Multi-UAV Doppler-SAR Interferometry

• Multi-UAV Doppler-SAR interferometry is a radar imaging technique that uses narrowband continuous waves and varying UAV velocities to recover terrain elevation.
• It achieves precise meter-level height estimates by leveraging fine Doppler and Doppler-rate differences rather than traditional range measurements.
• The method offers practical advantages such as low-power, lightweight hardware and the potential for passive operation using ambient narrowband signals.

Doppler-SAR interferometry is an imaging and height-mapping radar modality that operates using ultra-narrowband continuous waveforms (UNCW), leveraging high-resolution Doppler and Doppler-rate information rather than the direct range resolution achieved by conventional wideband synthetic aperture radar (SAR) interferometry. Unlike conventional approaches, which exploit the spatial baseline between displaced sensor positions, Doppler-SAR interferometry utilizes differences in antenna velocities to retrieve topography, making it suitable for long-range, low-cost, passive, and lightweight implementations (Yazici et al., 2017).

1. Imaging Paradigms: Doppler-SAR versus Wideband SAR

Conventional wideband SAR forms high range resolution images through the measurement of two-way time delay ("range") and pulse-to-pulse Doppler shifts as the sensor traverses a trajectory. Interferometric wideband SAR recovers terrain elevation by measuring the phase difference between images collected from antennas at spatially distinct positions, with topography directly tied to range differences.

In contrast, Doppler-SAR transmits a UNCW, producing images by backprojection along iso-Doppler and iso-Doppler-rate surfaces, which sharply focus only where both conditions are met. The innate range resolution of Doppler-SAR is coarse due to bandwidth limitations, but it offers extremely fine Doppler and Doppler-rate resolution. Doppler-SAR interferometric phase is thus governed by differences in Doppler signatures obtained by sensors with different velocities, rather than position, allowing for height recovery through analysis of velocity baselines.

2. Mathematical Foundations and Sensor Kinematics

Ground reflectors are modeled as $$\mathbf x = [x_1, x_2, h(x_1, x_2)] \in \mathbb R^3$$, where $h(x_1, x_2)$ is the unknown terrain height. Two monostatic antennas follow parametric trajectories $\mathbf r_i(s)$ with slow time $s\in [S_1, S_2]$. Their velocities and accelerations are denoted $\dot{\mathbf r}_i(s)$ and $\ddot{\mathbf r}_i(s)$, respectively. The range and look-direction are

$$R_i(\mathbf x, s) = \|\mathbf x - \mathbf r_i(s)\|, \qquad \mathbf L_i(\mathbf x, s) = \frac{\mathbf x - \mathbf r_i(s)}{R_i(\mathbf x, s)}$$

where $i=1,2$ indexes the antennas.

The instantaneous monostatic Doppler for narrowband transmission is

$$f_i^d(\mathbf x, s) = -\frac{\omega_0}{c}\, \mathbf L_i(\mathbf x, s) \cdot \dot{\mathbf r}_i(s)$$

where $\omega_0$ is the center frequency and $c$ the speed of light. For image formation, one expands around the slow time $s=s_d^i$ at which the Doppler-rate vanishes, defined by

$$\frac{\partial f_i^d(\mathbf x,s)}{\partial s}\bigg|_{s=s_d^i} = 0 \implies \mathbf L_i(\mathbf x, s_d^i) \cdot \ddot{\mathbf r}_i(s_d^i) - \frac{\dot{\mathbf r}_i(s_d^i) \cdot \dot{\mathbf r}_i^\perp(s_d^i)}{R_i(\mathbf x, s_d^i)} = 0.$$

3. Doppler-SAR Image Formation and Interferometry

Doppler-SAR image formation is achieved by filtered backprojection of correlated data onto iso-Doppler and iso-Doppler-rate surfaces. The optimal focus for $(x_1,x_2,x_3)$ corresponds to simultaneous satisfaction of:

• Iso-Doppler: $$\mathbf L_i(\mathbf z, s)\cdot\dot{\mathbf r}_i(s) = \frac{c}{\omega_0}\,f_i^d(\mathbf x, s)$$
• Iso-Doppler-rate: $$\mathbf L_i(\mathbf z, s)\cdot\ddot{\mathbf r}_i(s) - \frac{\dot{\mathbf r}_i(s)\cdot\dot{\mathbf r}_i^\perp(s)}{R_i(\mathbf z, s)} = \frac{c}{\omega_0}\,\partial_s f_i^d(\mathbf x,s)$$
• Height-matching: $z_3 = h(\mathbf x)$

The absence of ground truth height ($h(\mathbf x)$) leads to so-called "lay-over," where the image is reconstructed on a fixed reference plane and displaced in ground range.

To extract elevation, two coregistered Doppler-SAR images are formed and their raw interferometric phase is computed as

$$\Phi^{\mathrm{UNB}}_{s_d}(\mathbf x) = \arg\left[ I_1^{\mathrm{UNB}}(\mathbf z_0^1)\, \overline{I_2^{\mathrm{UNB}}(\mathbf z_0^2)} \right] = 2\,s_d^1T_\phi\, [f_1^d(\mathbf x,s_d^1) - f_2^d(\mathbf x,s_d^2)]$$

where $T_\phi$ is the phase processing window and $s_d^1$ the zero Doppler-rate slow time.

With baseline velocity $$\mathbf v = \dot{\mathbf r}_2(s_d^2) - \dot{\mathbf r}_1(s_d^1)$$ and spatial baseline $$\mathbf b = \mathbf r_2(s_d^2) - \mathbf r_1(s_d^1)$$, a small-baseline, large-range approximation yields

$$-\frac{c}{2\,s_d^1T_\phi\,\omega_0} \Phi^{\mathrm{UNB}}_{s_d}(\mathbf x) \approx \mathbf L_1(\mathbf x,s_d^1) \cdot \mathbf v + \frac{\mathbf b_1^\perp \cdot \dot{\mathbf r}_2(s_d^2)}{R_1(\mathbf x,s_d^1)}$$

with $$\mathbf b_1^\perp = \mathbf b - \mathbf L_1(\cdot)[\mathbf L_1(\cdot)\cdot\mathbf b]$$.

4. Height Mapping Equations

The unknown scatterer location $\mathbf x$ is determined from three nonlinear constraints involving the image parameter $\mathbf z$:

• Iso-Doppler: $$\widehat{(\mathbf z-\mathbf r_1(s_d^1))}\cdot\dot{\mathbf r}_1(s_d^1) = \frac{c}{\omega_0} f_1^d(\mathbf x, s_d^1)$$
• Iso-Doppler-rate: $$\mathbf L_1(\mathbf z, s_d^1)\cdot\ddot{\mathbf r}_1(s_d^1) - \frac{ \dot{\mathbf r}_1(s_d^1)\cdot\dot{\mathbf r}_1^\perp(s_d^1) }{ R_1(\mathbf z, s_d^1) } = \partial_s f_1^d(\mathbf x, s_d^1)$$
• Interferometric Doppler-rate: $$\mathbf L_1(\mathbf z, s_d^1)\cdot\mathbf v + \frac{ \mathbf b_1^\perp\cdot\dot{\mathbf r}_2(s_d^2) }{ R_1(\mathbf z, s_d^1) } = -\frac{c}{2\,s_d^1T_\phi\,\omega_0} \Phi^{\mathrm{UNB}}_{s_d}(\mathbf x)$$

Solving these equations for $\mathbf z = (x_1, x_2, x_3)$ yields the terrain elevation $h = x_3$.

A linearized, “flattened” phase representation introduces a reference $\mathbf z_0 = [z_1, z_2, 0]$ with $\mathbf x = \mathbf z_0 + \boldsymbol\ell$. Under a far-field look-direction approximation, the flattened phase is

$$\Phi^{\mathrm{UNB}}_{\mathrm{flat}}(\mathbf x) = \Phi^{\mathrm{UNB}}_{s_d}(\mathbf x) - \Phi^{\mathrm{UNB}}_{s_d}(\mathbf z_0) \approx \frac{ \mathbf v_1^\perp \cdot \boldsymbol\ell }{ R_1(\mathbf z_0, s_d^1) }$$

where $$\mathbf v_1^\perp = \mathbf v - (\mathbf L_1 \cdot \mathbf v)\mathbf L_1$$.

5. Comparative Analysis with Conventional SAR Interferometry

A direct methodological comparison illuminates several operational distinctions:

	Wideband SAR	Doppler-SAR
Primary measurement	Range difference $\Delta R = R_1 - R_2$	Doppler difference $\Delta f^d = f_1^d - f_2^d$
Raw interferometric phase	$$\Phi^{\mathrm{WB}} = 2\frac{\omega_0}{c} \mathbf L_1\cdot\mathbf b$$	$\Phi^{\mathrm{UNB}} = 2\,s_dT_\phi(f_1^d - f_2^d)$
Flattened phase	$$2\frac{\omega_0}{c}\frac{\mathbf b_1^\perp\cdot\boldsymbol\ell}{R_1}$$	$$-2\frac{\omega_0}{c}\frac{s_dT_\phi}{R_1}\mathbf v_1^\perp\cdot\boldsymbol\ell$$
Degrees of freedom	Two look locations (baseline $\mathbf b$)	Two look velocities (velocity baseline $\mathbf v$)
Key approximation	Small baseline $\|\mathbf b\|\ll R_1$	Small baseline and small velocity difference versus range

This comparative structure highlights that Doppler-SAR shifts interferometric sensitivity from spatial to velocity baselines. The primary metric becomes the Doppler and Doppler-rate differences rather than direct range differences.

6. Simulation-Based Validation

Numerical simulations were conducted with both modalities over a 128×128 m scene with 1 m pixel resolution and a single point scatterer at (–20 m, –31 m, 50 m):

• Wideband SAR configuration:
  • Antennas at 3 km and 4 km altitudes, each moving at 100 m/s along 1 km straight tracks.
  • 100 MHz bandwidth, 8 GHz center frequency.
  • 512 frequency × 1024 slow-time samples.
  • Lay-over positions: (–41,–31,0) and (–48,–31,0).
  • Interferometric height estimate: 50 m (exact).
• Doppler-SAR configuration:
  • Antennas at 2 km and 4 km altitudes, velocities 100 m/s and 400 m/s.
  • Single-frequency, 8 GHz, $T_\phi=0.01\,$s.
  • 512 fast-time × 1024 slow-time samples.
  • Lay-over positions: (–34,–31,0) and (–48,–31,0).
  • Interferometric height estimate: 50 m (exact).

Both methods achieved meter-level horizontal reconstruction and precise height recovery in noiseless scenarios (Yazici et al., 2017). This demonstrates functional equivalence in interferometric mapping despite the fundamentally different measurement approaches.

7. Practical Advantages and Implementation Considerations

Doppler-SAR interferometry presents several operational benefits:

• Ultra-narrowband continuous-wave hardware is low-cost, lightweight, low-power, and straightforward to calibrate.
• High Doppler resolution enables high angular precision over long distances despite poor inherent range resolution.
• Hardware complexity is reduced—no wideband pulse, no high-throughput analog-to-digital conversion required.
• May be implemented passively using ambient UNCW "sources of opportunity" such as FM radio or digital television broadcasts.
• Well-suited to small platforms (micro-satellites, UAVs) with strict size, weight, and power (SWaP) constraints.
• Minimal spectral occupancy and no reliance on high-power wideband emissions allows for environmentally friendly operation.

These characteristics position Doppler-SAR interferometry as a compelling alternative to conventional SAR modalities, exchanging range resolution for fine Doppler-based measurement and enabling new architectures for terrain elevation mapping without the cost or logistical complexity of wideband transmission (Yazici et al., 2017).