\DeclareAcronym

snrshort = SNR,long = signal-to-noise ratio,\DeclareAcronympdfshort = PDF,long = probability density function,\DeclareAcronymsarshort = SAR,long = synthetic aperture radar,\DeclareAcronyminsarshort = InSAR,long = interferometric synthetic aperture radar,\DeclareAcronymaoshort = AO,long = alternating optimization,long-plural-form = alternating optimizations\DeclareAcronymmimoshort = MIMO,long = multiple-input multiple-output\DeclareAcronymuavshort = UAV,long = unmanned aerial vehicle,long-plural-form = unmanned aerial vehicles\DeclareAcronymfdmashort = FDMA,long = frequency-division multiple-access,\DeclareAcronym1dshort = 1D,long = one-dimensional,\DeclareAcronymislrshort = ISLR,long = integrated sidelobe ratio,\DeclareAcronympslrshort = PSLR,long = peak sidelobe ratio,\DeclareAcronym3dshort = 3D,long = three-dimensional,\DeclareAcronympsoshort = PSO,long = particle swarm optimization,\DeclareAcronym2dshort = 2D,long = two-dimensional,\DeclareAcronymdemshort = DEM,long = digital elevation model,\DeclareAcronymgsshort = GS,long = ground station,long-plural-form = ground stations\DeclareAcronymlosshort = LOS,long = line-of-sight,\DeclareAcronymscashort = SCA,long = successive convex approximation,\DeclareAcronymneszshort = NESZ,long = noise equivalent sigma zero,\DeclareAcronymwrtshort = w.r.t.,long = with respect to ,\DeclareAcronymrhsshort = r.h.s,long = right-hand side ,\DeclareAcronymgmtishort = GMTI,long = ground moving target indication,\DeclareAcronymlhsshort = l.h.s,long = left-hand side ,\DeclareAcronymbcdshort = BCD,long = block coordinate descent,\DeclareAcronymhoashort = HoA,long = height of ambiguity,

Mohamed-AmineLahmeri1, Víctor Mustieles-Pérez12, Martin Vossiek1, Gerhard Krieger12, andRobert Schober1

1Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Germany

2German Aerospace Center (DLR), Microwaves and Radar Institute, Weßling, Germany

###### Abstract

In this paper, we study the optimization of the sensing accuracy of unmanned aerial vehicle (UAV)-based dual-baseline interferometric synthetic aperture radar (InSAR) systems. A swarm of three UAV-synthetic aperture radar (SAR) systems is deployed to image an area of interest from different angles, enabling the creation of two independent digital elevation models (DEMs). To reduce the InSAR sensing error, i.e., the height estimation error, the two DEMs are fused based on weighted average techniques into one final DEM. The heavy computations required for this process are performed on the ground. To this end, the radar data is offloaded in real time via a frequency division multiple access (FDMA) air-to-ground backhaul link. In this work, we focus on improving the sensing accuracy by minimizing the worst-case height estimation error of the final DEM. To this end, the UAV formation and the power allocated for offloading are jointly optimized based on alternating optimization (AO), while meeting practical InSAR sensing and communication constraints. Our simulation results demonstrate that the proposed solution can improve the sensing accuracy by over 39% compared to a classical single-baseline UAV-InSAR system and by more than 12% compared to other benchmark schemes.

## I Introduction

The use of \acuav swarms for remote sensing has recently gained attention due to their flexibility and efficiency in data collection tasks [1]. This has led to an increased use of drones in diverse applications, such as mapping, monitoring traffic, and addressing climate change [2]. For these tasks, a variety of sensors can be deployed onboard, including cameras, LiDARs, and radars. In particular, the deployment of \acsar on \acpuav has attracted significant interest due to its ability to provide very high-resolution \acsar images over local areas, even under challenging conditions, overcoming the limitations of traditional airborne and spaceborne systems. This integration has sparked multiple recent studies focusing on system design [3], trajectory and resource allocation optimization [4, 5, 6], and experimental measurement campaigns for \acuav-\acsar systems [7].

An interesting remote sensing application of \acuav swarms is \ac3d radar imaging, which can be realized using techniques, such as \acmimo radar, tomography, and interferometry [8]. In particular, \acinsar leverages the phase differences between at least two \acsar images, captured from different angles, to extract topographic information and generate \acpdem. Key performance metrics in interferometry include \acsnr, coverage, coherence, \achoa, and height error [9, 10], which are affected by the interferometric baseline, i.e., the distance between the sensing platforms. While \acinsar has been extensively studied for spaceborne and airborne platforms [8], the optimization of \acinsar performance for \acuav-based systems remains largely unexplored. In our recent research work [11], we investigated formation and resource allocation optimization for maximizing the \acinsar coverage, but for a single-baseline \acuav-\acinsar system. In contrast, dual-baseline \acinsar systems offer advantages, such as enhanced phase unwrapping and improved sensing accuracy [12]. However, results from single-baseline systems [11] do not apply to dual-baseline systems due to the different expressions for the height error and the use of multiple acquisition geometries.

In this work, we study a dual-baseline \acuav-based \acinsar system, where a swarm consisting of one master and two slave \acpuav is deployed to generate two independent \acpdem of a target area, which are then fused into a single \acdem based on weighted averaging [12]. Additionally, the radar data is offloaded to the ground in real time. We investigate the joint optimization of the \acuav formation and communication power allocation for minimization of the worst-case height error in the final \acdem under communication and sensing constraints. Our main contributions can be summarized as follows:

- •
We propose an approximate bi-static \acsnr expression valid for the considered sensing application.

- •
Based on the Cramér–Rao bound of the phase error, we derive a tractable upper bound for the complex expression of the height error of the final \acdem.

- •
We formulate and solve a joint optimization problem for \acuav formation and communication power allocation to minimize the derived upper bound on the height error, while satisfying sensing and communication constraints.

- •
Our simulation results demonstrate the effectiveness of the considered dual-baseline \acinsar system compared to single-baseline systems and other benchmark schemes.

Notations:In this paper, lower-case letters $x$ refer to scalar variables, while boldface lower-case letters $\mathbf{x}$ denote vectors. $\{a,...,b\}$ denotes the set of all integers between $a$ and $b$. $|\cdot|$ denotes the absolute value operator. $\mathbb{R}^{N}$ represents the set of all $N$-dimensional vectors with real-valued entries. For a vector $\mathbf{x}=(x_{1},...,x_{N})^{T}\in\mathbb{R}^{N}$, $||\mathbf{x}||_{2}$ denotes the Euclidean norm, whereas $\mathbf{x}^{T}$ stands for the transpose of $\mathbf{x}$. For real-valued multivariate functions $f(\mathbf{x})$, $\frac{\partial f}{\partial\mathbf{x}}(\mathbf{a})=\Big{(}\frac{\partial f}{%\partial x_{1}}(\mathbf{a}),...,\frac{\partial f}{\partial x_{N}}(\mathbf{a})%\Big{)}^{T}$ denotes the partial derivative of $f$ \acwrt $\mathbf{x}$ evaluated for an arbitrary vector $\mathbf{a}$. For any Boolean expression $\mathcal{S}$, $\mathds{1}\{\mathcal{S}\}$ denotes the indicator function, which equals 1 if $\mathcal{S}$ is true and 0 otherwise.

## II System Model

We consider three rotary-wing \acpuav, denoted by $U_{k},k\in\{0,1,2\}$, performing \acinsar sensing over a target area. $U_{0}$, the master drone, transmits and receives radar echoes, while $U_{1}$ and $U_{2}$, the slave drones, only receive. We use a \ac3d coordinate system, where the $x$-, $y$-, and $z$-axes represent the range direction, the azimuth direction, and the altitude, respectively. The mission time $T$ is divided into $N$ slots of duration $\delta_{t}$, with $T=N\cdot\delta_{t}$. The drone swarm forms a dual-baseline interferometer with two independent observations acquired by $(U_{0},U_{1})$ and $(U_{0},U_{2})$, respectively. The considered \acuav-\acsar systems operate in stripmap mode [13] and fly at a constant velocity, $v_{y}$, following a linear trajectory that is parallel to a line, denoted by $l_{t}$, which is parallel to the $y$-axis and passes in time slot $n$ through reference point $\mathbf{p}_{t}[n]=(x_{t},y[n],0)^{T}\in\mathbb{R}^{3}$, see Figure 1. The position of $U_{k}$ in time slot $n\in\{1,...,N\}$ is $\mathbf{q}_{k}[n]=(x_{k},y[n],z_{k})^{T}$, with the $y$-axis position vector $\mathbf{y}=(y[1]=0,y[2],...,y[N])^{T}\in\mathbb{R}^{N}$ given by:

$\displaystyle y[n+1]=y[n]+v_{y}\delta_{t},\forall n\in\{1,N-1\}.$ | (1) |

For simplicity, we denote the position of $U_{k}$ in the across-track plane (i.e., $xz-$plane) by $\mathbf{q}_{k}=(x_{k},z_{k})^{T}\in\mathbb{R}^{2},\forall k\in\{0,1,2\}$. The interferometric baseline, $b_{k}$, which refers to the distance between sensors $U_{0}$ and $U_{k}$, is given by:

$\displaystyle b_{k}(\mathbf{q}_{0},\mathbf{q}_{k})=||\mathbf{q}_{k}-\mathbf{q}%_{0}||_{2},\forall k\in\{1,2\}.$ | (2) |

The perpendicular baseline, denoted by $b_{\bot,k}$, is the magnitude of the projection of $U_{k}$’s baseline vector perpendicular to $U_{0}$’s \aclos to $\mathbf{p}_{t}[n]$ and is given by:

$b_{\bot,k}(\mathbf{q}_{0},\mathbf{q}_{k})=b_{k}(\mathbf{q}_{0},\mathbf{q}_{k})%\cos\Big{(}\theta_{0}-\alpha_{k}(\mathbf{q}_{0},\mathbf{q}_{k})\Big{)},\forallk%\in\{1,2\},$ | (3) |

where $\theta_{0}$ is the fixed look angle that $U_{0}$’s \aclos has with the vertical, and $\alpha_{k}$ is the angle between the interferometric baseline $b_{k}$ and the horizontal plane.

### II-A \acinsar Performance

Next, we introduce the relevant \acinsar sensing performance metrics.

#### II-A1 \acinsar Coverage

Let $r_{k},k\in\{0,1,2\}$, denote $U_{k}$’s slant range \acwrt $\mathbf{p}_{t}[n]$. The slant range is independent of time and is given by:

$\displaystyle r_{k}(\mathbf{q}_{k})=\sqrt{(x_{k}-x_{t})^{2}+z_{k}^{2}},\forallk%\in\{0,1,2\}.$ | (4) |

The radar swath is designed to be centered \acwrt $l_{t}$. To this end, the look angle of the slave \acpuav, denoted by $\theta_{k}(\mathbf{q}_{k}),k\in\{1,2\}$, is adjusted such that the beam footprint is centered around $\mathbf{p}_{t}$, i.e., $\theta_{k}(\mathbf{q}_{k})=\arctan\left(\frac{x_{k}-x_{t}}{z_{k}}\right)$. The swath width of $U_{k}$ can be approximated as follows [13]:

$\displaystyle S_{k}(\mathbf{q}_{k})=\frac{\Theta_{\rm 3dB}r_{k}(\mathbf{q}_{k}%)}{\cos(\theta_{k}(\mathbf{q}_{k}))},\forall k\in\{0,1,2\},$ | (5) |

where $\Theta_{\mathrm{3dB}}$ is the -3 dB beamwidth in elevation.

#### II-A2 \Acinsar Coherence

A key performance metric for \acinsar is coherence, representing the cross-correlation between two \acsar images. For the images acquired by $(U_{0},U_{k}),k\in\{1,2\}$, the total coherence can be decomposed into several decorrelation sources as follows:

$\gamma_{k}(\mathbf{q}_{0},\mathbf{q}_{k})=\gamma_{\mathrm{Rg},k}(\mathbf{q}_{k%})\gamma_{\mathrm{SNR},k}(\mathbf{q}_{0},\mathbf{q}_{k})\gamma_{\rm other},%\forall k\in\{1,2\},$ | (6) |

where $\gamma_{\mathrm{Rg},k}$ is the baseline decorrelation, $\gamma_{\mathrm{SNR},k}$ is the \acsnr decorrelation, and $\gamma_{\rm other}$ represents the contribution from all other decorrelation sources. The \acsnr decorrelation of pair $(U_{0},U_{k})$ is affected by the \acpsnr of both \acpuav and is given by [10]:

$\displaystyle\gamma_{\mathrm{SNR},k}(\mathbf{q}_{0},\mathbf{q}_{k})=\frac{1}{%\sqrt{1+\mathrm{SNR}^{-1}_{0}(\mathbf{q}_{0})}}\frac{1}{\sqrt{1+\mathrm{SNR}^{%-1}_{k}(\mathbf{q}_{0},\mathbf{q}_{k})}},$ | (7) |

where $\mathrm{SNR}_{0}$ denotes the \acsnr of the mono-static acquisition by $U_{0}$ given by [10]:

$\mathrm{SNR}_{0}(\mathbf{q}_{0})=\frac{\gamma_{m}}{r_{0}^{3}(\mathbf{q}_{0})},$ | (8) |

where $\gamma_{m}=\frac{\sigma_{0}P_{t}\;G_{t}\;G_{r}\lambda^{3}c\tau_{p}\mathrm{PRF}%}{4^{4}\pi^{3}v_{y}\sin(\theta_{0})k_{b}T_{\mathrm{sys}}\;B_{\mathrm{Rg}}\;F\;L}$. Here, $\sigma_{0}$ is the normalized backscatter coefficient, $P_{t}$ is the radar transmit power, $G_{t}$ and $G_{r}$ are the transmit and receive antenna gains, respectively, $\lambda$ is the radar wavelength, $c$ is the speed of light, $\tau_{p}$ is the pulse duration, $\mathrm{PRF}$ is the pulse repetition frequency, $k_{b}$ is the Boltzmann constant, $T_{\mathrm{sys}}$ is the receiver temperature, $B_{\mathrm{Rg}}$ is the bandwidth of the radar pulse, $F$ is the noise figure, and $L$ represents the total radar losses. The derivation of the bi-static \acsnr for $U_{k}$, denoted by $\mathrm{SNR}_{k}$, is more complicated. Here, assuming a small bi-static angle $|\theta_{0}-\theta_{k}|$, which holds for \acinsar applications [8], we propose the following approximation^{1}^{1}1Please find a detailed derivation of the approximated bi-static \acsnr expression in Appendix A.:

$\mathrm{SNR}_{k}(\mathbf{q}_{0},\mathbf{q}_{k})\approx\frac{\gamma_{m}}{r_{0}^%{2}(\mathbf{q}_{0})r_{k}(\mathbf{q}_{k})},\forall k\in\{1,2\}.$ | (9) |

Furthermore, the baseline decorrelation reflects the loss of coherence caused by the different angles used for the acquisition of both \acinsar images [14]:

$\gamma_{\mathrm{Rg},k}(\mathbf{q}_{k})=\frac{1}{B_{p}}\left[\frac{2+B_{p}}{1+%\mathcal{X}(\mathbf{q}_{k})}-\frac{2-B_{p}}{1+\mathcal{X}^{-1}(\mathbf{q}_{k})%}\right],$ | (10) |

where $B_{p}=\frac{B_{\mathrm{Rg}}}{f_{0}}$ is the fractional bandwidth, $f_{0}$ is the radar center frequency, and function $\mathcal{X}$ is given by [14]:

$\mathcal{X}(\mathbf{q}_{k})=\frac{2\Big{(}\sin(\theta_{0})\mathds{1}\{\theta_{%0}>\theta_{k}(\mathbf{q}_{k})\}+\sin(\theta_{k}(\mathbf{q}_{k}))\mathds{1}\{%\theta_{0}\leq\theta_{k}(\mathbf{q}_{k})\}\Big{)}}{\sin(\theta_{0})+\sin(%\theta_{k}(\mathbf{q}_{k}))}.$ | (11) |

#### II-A3 Height of Ambiguity (HoA)

The \achoa is related to the sensitivity of the radar system to topographic height variations [10]. The \achoa of pair $(U_{0},U_{k})$ is given by [10]:

$\displaystyle h_{\mathrm{amb},k}(\mathbf{q}_{0},\mathbf{q}_{k})=\frac{\lambda r%_{0}(\mathbf{q}_{0})\sin(\theta_{0})}{b_{\perp,k}(\mathbf{q}_{0},\mathbf{q}_{k%})},\forall k\in\{1,2\}.$ | (12) |

#### II-A4 \acdem Height Accuracy

The height error of the \acdem acquired by the \acinsar pair $(U_{0},U_{k})$ is given by [10]:

$\sigma_{h_{k}}(\mathbf{q}_{0},\mathbf{q}_{k})=h_{\mathrm{amb},k}(\mathbf{q}_{0%},\mathbf{q}_{k})\frac{\sigma_{\Phi_{k}}(\mathbf{q}_{0},\mathbf{q}_{k})}{2\pi}%,\forall k\in\{1,2\},$ | (13) |

where $\sigma_{\Phi_{k}}$ is the random error in the interferometric phase and can be approximated in the case of high interferometric coherences by the Cramér–Rao bound [8]:

$\sigma_{\Phi_{k}}(\mathbf{q}_{0},\mathbf{q}_{k})=\frac{1}{\gamma_{k}(\mathbf{q%}_{0},\mathbf{q}_{k})}\sqrt{\frac{1-\gamma_{k}^{2}(\mathbf{q}_{0},\mathbf{q}_{%k})}{2n_{L}}},\forall k\in\{1,2\},$ | (14) |

where $n_{L}$ is the number of independent looks employed, i.e., $n_{L}$ adjacent pixels of the interferogram are averaged to improve phase estimation [8]. The fusion of the two independent \acinsar \acpdem is performed based on inverse-variance weighting, such that the height of an arbitrary target estimated by \acinsar pair $(U_{0},U_{k})$, denoted by $h_{k}$, is weighted by $w_{k}(\mathbf{q}_{0},\mathbf{q}_{k})=\frac{1}{\sigma^{2}_{h_{k}}(\mathbf{q}_{0%},\mathbf{q}_{k})}$, $k\in\{1,2\}$, and averaged as $\frac{h_{1}w_{1}+h_{2}w_{2}}{w_{1}+w_{2}}$. The final height error of the fused \acdem is characterized by [12]:

$\displaystyle\sigma_{h}(\mathbf{q}_{0},\mathbf{q}_{1},\mathbf{q}_{2})=\sqrt{%\frac{\sum\limits_{k\in\{1,2\}}w_{k}^{2}(\mathbf{q}_{0},\mathbf{q}_{k})\sigma^%{2}_{h_{k}}(\mathbf{q}_{0},\mathbf{q}_{k})}{\left(\sum\limits_{k\in\{1,2\}}w_{%k}(\mathbf{q}_{0},\mathbf{q}_{k})\right)^{2}}}.$ | (15) |

### II-B Communication Performance

We consider real-time offloading of the radar data to a \acgs, where the master and slave \acpuav employ \acfdma. The instantaneous communication transmit power consumed by \acuav $U_{k}$ is given by $\mathbf{P}_{\mathrm{com},k}=(P_{\mathrm{com},k}[1],...,P_{\mathrm{com},k}[N])^%{T}\in\mathbb{R}^{N},k\in\{0,1,2\}$.We denote the location of the \acgs by $\mathbf{g}=(g_{x},g_{y},g_{z})^{T}\in\mathbb{R}^{3}$ and the distance from $U_{k}$ to the \acgs by $d_{k,n}(\mathbf{q}_{k})=||\mathbf{q}_{k}[n]-\mathbf{g}||_{2},\forall k\in\{0,1%,2\},\forall n.$ Thus, adopting the free-space path loss model and \acfdma, theinstantaneous throughput from $U_{k},\forall k\in\{0,1,2\},$ to the \acgs is given by:

$\displaystyle R_{k,n}(\mathbf{q}_{k},\mathbf{P}_{\mathrm{com},k})=B_{c,k}\;%\log_{2}\left(1+\frac{P_{\mathrm{com},k}[n]\;\beta_{c,k}}{d_{k,n}^{2}(\mathbf{%q}_{k})}\right),\forall n,$ | (16) |

where $B_{c,k}$ is $U_{k}$’s fixed communication bandwidth and $\beta_{c,k}$ is the reference channel gain^{2}^{2}2The reference channel gain is the channel power gain at a reference distance of 1 m. divided by the noise variance.

## III Problem Formulation

In this paper, we aim to minimize the height error of the final \acdem $\sigma_{h}$ by jointly optimizing the \acuav formation $\mathcal{Q}=\{\mathbf{q}_{k},\forall k\in\{0,1,2\}\}$ and the instantaneous communication transmit powers $\mathcal{P}=\{\mathbf{P}_{\mathrm{com},k},\forall k\in\{0,1,2\}\}$, while satisfying communication and sensing quality-of-service constraints. To this end, we formulate the following optimization problem:

$\displaystyle(\mathrm{P}):\min_{\mathcal{Q},\mathcal{P}}\hskip 8.53581pt\sigma%_{h}(\mathcal{Q})$ | |||

s.t. | $\displaystyle\mathrm{C1:}\;z_{\mathrm{min}}\leq z_{k}\leq z_{\mathrm{max}},%\forall k\in\{0,1,2\},$ | ||

$\displaystyle\mathrm{C2}:\;x_{0}=x_{t}-z_{0}\tan(\theta_{0}),$ | |||

$\displaystyle\mathrm{C3}:\;\theta_{\mathrm{min}}\leq\theta_{k}(\mathbf{q}_{k})%\leq\theta_{\mathrm{max}},\forall k\in\{1,2\},$ | |||

$\displaystyle\mathrm{C4}:||\mathbf{q}_{i}-\mathbf{q}_{j}||_{2}\geq d_{\mathrm{%min}},\forall i\neq j\in\{0,1,2\},$ | |||

$\displaystyle\mathrm{C5}:S_{k}(\mathbf{q}_{k})\geq S_{\rm min},\forall k\in\{0%,1,2\},$ | |||

$\displaystyle\mathrm{C6}:\gamma_{\mathrm{SNR},k}(\mathbf{q}_{0},\mathbf{q}_{k}%)\geq\gamma_{\rm SNR}^{\mathrm{min}},\forall k\in\{1,2\},$ | |||

$\displaystyle\mathrm{C7}:\gamma_{\mathrm{Rg},k}(\mathbf{q}_{k})\geq\gamma_{\rmRg%}^{\mathrm{min}},\forall k\in\{1,2\},$ | |||

$\displaystyle\mathrm{C8}:\;h_{\mathrm{amb},k}(\mathbf{q}_{0},\mathbf{q}_{k})%\geq h_{\mathrm{amb}}^{\mathrm{min}},\forall k\in\{1,2\},$ | |||

$\displaystyle\mathrm{C9}:0\leq P_{\mathrm{com},k}[n]\leq P_{\mathrm{com}}^{%\mathrm{max}},\forall\;k\in\{0,1,2\},\forall n,$ | |||

$\displaystyle\mathrm{C10}:R_{k,n}(\mathbf{q}_{k},\mathbf{P}_{\mathrm{com},k})%\geq R_{\mathrm{min},k},\forall\;k\in\{0,1,2\},\forall n,$ | |||

$\displaystyle\mathrm{C11}:\sum_{n=1}^{N}P_{\mathrm{com},k}[n]\leq E_{\rm com}^%{\rm max},\forall\;k\in\{0,1,2\}.$ |

Constraint $\mathrm{C1}$ specifies the maximum and minimum allowed flying altitude, denoted by $z_{\mathrm{max}}$ and $z_{\mathrm{min}}$, respectively. Constraint $\mathrm{C2}$ ensures that the beam footprint of the master \acuav is centered around $\mathbf{p}_{t}[n]$. Constraint $\mathrm{C3}$ specifies the minimum and maximum slave look angle, denoted by $\theta_{\rm min}$ and $\theta_{\rm max}$, respectively. Constraint $\mathrm{C4}$ ensures a minimum safety distance $d_{\mathrm{min}}$ between any two \acpuav. Constraint $\mathrm{C5}$ imposes a minimum radar swath width $S_{\mathrm{min}}$. Constraints $\mathrm{C6}$ and $\mathrm{C7}$ ensure minimum thresholds for \acsnr and baseline decorrelation, $\gamma_{\rm SNR}^{\mathrm{min}}$ and $\gamma_{\rm Rg}^{\mathrm{min}}$, respectively. Constraint $\mathrm{C8}$ imposes a minimum \achoa, $h_{\mathrm{amb}}^{\mathrm{min}}$, required for phase unwrapping [9]. Constraint $\mathrm{C9}$ imposes a maximum communication transmit power, $P_{\mathrm{com}}^{\mathrm{max}}$. Constraint $\mathrm{C10}$ ensures the minimum required sensing data rate for $U_{k}$, $R_{\mathrm{min},k},\forall k\in\{0,1,2\}$. Constraint $\mathrm{C11}$ limits the total communication energy of $U_{k}$ to $E_{\mathrm{com}}^{\rm max},\forall k\in\{0,1,2\}$.

Problem $\mathrm{(P)}$ is a non-convex optimization problem. The non-convexity is caused by the objective function and constraints $\mathrm{C4},\mathrm{C5}$, $\mathrm{C7}$, and $\mathrm{C8}$. In fact, the height error is simultaneously \achoa- and coherence-dependent, making the objective function challenging. Moreover, the lower bound on an Euclidean distance in $\mathrm{C4}$ and the trigonometric functions in $\mathrm{C5}$, $\mathrm{C7}$, and $\mathrm{C8}$ make these constraints non-convex and difficult to handle.

## IV Solution of the Optimization Problem

To balance performance and complexity, we propose a low-complexity sub-optimal solution that minimizes an upper bound on problem $\mathrm{(P)}$ based on \acao. We divide problem $\mathrm{(P)}$ into 3 sub-problems: $\mathrm{(P.0)}$, $\mathrm{(P.1)}$, and $\mathrm{(P.2)}$. In $\mathrm{(P.0)}$, we optimize the position and communication power of $U_{0}$, whereas in $\mathrm{(P.1)}$ and $\mathrm{(P.2)}$, we optimize the positions and communication powers of $U_{1}$ and $U_{2}$, respectively. Due to symmetry, we focus on $\mathrm{(P.0)}$ and $\mathrm{(P.1)}$, as $\mathrm{(P.2)}$ can be solved similarly to $\mathrm{(P.1)}$, see Figure 2.

### IV-A Master \acuav Optimization

In this sub-section, we optimize the position and communication transmit power of the master \acuav, denoted by $\mathbf{q}_{0}$ and $\mathbf{P}_{\rm com,0}$, respectively, for fixed $\{\mathbf{q}_{1},\mathbf{q}_{2},\mathbf{P}_{\rm com,1},\mathbf{P}_{\rm com,2}\}$. The resulting sub-problem, denoted by $\mathrm{(P.0)}$, is still non-convex due to its objective function and $\mathrm{C4}$. Yet, we leverage \acsca to provide a low-complexity solution for $(\mathrm{P.0})$.

As the master look angle $\theta_{0}$ is fixed, the perpendicular baseline is independent of $\mathbf{q}_{0}$ and is given by [11]:

$b_{\bot,k}(\mathbf{q}_{k})=\frac{\Big{|}(x_{t}-x_{k})-\tan(\theta_{0})z_{k}%\Big{|}}{\sqrt{\tan(\theta_{0})^{2}+1}},\forall k\in\{1,2\}.$ | (17) |

###### Proposition 1.

The height error of the final \acdem, $\sigma_{h}$, can be upper bounded based on the worst-case coherence as follows:

$\sigma_{h}(\mathcal{Q})\leq\overline{\sigma_{h}}(\mathcal{Q})\triangleq\frac{}%{}\sqrt{\frac{\lambda^{2}r^{2}_{0}(\mathbf{q}_{0})\sin^{2}(\theta_{0})(1-%\mathcal{A}^{2})}{8\pi^{2}\mathcal{A}^{2}n_{L}\left(b^{2}_{\bot,1}(\mathbf{q}_%{1})+b^{2}_{\bot,2}(\mathbf{q}_{2})\right)}},$ | (18) |

where $\mathcal{A}=\gamma_{\rm Rg}^{\mathrm{min}}\gamma_{\rm SNR}^{\mathrm{min}}%\gamma_{\rm other}$.

###### Proof.

Please refer to Appendix B.∎

Therefore, we relax the complex objective function of $\mathrm{(P.0)}$ by minimizing instead the upper bound on the height error, denoted by $\overline{\sigma_{h}}$ and provided in Proposition 1. In iteration $i$ of the \acsca algorithm, constraint $\mathrm{C4}$ is tackled using a surrogate function for the Euclidean distance $||\mathbf{q}_{0}-\mathbf{q}_{k}||_{2}$ around $\mathbf{q}_{0}^{(i)}$ as follows [15]:

$\mathrm{\widetilde{C4}}:2\mathbf{q}_{0}^{T}(\mathbf{q}_{0}^{(i)}-\mathbf{q}_{k%})-||\mathbf{q}_{0}^{(i)}||_{2}^{2}+||\mathbf{q}_{k}||_{2}^{2}\geq d_{\rm min}%,{\forall i,}\forall k\in\{1,2\}.$ | (19) |

The resulting sub-problem is denoted by $\widetilde{\mathrm{(P.0)}}$ and is given by:

$\displaystyle\widetilde{\mathrm{(P.0)}}:\min_{\mathbf{q}_{0},\mathbf{P}_{\rmcom%,0}}\hskip 8.53581pt\overline{\sigma_{h}}(\mathcal{Q})$ | |||

s.t. | $\displaystyle\mathrm{C1-C3,\widetilde{\mathrm{C4}},C5,C6,C8-C11}.$ |

Problem $\widetilde{\mathrm{(P.0)}}$ is convex and can be solved using the Python CVXPY library [16]. The solution procedure to solve $\mathrm{(P.0)}$ is summarized in Algorithm 1, which converges to a local optimum of the upper bound on problem $\mathrm{(P.0)}$, $\overline{\sigma_{h}}$, in polynomial computational timecomplexity [17]. Algorithm 1 involves $N+2$ optimization variables, resulting in a computational complexity of $\mathcal{O}(M_{0}(N+2)^{3.5})$, where $M_{0}$ is the number of iterations needed for convergence [17].

### IV-B Slave \acuav Optimization

Next, we optimize the position and communication transmit power of slave \acuav $U_{1}$, denoted by $\mathbf{q}_{1}$ and $\mathbf{P}_{\rm com,1}$, respectively, for fixed $\{\mathbf{q}_{0},\mathbf{q}_{2},\mathbf{P}_{\rm com,0},\mathbf{P}_{\rm com,2}\}$. The resulting problem, denoted by $\mathrm{(P.1)}$,is non-convex due to the objective function and constraints $\mathrm{C4},\mathrm{C5},\mathrm{C7},$ and $\mathrm{C8}$. To tackle this sub-problem, we employ again \acsca. First, we adopt the upper bound $\overline{\sigma_{h}}$ provided by Proposition 1. Furthermore, it can be shown that minimizing $\overline{\sigma_{h}}$ for fixed $\mathbf{q}_{0}$ and $\mathbf{q}_{2}$ is equivalent to maximizing the perpendicular baseline $b_{\bot,1}(\mathbf{q}_{1})$. Moreover, in each \acsca iteration $j$, non-convex constraint $\mathrm{C4}$ is replaced with convex constraint $\widetilde{\widetilde{\mathrm{C4}}}$ based on a surrogate function similar to (19). Constraint $\mathrm{C5}$ is convexified based on a Taylor approximation around point $\mathbf{q}_{1}^{(j)}$ as follows:

$\mathrm{\widetilde{C5}}:r_{1}^{2}(\mathbf{q}_{1}^{(j)})+\left(\frac{\partial r%_{1}^{2}}{\partial\mathbf{q}_{1}}(\mathbf{q}_{1}^{(j)})\right)^{T}(\mathbf{q}_%{1}-\mathbf{q}_{1}^{(j)})\geq\frac{S_{\rm min}z_{1}}{\Theta_{\rm 3dB}},{%\forall j}.$ | (20) |

The resulting problem is given by:

$\displaystyle\mathrm{\widetilde{(P.1)}}:\max_{\mathbf{q}_{1},\mathbf{P}_{\rmcom%,1}}\hskip 8.53581ptb_{\bot,1}(\mathbf{q}_{1})$ | |||

s.t. | $\displaystyle\mathrm{C1,C3,\widetilde{\widetilde{C4}},\widetilde{C5},{C6}-C11}.$ |

Yet, the expressions for the perpendicular baseline in (17) and for the baseline decorrelation still present obstacles for solving $\mathrm{\widetilde{(P.1)}}$. Thus, we divide the search space of problem $\mathrm{\widetilde{(P.1)}}$ into two disjoint sets, denoted by $\mathcal{I}_{a}^{k}=\{\mathbf{q}_{k};\theta_{0}\geq\theta_{k}(\mathbf{q}_{k})\}$ and $\mathcal{I}^{k}_{b}=\{\mathbf{q}_{k};\theta_{0}<\theta_{k}(\mathbf{q}_{k})\},k%\in\{1,2\}$. The solution that maximizes $\overline{\sigma_{h}}$ over $\mathcal{I}^{1}_{a}$ and $\mathcal{I}^{1}_{b}$ is selected, see Figure 2.

###### Proposition 2.

Constraint $\mathrm{C7}$ is equivalent to the following convex constraints:

$\displaystyle\quad\begin{cases}\mathrm{C7a}:z_{k}\alpha_{a}-(x_{t}-x_{k})\leq 0%,\text{ if }\mathbf{q}_{k}\in\mathcal{I}_{a}^{k}\\\mathrm{C7b}:(x_{t}-x_{k})-z_{k}\alpha_{b}\leq 0,\text{ if }\mathbf{q}_{k}\in%\mathcal{I}_{b}^{k}\end{cases},\forall k\in\{1,2\},$ | (21) |

where $\alpha_{a}=\tan\left(\arcsin\left(\frac{2-h(\gamma_{\rm Rg}^{\rm min})}{h(%\gamma_{\rm Rg}^{\rm min})}\sin(\theta_{0})\right)\right)$, $\alpha_{b}=\tan\left(\arcsin\left(\frac{h(\gamma_{\rm Rg}^{\rm min})}{2-h(%\gamma_{\rm Rg}^{\rm min})}\sin(\theta_{0})\right)\right)$, and function $h(x)=\frac{xB_{p}-2-B_{p}}{B_{p}-2-xB_{p}}$.

###### Proof.

Please refer to Appendix C.∎

Constraint $\mathrm{C7}$ is transformed based on Proposition 2. Then, for $\mathbf{q}_{1}\in\mathcal{I}^{1}_{a}$, constraint $\mathrm{C7}$ is replaced by $\mathrm{C7a}$ and the resulting problem is denoted by $\mathrm{\widetilde{(P.1.a)}}$. Simlarly, sub-problem $\mathrm{\widetilde{(P.1.b)}}$ denotes sub-problem $\mathrm{\widetilde{(P.1)}}$ for $\mathbf{q}_{1}\in\mathcal{I}^{1}_{b}$, where constraint $\mathrm{C7b}$ replaces $\mathrm{C7}$.

The proposed \acsca algorithm to solve $\mathrm{(P.1)}$ is omitted due to space limitation, but is similar to Algorithm 1, where the convex approximations $\mathrm{(\widetilde{P.1.a})}$ and $\mathrm{(\widetilde{P.1.b})}$ are solved in parallel using CVXPY [16], with precision $\epsilon_{1}=\epsilon_{0}$. The algorithm converges to a local optimum of the upper bound on sub-problem $\mathrm{(P.1)}$ entailing computational complexity $\mathcal{O}(2M_{1}(N+2)^{3.5})$, where $M_{1}$ is the required number of iterations [17].

### IV-C Solution to Problem $\mathrm{(P)}$

To solve problem $\mathrm{(P)}$, we use \acao by solving sub-problems $\mathrm{(P.0)}$, $\mathrm{(P.1)}$, and $\mathrm{(P.2)}$ iteratively, see Figure 2. In Algorithm 2, we summarize all steps used to solve problem $\mathrm{(P)}$. Algorithm 2 converges to a local optimum of the worst-case height error, $\overline{\sigma_{h}}$, with time complexity $\mathcal{O}(M_{2}(2M_{1}+M_{0})(N+2)^{3.5})$, where $M_{2}$ is the required number of iterations [17].

## V Simulation Results

This section presents simulation results for Algorithm 2, using parameters from Table I, unless stated otherwise. To evaluate performance, we adopt the next benchmark schemes:

- •
Benchmark scheme 1: Here, a single-baseline \acuav-\acinsar system consisting only of $U_{0}$ and $U_{1}$ is considered [11]. The upper bound on the height error $\sigma_{h_{1}}$ is minimized based on a two-step \acao algorithm.

- •
Benchmark scheme 2: In this scheme, we fix the position of the master \acuav at $\mathbf{q}_{0}=\mathbf{q}_{0}^{\rm fixed}$, which is feasible for $\mathrm{(P)}$, and optimize the remaining variables.

- •
Benchmark scheme 3: Here, we apply a static and constant communication power allocation (i.e., $P_{\mathrm{com},k}[n]=\frac{E_{\rm com}^{\rm max}}{N},\forall n,\forall k$), and optimize the remaining variables.

In Figure 3, we present the height error of the final \acdem $\sigma_{h}$ and its upper bound $\overline{\sigma_{h}}$ versus the minimum \achoa, $h_{\rm amb}^{\rm min}$. The figure shows that the sensing accuracy degrades with stricter requirements on the minimum \achoa, which is due to the relation between the \achoa and the height error, see (13). We note that the tightness of the upper bound on the height error in (18) increases with stricter sensing requirements, i.e., if $\gamma_{\rm SNR}^{\rm min}\gamma_{\rm Rg}^{\rm min}\gamma_{\rm other}\to 1$, then $\sigma_{h}\to\overline{\sigma_{h}}$. Nevertheless, Figure 3reveals that even for system parameters, for which the upper bound is not tight, it is a useful metric for optimization. In fact, the proposed scheme consistently achieves a gain of at least 39.3% compared to benchmark scheme 1. This gain is due to averaging the height error, which improves the sensing accuracy and highlights the importance of using multiple \acpuav for acquisition. Additionally, optimizing the \acuav formation and the communication power allows the proposed solution to outperform both benchmark schemes 2 and 3, with a gain that varies with the \achoa. For instance, for $h_{\rm amb}^{\rm min}=1.2$ m, respective gains of 20.5% and 12.5% are observed.

Figure 4 depicts the final height error and its upper bound versus the minimum data rate of $U_{1}$. The figure shows that higher data rate requirements lead to worse accuracy. This can be explained by $U_{1}$’s extended range when lower data rates are sufficient. In contrast, increasing the data rate requirement limits the $U_{1}$-\acgs distance, which affects the perpendicular baseline and increases the height estimation error. Furthermore, Figure 4 highlights the need to properly allocate the communication power to ensure real-time data offloading to the \acgs. A static power allocation, as in benchmark scheme 3, negatively affects the height error and eventually leads to infeasibility of problem $\mathrm{(P)}$ starting from $R_{\rm min,1}=17.1$ Mbit/s, indicated by the red colored region in Figure 4. Figure 4 confirms that the proposed scheme outperforms all benchmark schemes.

## VI Conclusion

In this work, we studied a dual-baseline \acuav-based \acinsar system using a swarm of three drones to generate two independent \acpdem of a target area. The final \acdem is obtained with a weighted averaging technique, improving sensing precision. We proposed a low-complexity algorithm that minimizes an upper bound on the height estimation error of the final \acdem by jointly optimizing the \acuav formation and communication power allocation, while meeting sensing and communication constraints. Simulation results showed that the proposed scheme significantly improves sensing accuracy compared to single-baseline systems and other benchmark schemes.

## Appendix A Bi-static \acsnr Approximation

In this appendix, we provide the detailed steps for deriving the bi-static \acsnr approximation in (9). To this end, we start with the bi-static radar \acsnr expression for distributed targets, denoted by $\mathrm{SNR}_{r}$ and given by [13]:

$\mathrm{SNR}_{r}=\frac{\sigma_{0}A_{\rm res}P_{t}G_{t}G_{r}\lambda^{2}}{(4\pi)%^{3}R^{2}_{\rm Tx}R^{2}_{\rm Rx}k_{b}B_{\rm noise}T_{\rm sys}FL},$ | (22) |

where $R_{\rm Tx}$ and $R_{\rm Rx}$ represent the radar transmit and receive slant ranges, respectively, $A_{\rm res}$ is the size of one resolution cell, i.e., the size of the smallest area the \acsar system can distinguish, $B_{\rm noise}$ is the effective noise bandwidth, and $L$ denotes the total radar losses. After coherent integration of the radar signal to form the \acsar image, the radar \acsnr is improved by a factor $N_{\rm proc}$ as follows:

$\mathrm{SNR}=\mathrm{SNR}_{r}N_{\rm proc}=\mathrm{SNR}_{r}N_{\rm pulse}N_{a},$ | (23) |

where $N_{\rm pulse}$ is the number of samples per pulse and $N_{a}$ is the number of samples within the synthetic aperture length, which is denoted by $L_{a}$. Note that $N_{\rm pulse}=\tau_{p}B_{\rm Rg}\approx\tau_{p}B_{\rm noise}$ and $N_{a}=\frac{L_{a}\mathrm{PRF}}{v_{y}}$. Thus, the image \acsnr can be written as:

$\mathrm{SNR}=\frac{\sigma_{0}A_{\rm res}P_{t}G_{t}G_{r}\lambda^{2}\tau_{p}%\mathrm{PRF}L_{a}}{(4\pi)^{3}R^{2}_{\rm Tx}R^{2}_{\rm Rx}v_{y}k_{b}T_{\rm sys}%FL}.$ | (24) |

The \acsnr expression depends on the resolution cell size $A_{\rm res}$, which is related to the range and azimuth resolutions, denoted by $\delta_{r}$ and $\delta_{a}$, respectively. For a mono-static system with slant range $R_{\rm Tx}$ and incidence angle $\theta$, $A_{\rm res}=\delta_{r}\delta_{a}=\frac{\lambda R_{\rm Tx}c}{4B_{\rm Rg}L_{a}%\sin(\theta)}$ leads to the expression in (8) [10]. However, calculating $A_{\rm res}$ for the bi-static case is more complex. According to [18], in this case, the ground range resolution is given by:

$\delta_{r}=\frac{c}{B_{\rm Rg}\left(\sin(\theta_{\rm Tx})+\sin(\theta_{\rm Rx}%)\right)},$ | (25) |

where $\theta_{\rm Tx}$ and $\theta_{\rm Rx}$ denote the transmit and receive incidence angles, respectively. Furthermore, the azimuth resolution is given by:

$\delta_{a}=l\frac{R_{\rm Rx}}{R_{\rm Tx}+R_{\rm Rx}}.$ | (26) |

In the case of across-track interferometry applications, since a lower bound is imposed on the baseline decorrelation by constraint $\mathrm{C7}$, the bi-static angle $\Delta\theta=|\theta_{\rm Tx}-\theta_{\rm Rx}|$ is relatively small (i.e., on the order of few degrees). Thus, the range resolution can be approximated by $\delta_{r}\approx\frac{c}{2B_{\rm Rg}\sin(\theta_{\rm Tx})}$ and $\sigma_{0}(\theta_{\rm Rx})\approx\sigma_{0}(\overbrace{\theta_{\rm Tx}}^{%\text{fixed angle}})=\sigma_{0}$. Based on the bi-static effective range $R_{\rm eff}=\frac{R_{\rm Rx}+R_{\rm Tx}}{2}$, we approximate the azimuth resolution as:

$\delta_{a}\approx\frac{\lambda R_{\rm eff}R_{\rm Rx}}{L_{a}(R_{\rm Tx}+R_{\rmRx%})},$ | (27) |

which leads to the following approximation of the resolution cell size:

$A_{\rm res}=\delta_{r}\delta_{a}\approx\frac{\lambda R_{\rm Rx}c}{4L_{a}B_{\rmRg%}\sin(\theta_{\rm Tx})}.$ | (28) |

Finally, the approximated bi-static \acsnr expression is obtained by inserting (28) in (24).

## Appendix B Proof of Proposition 1

It can be shown that $\sigma_{\Phi_{k}}$ is monotonically decreasing \acwrt the total coherence $\gamma_{k}$. Therefore, given the worst-case coherence value $\mathcal{A}=\gamma_{\rm Rg}^{\mathrm{min}}\gamma_{\rm SNR}^{\mathrm{min}}%\gamma_{\rm other}$, the following inequality holds:

$\sigma_{\Phi_{k}}(\mathbf{q}_{0},\mathbf{q}_{k})\leq\frac{1}{\mathcal{A}}\sqrt%{\frac{1-\mathcal{A}^{2}}{2n_{L}}},\forall k\in\{1,2\}.$ | (29) |

Based on (13), (17), and (29), we construct an upper bound on the height error $\sigma_{h_{k}}$ of interferometric pair $(U_{0},U_{k})$ as follows:

$\sigma_{h_{k}}(\mathbf{q}_{0},\mathbf{q}_{k})\leq\frac{\lambda r_{0}(\mathbf{q%}_{0})\sin(\theta_{0})}{2\pi b_{\bot,k}(\mathbf{q}_{k})\mathcal{A}}\sqrt{\frac%{1-\mathcal{A}^{2}}{2n_{L}}},\forall k\in\{1,2\}.$ | (30) |

The proof is concluded by inserting the upper bound (30) into the expression for the height error of the final \acdem, given in (15).

## Appendix C Proof of Proposition 2

Let function $f(x)=\frac{1}{B_{p}}\left[\frac{2+B_{p}}{1+x}-\frac{2-B_{p}}{1+\frac{1}{x}}\right]$ such that $f(\mathcal{X}(\mathbf{q}_{k}))$ represents the baseline decorrelation. Note that $\forall\mathbf{q}_{k},0\leq\mathcal{X}(\mathbf{q}_{k})\leq 2$, therefore, we focus on function $f$ in the domain $[0,2]$. It can be shown that function $f$ is a decreasing and invertible function such that:

$f^{-1}(x)=h(x)=\frac{B_{p}x-2-B_{p}}{B_{p}-2-xB_{p}}.$ | (31) |

Therefore, constraint $\mathrm{C7}$ is equivalent to the following constraint:

$\mathrm{C7}:f(\mathcal{X}(\mathbf{q}_{k}))\geq\gamma_{\rm Rg}^{\rm min}\iff%\mathcal{X}(\mathbf{q}_{k})\leq h(\gamma_{\rm Rg}^{\rm min}).$ | (32) |

Based on (11) and (32), constraint $\mathrm{C7}$ is equivalent to the following constraints:

$\begin{dcases}\mathrm{C7a}:\sin(\theta_{k}(\mathbf{q}_{k}))\geq\frac{2-h(%\gamma_{\rm Rg}^{\rm min})}{h(\gamma_{\rm Rg}^{\rm min})}\sin(\theta_{0}),%\text{if }\mathbf{q}_{k}\in\mathcal{I}_{a}^{k},\\\mathrm{C7b}:\sin(\theta_{k}(\mathbf{q}_{k}))\leq\frac{h(\gamma_{\rm Rg}^{\rmmin%})}{2-h(\gamma_{\rm Rg}^{\rm min})}\sin(\theta_{0}),\text{if }\mathbf{q}_{k}%\in\mathcal{I}_{b}^{k}.\end{dcases}$ | (33) |

Note that constraints $\mathrm{C7a}$ and $\mathrm{C7b}$ are in general non-convex, however, knowing that $0\leq\theta_{k}(\mathbf{q}_{k})\leq\frac{\pi}{2},\forall\mathbf{q}_{k},\forallk$, then we can apply increasing functions $\arcsin$ and $\tan$ to (33), which results in the following equivalent convex constraints:

$\begin{dcases}\mathrm{C7a}:\frac{x_{t}-x_{k}}{z_{k}}\geq\tan\left(\arcsin\left%(\frac{\left(2-h(\gamma_{\rm Rg}^{\rm min})\right)\sin(\theta_{0})}{h(\gamma_{%\rm Rg}^{\rm min})}\right)\right),\text{if }\mathbf{q}_{k}\in\mathcal{I}_{a}^{%k},{\forall k,}\\\mathrm{C7b}:\frac{x_{t}-x_{k}}{z_{k}}\leq\tan\left(\arcsin\left(\frac{h(%\gamma_{\rm Rg}^{\rm min})\sin(\theta_{0})}{2-h(\gamma_{\rm Rg}^{\rm min})}%\right)\right),\text{if }\mathbf{q}_{k}\in\mathcal{I}_{b}^{k},{\forall k.}\end%{dcases}$ | (34) |

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