate the top K genes and introduce randomness into the mutated genes.
The knowledge-guided mutation operator consists of two main parts: evaluate genes based on the knowledge (Steps 1-6) and mutate genes guided by the knowledge (Steps 7-10), which is illustrated in Algorithm 1. Specifically, GA seq denotes the individuals in the population after the operation of selection. The number of individuals in the population is N p . Δ = PRI max − PRI min represents the PRI span of the sequence.
The maximum and minimum slant ranges in the whole swath are R max and R min , respectively.

1) Evaluating Genes Based on the Knowledge:
First, Steps 1-4 in Algorithm 1 repeat the process of Steps 1 and 2 in Section III. The blind area distribution is calculated for each individual (the sequence of variable PRI) in the population. The map of the blind area distribution is converted into a matrix I(i, j). Spatial filtering is performed to measure the number of consecutive blind areas along azimuth in matrix I(i, j). Thus, we where w is the spatial filter as illustrated in (14).
Then, in Step 5 of Algorithm 1, each gene is evaluated by calculating the number of consecutive blind areas (L(i)). In order to know which transmitted pulses causes a large number of consecutive blind areas, we count the numbers of matrix elements along the columns satisfying g(i, j) = 0, 1, . . . , N a − 1, as where the operator N j (·) is used to count the numbers of matrix elements along the columns. The number in the ith row satisfying g(i, j) = k − 1 > 0 is denoted as L(i), which can be expressed as A large value of L(i) indicates that there are a lot of consecutive blind areas for the ith transmitted pulse. Subsequently, in Step 6, instead of randomly mutating genes, we select the worse genes with large L(i) values in a greedy strategy. In the knowledge-guided mutation operator, our purpose is to assign a rule to mutate the genes (the PRIs of the transmitted pulses) to separate the consecutive blind areas. To this end, we need to obtain the locations of the genes that need to be mutated. The L(i) provides a feasible solution. Assuming that the operator R[·] is used to sort the vector in descending order, as Thus, the indexes of the top K largest values of L(i) can be denoted by where k ∈ [1, . . . , K]. H(k) denotes the indexes of the transmitted pulses to be mutated. The number of mutated genes is K to improve the diversity. The randomness is introduced by the mutation probability. The mutation probability determines whether the H(k)th transmitted pulse is mutated.

2) Mutating Genes Guided by the Knowledge:
The retrieved knowledge is then used during mutation to guide the GA toward promising solution areas. These selected worse genes PRI H(k) are mutated to get the most rewards of separating Algorithm 1: Knowledge-Guided Mutation Operation.

Input:
The individuals of population : GA seq ; The individual number: N p ; Spatial filter: W ; System parameters: P RI min , P RI max , R min , R max , T p ; Output: GA seqnew 1: for n = 1 : N p , each GA seq , do 2: Calculating the blind area distribution 3: Converting blind area distribution into a matrix I(i, j) 4: Spatial filtering W N a ×1 to evaluate the discontinuity of blind area in azimuth in matrix Counting the numbers of elements in each row satisfying g(i, j) > 0 as L(i) 6: Selecting the genes in the individual to be mutated: Mutating the chosen genes in the individual: for k = 1 : K, each index of gene to be mutated, do Determining whether the genes are mutated or not by the mutation probability for q = 1 : N q , do Generating a random value ξ q , ξ q ∼ U (0, 1) Computing the mutated gene as where ξ q ∼ U (0, 1) ∀q ∈ [1, N q ]. ξ q is a random variable following a uniform distribution to introduce more randomness.
Considering the computational burden, we set N q as 1/10 of the number of individuals in the population in the following simulation. D is a constant value sufficiently smaller than Δ = PRI max − PRI min . Since the choice of PRI value of the current transmitted pulse will affect the blind area locations of the subsequent transmitted pulses, we hope that the single mutation process will not significantly change the blind area distribution, and at the same time, eliminate the continuity of the blind area along the azimuth. Therefore, we choose D = T p . After that, the PRI H(k) is replaced by PRI q H(k)new in the sequence. The diversity of mutated genes could be increased, since each original gene has 2N q mutated genes to generate 2N q new individuals. The probability of fitter individuals tends to increase in each iteration. The PRI H(k)new should satisfy the constrained condition, i.e., PRI H(k)new ∈ [PRI min , PRI max ]; otherwise, this individual will be discarded.
As a result, the GA is guided toward promising solution areas. The knowledge-guided mutation operator effectively improves the possibility of eliminating consecutive blind areas. Compared with the traditional random mutation operator, it can provide a sequence with a lower objective function value and smaller number of iterations, which can further improve the imaging performance after reconstruction.
It should be noted that one drawback of a GA is that it sometimes suffers from premature convergence. Premature convergence occurs when a population has converged to a single solution, but the quality of that solution is not as high as expected. One possible method to avoid premature convergence is to introduce a variety of diversity. In order to increase the diversity in the population, we take two measures to introduce the randomness in the proposed knowledge-guided mutation operator, as follows. 1) In Step 6 of Algorithm 1, instead of mutating all the genes, the top K genes are mutated, and the mutation probability determines whether the individual is mutated. 2) In Step 7 of Algorithm 1, a random variable is used following a uniform distribution when mutating genes guided by the knowledge. These two measures introduce much more randomness to increase the diversity of the individuals, which is beneficial to provide a stable and convergent solution.

B. Optimization for Sequence With Nonlinear PRI Variation Strategy
Based on the knowledge-guided mutation operator, the flowchart of the optimization for the nonlinear PRI variation sequence is shown in Fig. 3. The individual denotes the sequence of variable PRI. The gene represents the PRI of the transmitted pulse in the sequence. The optimization procedure starts with the initialization of the population. The population size can be denoted as N p . The initial population is generated by the sequences of random PRI. After the initial population is generated, the objective function is calculated to evaluate the performance of these initial individuals.
Next, the method of fitness proportionate selection is adopted in the selection operation. The selection probability is proportional to the objective function value of the individual. The probability of choosing ith individual is equal to where F i = 1/R i is the reciprocal of the objective value for the ith individual, and N R denotes the size of the current generation. Individuals are selected by an objective-based process, where sequences with lower objective function values are typically more likely to be selected. Even though the weaker individuals with a higher value will be less likely to be selected, it is still possible they will survive because the probability is not zero.
Subsequently, a combination of genetic operators, i.e., crossover and the knowledge-guided mutation, is performed to generate a new population with suboptimal performance. The constrained condition is that the value of PRI in each individual should satisfy PRI ∈ [PRI min , PRI max ]; otherwise, this individual will be discarded in the operations. The optimization is an iterative process with the population in each iteration called a generation. The generational process is repeated until a solution satisfying minimum criteria is found or a fixed number of iterations is reached. After that, we can obtain the optimized sequence of the proposed strategy.

V. PERFORMANCE OF THE NONLINEAR PRI VARIATION STRATEGY
This section illustrates the performance of the proposed nonlinear strategy. In Section V-A, the sequences of the nonlinear PRI variation is provided. The effectiveness and reliability of the strategy are further verified by Monte Carlo simulations in Section V-B. In Section V-C, the effect of the scene size is analyzed to verify the suitability of the proposed strategy for a given scenario. Finally, the computational complexity is provided in Section V-D.

A. Proposed Nonlinear PRI Variation Strategy
The sequence of nonlinear PRI variation is optimized as the flowchart shown in Fig. 3. The maximum and minimum slant ranges are set as R min = 868 km and R max = 1097 km. The maximum and minimum values of PRI are set as PRI min = 0.560 ms and PRI max = 0.703 ms. The duration of transmitted pulse is T p = 20 μs. The period time is required to be set as T sw = T a (R min )/5, which decreases the strong dependence of the imaging performance on the target azimuth position [27]. T a (R min ) denotes the shortest illumination time corresponding to the minimum slant range. Thus, the number of the transmitted pulses in one period is calculated to be 252 in the simulation. The initial individuals are generated by the random PRI sequences. The values of PRIs in each sequence are chosen in [PRI min , PRI max ]. The size of the spatial filter is N a = 5. The performance of sequences with different PRI variation strategies are provided in Figs. 4 and 5, showing the PRI trend, the blind area distribution, the percentage of lost pulses, and the maximum, mean, and minimum pulses separations. A limited pulse separation may facilitate the reconstruction of lost data by interpolation approaches. Fig. 4 shows the results of the random PRI variation strategy. The results of the slow linear PRI variation strategy, the fast linear PRI variation strategy, the stepwise PRI strategy, the elaborated linear PRI variation strategy, and the proposed nonlinear PRI variation strategy are, respectively, presented in the columns from left to right of Fig. 5.
These five sequences with different PRI variation strategies have the same parameter of PRI max = 0.703 ms. The sequences of slow linear PRI change, fast linear PRI change, and elaborated linear PRI change are generated, as described in [24]. The blind area distribution is extremely sensitive to the parameters of the sequence. The slow linear PRI variation strategy meets the criterion of the minimum PRI span so that the blind areas are distributed across the slant ranges in the whole swaths. There are a large gap made of several consecutive blind areas and a short gap resulting from the periodic jump from PRI min to PRI max . The main advantage is that it has the largest value of PRI min compared with the other strategies, which brings good performance in terms of RASR. The cost is that the large gap will cause high sidelobes in the vicinity of the main lobe in the azimuth impulse response after reconstruction.
The fast linear PRI variation strategy adopts a shorter sequence with considerably faster PRI change, which meets the criterion that two consecutive pulses are never lost for all slant ranges in the desired swath, as shown in Fig. 5. Compared with the slow linear PRI variation strategy, instead of occurring in the vicinity of the main lobe, high sidelobes are presented more distance in the azimuth impulse response. It is because the blind areas have obvious periodicity along the azimuth. The stepwise PRI strategy using multiple constant PRIs can achieve nonconsecutive blind area. The number of PRIs in one cycle is 18. Different from other strategies, the pulse duration is 30 μs to avoid the discrepancy of the imaging result along the slant ranges. The blind areas are less compact compared with the fast linear variation. However, the same as the fast linear strategy, due to the periodicity of the blind areas, it tends to cause more distant high sidelobes in the azimuth impulse response after BLU reconstruction.
The elaborated linear PRI variation strategy is an advanced approach to suppress these high sidelobes. By splicing several short sequences of fast linear PRI variation into a long sequence, the sequence is designed elaborately to eliminate the periodicity of blind areas. Accordingly, the periodic blind areas in the sequence of fast linear PRI are shifted. The random PRI variation strategy can introduce more irregularity in the sequences. The serious drawback is that there are numerous consecutive lost pulses at some specific slant ranges, for instance, at the slant ranges from 930 to 970 km in Fig. 4(b), and there are few blind areas at the other slant ranges, for instance, at the slant ranges from 970 to 1100 km. Consequently, this will lead to a significant discrepancy in the imaging performance after azimuth focusing.
The results of the proposed nonlinear PRI variation strategy are given in the right column of Fig. 5. As shown in Fig. 5(b), the blind area distribution can meet the two requirements, i.e., the blind areas are uniformly distributed along the slant range and there are no consecutive blind areas along the azimuth at each slant range. Since the number of changing PRI is large enough to be equivalent to the number of changing PRI in the sequence of slow linear variation, the periodicity of azimuth sampling within the synthetic aperture time is adequately reduced. Moreover, in the case of determined PRI max , the linear strategy has two degrees of freedom. The values of PRI min and δ are free to vary. Keeping PRI variation nonlinear introduces much more degrees of freedom into the strategy, as each PRI value in the sequence can be changed. The number of degrees of freedom equals the number of PRIs for one period in the sequence. The nonlinear PRI variation sequences introduce much more irregularity, which provides the potential to spread the energy of high sidelobes along azimuth after azimuth focusing. The performance of suppressing high sidelobes will be further discussed and verified in detail in Section VI.
As shown in Fig. 5(c), the percentages of lost pulses in the random PRI strategy, the slow linear PRI strategy, the fast linear PRI strategy, and the stepwise PRI strategy vary dramatically with the slant ranges. However, the percentages of lost pulses in the elaborated PRI variation strategy and proposed nonlinear PRI variation strategy are lower and more uniform along the slant range compared with the other strategies. Especially, in the proposed nonlinear PRI variation strategy, the percentage of lost pulses changes slightly around the value 3%. This phenomenon can also be observed in the pulse separation versus slant ranges, as shown in Fig. 5(d).
Besides, the sequence with proposed nonlinear PRI variation strategy can achieve superior bind area distribution with larger PRI min = 0.560 ms compared with the minimum PRI of PRI min = 0.500 ms in the example of the sequences with elaborated linear PRI variation strategy. This benefit can improve the performance in terms of RASR.
The objective function (Q P ) versus iteration number is plotted as a solid curve in Fig. 6. The convergence rate of Fig. 6 shows a declining trend, illustrating that the algorithm can converge to a set of solutions within a minimum. At the beginning of the iteration, the curve decreases dramatically. As the number of iterations increases, the curve gradually flattens out. The objective values of different PRI variation strategies are also marked in Fig. 6 for comparisons. The objective value of the random PRI variation strategy is equal to 11.67. Since the random PRI is the initial individual in the optimization, we mark it as purple point (0, 11.67) at the beginning of the iteration in Fig. 6. The objective value of the elaborated linear PRI strategy is 0.167, illustrating that in the processing of the proposed algorithm, 34 iterations are required to achieve an equivalent effect as the elaborated linear PRI variation strategy, as marked as yellow point (34, 0.167) in Fig. 6. The dashed line y = 0.167 is plotted to intuitively observe the comparison after 34 iterations. For the optimized nonlinear PRI variation, the minimum of the objective function reaches 0.0293 after 100 iteration, which is marked as red point (100, 0.0293). The low objective function value means that the blind areas can meet the two requirements: the blind areas are distributed uniformly along the slant range and discontinuously along the azimuth. The objective function value of the optimized nonlinear PRI variation strategy is much lower than its competitors. This conclusion is consistent with the results in Fig. 5, indicating that the optimized strategy has the potential to provide superior imaging performance after reconstruction.

B. Monte Carlo Simulation
In order to further verify the stability and the effectiveness of the proposed strategy, the Monte Carlo simulations are applied. A total of 50 trials are performed for the design of the sequences with nonlinear PRI variation strategy. Each trial can obtain the curve of the objective function changing with the number of iterations. The mean square errors (mses) versus the number of iterations are plotted in Fig. 7. In the first iteration, the individuals are sequences of random PRI, which have large mses of the objective function. With the increase of the iterations, the mse curve has a declining trend, as shown in Fig. 7. Especially, when the number of iterations exceeds 80, the curve gradually flattens out, and finally achieves a mse value of 0.00081 at 100 iterations. The results show that the mse curve of the objective function converges with the increase of the iteration numbers, which further verifies the effectiveness and stability of the proposed strategy.

C. Effect of the Scene Size
The proposed strategy can generate an optimized sequence of nonlinear PRIs when the scene size is large. Especially, the scope of the slant ranges is from 868 to 1097 km in Section V-A, and the slant range coverage can reach 229 km (the ground range coverage is 335 km with the geometry in Table I). In order to verify the suitability of the proposed strategy for a given scenario, the effect of the scene size on the sequences of nonlinear PRI variation is analyzed.
A smaller ground range coverage is adopted, which means that the number of columns in the matrix of blind area distribution is fewer in the calculation of the objective function.  The maximum and minimum slant ranges are set as R min = 900 km and R max = 1050 km. The ground range coverage is 242 km. After 100 iterations, the PRI trend, the blind area distribution, the percentage of lost pulses, and the maximum, mean, and minimum pulses separations for the sequences with nonlinear PRI variation strategy are shown in Fig. 8. The minimum of the objective function can reach to 0.0030, illustrating that the proposed strategy has the robustness and suitability of the scene size for a given scenario.

D. Computational Complexity
The computation complexity mainly depends on the calculation of the objective function and the genetic operators including selection, crossover, and mutation. The computation complexity is T total = P NA (T object + T genetic ), where P NA is the number of the generations related to M (i.e., P NA ∝ M ). Especially, for the proposed objective function, the computational complexity is T object = O (GN M N a ), where the three subterms are caused by the conversion of the matrix of the blind area distribution, the spatial filtering and the relative entropy, respectively. G is the population size. The size of individual M is the number of PRIs in one period. N denotes the number of slant range units. N a represents the size of the spatial filter in (14). The complexity of the GA is in the order of T genetic = P M O (GN M N a ) + P c O(G) + P s O(G log G), where P M , P C , and P S represent the probabilities of the mutation, crossover, and selection. Thus, the total computation complexity is in the order of T total = P NA (O (GN M N a ) GN M N a +GP s log G). The abovementioned complexity is calculated on the condition that a fixed number of iterations is reached. A rigorous runtime analysis is proposed, and the nontrivial lower bounds of the runtime of the standard GA are deduced rigorously in [41].
Moreover, in this article, the proposed algorithm has been implemented by a personal computer with an Intel Core i7-9750H (2.6 GHz) and 16-GB memory. The operation time for 100 iterations is 11 881.7 s. The sequence of variable PRI is completed in the early stage of the radar system design on the ground, which is only designed once or a few times within a mission lifetime. The modern SAR satellites support many wave positions sets, such as hundreds. As one of the important wave position parameters, the PRI values of sequence need to be calculated on the ground in advance and stored in on-board memory.

VI. IMAGING RESULTS WITH THE NONLINEAR PRI VARIATION STRATEGY
From the abovementioned analysis, the nonlinear PRI variation strategy ensures a low objective function value. It has the potential to suppress high sidelobes in the azimuth pulse response, providing good imaging performance after reconstruction. To further verify this, simulations and experiments on point target, extended target, and equivalent SAR data are performed using BLU reconstruction.

A. Results and Analysis on Point Target
1) Imaging Performance on Point Target: The BLU interpolation algorithm is performed on the point targets to verify that the proposed PRI variation strategy is feasible and capable of improving the imaging performance. The parameters of the radar system are listed in Table I. The slant range of the simulated point target is at 935 km. The sequences with different PRI variation in Fig. 5 are utilized in the simulation.
After reconstructing by BLU interpolation, the imaging results of point targets with different PRI variation strategies are shown in Fig. 9. An azimuth Hamming window is applied to compensate for the azimuth pattern. The profiles of the azimuth impulse response in Fig. 10 display the details of the sidelobes. In order to visualize the sidelobes close to the target more in more detail, the zoomed-in versions of Figs. 9 and 10 are also provided with a span of 560 azimuth indexes in Figs. 11 and 12. For the imaging result of random PRI, numerous gaps at the slant range of 935 km lead to high sidelobes in azimuth. These high sidelobes appear throughout the azimuth, resulting in weak imaging performance. For the imaging result of the slow   PRI variation strategy, high sidelobes occur in the vicinity of the main lobe in azimuth, especially when the azimuth indexes are between 2600 and 2900. Instead of high sidelobes in the vicinity of the main lobe, the imaging results of the fast linear PRI strategy and stepwise PRI strategy have more distant high sidelobes far from the main lobe, as shown in Fig. 9(c) and (d). The elaborated linear PRI variation strategy suppress such high sidelobes caused by the two other linear strategies. Compared with the elaborated linear strategy, the proposed nonlinear PRI variation strategy further spread the energies of these high sidelobes along the azimuth, as shown in Figs. 9(e) and 10(b). This phenomenon demonstrates that the proposed nonlinear PRI variation strategy is feasible and capable of suppressing the high sidelobes. In general, compared with the other competitors, Fig. 12. Zoomed-in profiles of Fig. 10 with a span of 500 azimuth indexes. (a) Slow linear PRI variation strategy, fast linear PRI variation strategy, stepwise PRI variation, and elaborated PRI linear PRI variation strategy. (b) Random PRI variation strategy, elaborated PRI linear variation strategy, and the proposed nonlinear PRI variation strategy. In order to access the image quality quantitatively, the peak sidelobe ratio (PSLR), the integrated sidelobe ratio (ISLR), the azimuth ambiguity-to-signal, and the relative error are calculated. The AASR in staggered mode can be estimated as the difference of the attained ISLR and the reference ISLR, i.e., AASR ∼ = ISLR − ISLR ref , which was proposed and analyzed thoroughly in [16]. The reference ISLR ref refers to the ISLR of a constant PRI SAR with PRI mean , where the azimuth antenna pattern equals to zero if the Doppler frequency satisfies f a / ∈ [−PRI mean /2, PRI mean /2]. It should be noted that the values of PRI min differ slightly for the sequence examples of different PRI variation strategies, and thus, the values of PRI mean are also slightly different. The relative error can be defined as the ratio of the error energy to the reference signal energy [22], as where N s is the number of samples in azimuth,ŝ[l] denotes the imaging result in staggered mode, and s[l] represents the reference imaging result of uniform sampling with constant PRI, satisfying PRF mean = √ PRF max · PRF min . Results in Table II demonstrate that the proposed nonlinear PRI variation strategy has the better performance in terms of PSLR, ISLR, and relative error compared to its rivals, indicating that superior image quality could be achieved.
2) Effect of the Number of Iterations on Imaging Performance: With the increase of iterations, the sequences of the nonlinear PRI variation have lower values of objective function, as shown in Fig. 6. The lower objective values tend to provide superior imaging performance. In order to further verify the reliability of the proposed strategy, the imaging performance in terms of Fig. 13. Imaging performance in terms of the PSLR, the ISLR, and the relative error versus the number of iterations. the PSLR, the ISLR, and the relative error versus the number of iterations are shown in Fig. 13. Since the initial individuals are sequences of random PRIs, the imaging performance is poor. Especially the ISLR is higher than −10 dB, which is not acceptable in practical systems. As the number of iterations increases, the curve declines and flattens out gradually, illustrating that the solution of the nonlinear PRI sequence is effective and stable in providing good imaging performance. Moreover, after 60 iterations, increasing the number of iterations cannot improve the imaging performance significantly. This phenomenon is consistent with the curve of objective function in Fig. 6.
3) Effect of Slant Range on Imaging Performance: The imaging results in the previous subsections discuss the case where the target is located in the center of the scene. As shown in Fig. 5(c) and (d), the proposed nonlinear variation strategy can ensure that blind areas are uniformly distributed along the slant range, which means that it has the potential to decrease the sensitivity of the performance against the slant range positions of the targets. In order to further verify the sensitivity of the imaging performance,  the imaging performance in terms of the PSLR, the ISLR, and the relative error versus the target slant range position is shown in Fig 14. The scope of the slant range is from 880 to 1080 km. The differences of the PSLR and the ISLR are around −3 dB. The scope of the relative error is from 0.0009 to 0.0011. The imaging performance shows only little dependency on target position in slant ranges. 4) Effect of the SNR on Imaging Performance: In order to analyze the effect of the SNR on imaging performance, the PSLR, the ISLR, and the relative error are shown in Fig. 15. The SNR varies from −30 to 30 dB with an interval of 5 dB. In total, 13 groups of echo data are generated by adding GWN of different powers in the case of the nonlinear PRI variation strategy. It can be found that the reconstruction performance degrades gradually with the increase of the noise levels. The BLU reconstruction algorithm takes into account the effect of SNR, as illustrated in (9). Thus, the imaging performance is still acceptable for a low SNR of −15 dB and flatten out after that as shown in Fig. 15.

5) Effect on RASR:
The staggered SAR operation has a significant effect on range ambiguity, and it is necessary to analyze the effect of PRI sequences on RASR. Different from the constant PRI, the range ambiguities are located at different ranges for the targets at different range units. Villano et al. [16] provided a sophisticated method to evaluate RASR in staggered SAR mode. Here, we adopt the same method to calculate the RASR. The number of elevation channels is 32. The off-nadir angle varies from 23.3 to 40.5 • covering the ground range from 333 to 668 km. Assume that the antenna gain matches the pattern of 32 elevation elements and four azimuth elements after beamforming. Fig. 16 shows the RASR of different PRI variation strategies, using the example of sequences in Fig. 5. In Fig. 16(a), the constant PRI sequence causes three regions with blind ranges. With the same PRI span, the proposed nonlinear PRI sequence has lower RASR than the random PRI sequence. In Fig. 16(b), the examples of the fast linear PRI sequence and the slow linear PRI sequence have larger PRI min than other sequences, leading to the improvement of RASR to some extent. The examples of the proposed nonlinear PRI sequence and the elaborated linear PRI sequence have the same PRI max , whereas the PRI min of the former is 0.06 ms larger than the latter. The nonlinear PRI sequence has a more uniform gap distribution, providing a slight improvement in the performance of RASR. It should be mentioned that the improvement is limited, especially when the ground range varies from 400 to 450 km.

B. Results and Analysis on Extended Target
Imaging results and analysis on extended targets are provided to further demonstrate the robustness of the proposed nonlinear PRI variation strategy. The simulated parameters are the same as in Table I. The continuous scene consists of 400 × 300 (azimuth × range) point targets. The probability distribution of radar cross section (RCS) is assigned to the extended target to simulate the scattering properties. The simulated distribution model is the logarithmic normal distribution, which can describe   [42]. In the simulation, the expectation of lognormal distribution μ is set to 1.5, and the variance σ is set to 0.6. The SNR is equal to 35 dB for the different PRI sequences. Fig. 17 shows the imaging results of extended targets with different PRI variation strategies. From Fig. 17(a)-(d), there are several artifacts of the true extended target after reconstruction by the BLU interpolation. These artifacts are not only spread in azimuth, but also exist in slant-range dimension, which degrades the image quality. As shown in Fig. 17(d) and (e), the artifacts can be effectively suppressed by using the elaborated linear PRI variation strategy and the proposed nonlinear PRI variation strategy.
The profiles of azimuth impulse response with different PRI variation strategies are shown in Fig. 18. Fig. 18(a) shows that the elaborated linear PRI variation strategy can spread the energies of high sidelobes along azimuth. It provides superior imaging performance compared with other linear PRI variation strategies. In Fig. 18(b), the random PRI variation strategy has the poor performance in terms of high sidelobes and ambiguities, which is caused by the numerous consecutive lost pulses. The proposed nonlinear PRI variation strategy can effectively suppress these high sidelobes and eliminate the ambiguities by introducing much more irregularity into the sequences.
To access the image quality of extended targets quantitatively, the image contrast (IC) and the relative error are calculated, as given in Table III. As the reference sampled on uniform grid can also be generated, the relative measurement of the error is obtained by the ratio of the energy of the error to the energy of the reference, as    that the imaging results of the sequences with the proposed nonlinear PRI variation strategy have the largest IC and the smallest relative error, indicating that superior image quality can be achieved.

C. Experimental Results With SAR Data
The simulations are performed with Gaofen-3 data to obtain the SAR echo in staggered mode. The real spaceborne data were acquired by the Gaofen-3 satellite over the city of Dalian in China. The size of the scene is 35 km × 35 km (azimuth × range). The large scene is located on the coastline and consists of farmland, hills, villages, and multiple marine targets sailed on the sea. The targets in the scene have different scattering characteristics. The generation of echo in staggered mode was proposed and further analyzed thoroughly in [25]. Two-point linear interpolation is performed to sample the echo with different PRI variation strategies. The parameters of PRI variation strategies are the same as those in Section V.
The imaging results of the scene are shown in Fig. 19 with different PRI variation strategies. To better appreciate the advantage of the proposed strategy over the others, the differences in log-intensity between the reconstructed images and the reference image are also provided in Fig. 19 with different PRI variation strategies.
The reference image is obtained by constant PRI with uniformly sampled data, as PRI mean = 1/PRF mean = 1/ √ PRF max · PRF min . The yellow pixels in Fig. 19 indicate significant differences between the reference image and the focused images in staggered mode, and black pixels indicate little differences. Compared with the image of random PRI variation strategy, the images of the linear PRI variation strategies are better reconstructed. Among these results, the image of the proposed nonlinear PRI variation strategy has the minimal difference with the reference image, which shows the proposed nonlinear PRI variation strategy can provide superior imaging performance after reconstruction.
The scene can be divided into three regions, namely Region-1, Region-2, and Region-3, based on scattering characteristics for further analysis. Region-1 is the land surface, including farmland, village roads, and hills. Region-2 consists of the sea surface that is relatively close to the land. Region-3 in the bottom right of the image is the sea surface with large waves, which is farther away from the land. The fast linear PRI variation strategy, the stepwise PRI strategy, and the elaborated linear PRI variation strategy can achieve accurate imaging performance in Region-1, but the elaborated linear PRI variation strategy can achieve substantially better imaging performance in Region-3. The slow linear PRI variation strategy and the random PRI variation strategy have relatively poor imaging performance in Region-1. Especially, compared with other strategies, the random PRI variation strategy leads to the worst performance in Region-3. The elaborated linear PRI variation strategy and the proposed nonlinear PRI variation strategy show significantly superior imaging performance in Region-2 compared with other strategies. Furthermore, the proposed nonlinear strategy can obtain slightly better imaging performance than the elaborated linear PRI variation strategy, especially in Region-1.
To access the image quality quantitatively, the relative error and the IC are calculated. The reconstructed image with the proposed nonlinear PRI variation strategy achieves the smallest relative error and the largest IC, as given in Table IV. It can be concluded that the proposed nonlinear PRI variation strategy is feasible and capable of improving the performance of the reconstructed images in staggered SAR mode. The effectiveness and the robustness have been demonstrated with the simulation and raw data experiment results.

VII. DISCUSSION
This article evaluates the effect of PRI variation strategies on imaging performance. We reveal that the nonlinear PRI variation strategy has the potential to improve imaging performance in staggered SAR mode. The findings match those observed in existing studies that focus on linear PRI variation strategies. The advanced elaborated linear PRI variation strategy introduces more irregularity to spread the energy of high sidelobes along the azimuth achieving impressive results. Similar to the existing elaborated linear PRI strategy, we considered the random PRI strategy to introduce irregularity. It is not very surprising that it causes a significant discrepancy in the reconstructed images for different slant ranges. Thus, we transform the design of the PRI strategy into an optimization problem and use the random PRI sequences as initial individuals. The proposed nonlinear PRI strategy introduces much more irregularity than the existing linear strategies, and can eliminate the discrepancy compared with the random PRI strategy.
One limitation of our work is that the proposed strategy cannot be found a closed form since we optimize the PRI values of the sequences. The blind area cannot be controlled straightforwardly without a closed form. Actually, the optimized parameters and the objective function are flexible. To control the blind areas in a straightforward way, a possible method in future work is to optimize the parameters of a closed form instead of optimizing the PRI values. Furthermore, the objective functions can be designed according to the requirements of the specific scenarios. A possibility of adaptive sequence design is also provided for different scenarios in future studies.

VIII. CONCLUSION
In this article, an optimized nonlinear PRI variation strategy is proposed in staggered SAR mode. The nonlinear variation strategy is introduced into the design of the PRI sequence, which can overcome the limitations of the linear PRI variation strategy and eliminate the discrepancy caused by the random PRI variation strategy. A novel objective function is defined to quantitatively measure the uniformity and the discontinuity of the blind area distribution. Subsequently, we transform the design of the nonlinear PRI variation strategy into an optimization problem. Then, to effectively solve the optimization problem, an improved GA is proposed by modifying the mutation operator. Compared with random PRI variation strategy and linear PRI variation strategies, the proposed nonlinear PRI variation strategy can suppress the high sidelobes in the azimuth impulse response, which can obtain superior imaging performance after reconstruction. Simulations and experimental results demonstrate the effectiveness and feasibility of the proposed nonlinear PRI variation strategy in staggered SAR mode.