Doppler-SAR Interferometry

• Doppler-SAR interferometry is a radar remote sensing technique that exploits differences in sensor velocities to derive topographic information from Doppler phase variations.
• It employs ultra-narrowband continuous waveforms and coherent integration along iso-Doppler surfaces, offering fine angular discrimination and simplified hardware compared to wideband SAR.
• Numerical simulations validate its meter-level horizontal accuracy and exact height recovery, making it ideal for passive, resource-constrained, and environmentally sensitive applications.

Doppler-SAR interferometry is a radar remote sensing technique for height mapping that leverages ultra-narrowband continuous waveforms (UNCW) to achieve high-resolution imaging via Doppler and Doppler-rate measurements, rather than traditional wideband range resolution. The core innovation lies in extracting topographical information by exploiting differences in velocity (baseline velocity) between two sensors, as opposed to the conventional method reliant on spatial (positional) baselines and range differences. This technique facilitates fine height recovery from phase differences between Doppler-domain synthetic aperture radar (SAR) images and offers practical advantages in hardware simplicity, lightweight operation, and adaptability to passive setups using available radio frequency (RF) sources (Yazici et al., 2017).

1. Imaging Paradigms: Wideband vs. Doppler-SAR

Conventional wideband SAR interferometry utilizes high-bandwidth pulses and recovers terrain elevation via phase differences reflecting range offsets between slightly displaced antennas. The two critical SAR measurements are:

• Range: High-resolution determined by two-way travel time measurement.
• Doppler: Frequency shift as the antenna moves, enabling azimuthal image formation.

Doppler-SAR diverges fundamentally: it deploys single-frequency ultra-narrowband continuous waves (UNCW). Here, direct range resolution is inherently coarse; however, Doppler resolution becomes extremely fine due to the monochromatic source and coherent integration. Images are reconstructed via backprojection along iso-Doppler and iso-Doppler-rate surfaces. In Doppler-SAR interferometry, the interferometric phase encodes differences in Doppler and Doppler-rate arising from differing sensor velocities, replacing spatial-baseline-dependent range comparisons with velocity-baseline-dependent Doppler differences (Yazici et al., 2017).

2. Signal Model and Sensor Kinematics

Topographic mapping relies on accurate geometric modeling. Denoting the terrain point as

$$\mathbf x = [x_1,\,x_2,\,h(x_1, x_2)] \in \mathbb{R}^3,$$

with $h(x_1, x_2)$ as the unknown elevation, two monostatic sensors traverse trajectories

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where $s$ is the synthetic aperture "slow time". The platform velocities and accelerations are $\dot{\mathbf r}_i(s)$ and $\ddot{\mathbf r}_i(s)$. The look-direction vector and scalar range are

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

The received Doppler-SAR signal (after far-field and start-stop approximations) with a narrowband transmitter $p(t) \approx \tilde p(t) e^{i\omega_0 t}$ is modeled as

$$d_i^{\mathrm{UNB}}(\mu, s) \approx \int e^{-i t \left[ \omega_0 (1-\mu) - 2 f_i^d(\mathbf x, s) \right] }A_i(\ldots) V(\mathbf x) \; d\mathbf x \; dt$$

with instantaneous monostatic Doppler

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

The acquisition focuses around zero-Doppler-rate ($s = s_d^i$) satisfying

$$\left. \frac{\partial f_i^d(\mathbf x, s)}{\partial s} \right|_{s = s_d^i} = 0,$$

which yields imaging surfaces based on iso-Doppler and iso-Doppler-rate constraints.

3. Doppler-SAR Image Formation and Lay-Over

A Doppler-SAR image for a given sensor is most sharply focused at $\mathbf z$ satisfying:

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

If $h(\mathbf x)$ is unknown, reconstruction defaults to a reference ground plane ($z_3 = 0$), and the image point $\mathbf z_0$ may not coincide with $\mathbf x$ (lay-over effect), analogous to standard SAR.

4. Doppler-SAR Interferometric Phase and Height Recovery

After generating two Doppler-SAR images—typically for two apertures with different velocities—and co-registering their magnitudes, their complex phase difference yields the interferometric observable:

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

Defining the baseline velocity and spatial offset as

$$\mathbf v = \dot{\mathbf r}_2(s_d^2) - \dot{\mathbf r}_1(s_d^1),\qquad \mathbf b = \mathbf r_2(s_d^2) - \mathbf r_1(s_d^1),$$

and $$\mathbf b_1^\perp = \mathbf b - \mathbf L_1(\cdot) [\mathbf L_1(\cdot) \cdot \mathbf b]$$, the (small-baseline limit) phase approximates a Doppler-rate iso-surface:

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

Height ($h = x_3$) is recovered by solving the nonlinear system:

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

Flattening the interferometric phase by referencing to a ground-range-matched point $\mathbf z_0$ and applying the far-field approximation, the phase is given by:

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

where $$\mathbf v_1^\perp = \mathbf v - ( \mathbf L_1 \cdot \mathbf v ) \mathbf L_1$$ and $\boldsymbol\ell$ is the offset from reference.

5. Comparison with Conventional SAR Interferometry

Attribute	Wideband SAR	Doppler-SAR
Primary measurement	Range difference $\Delta R = R_1 - R_2$	Doppler difference $\Delta f^d = f_1^d - f_2^d$
Interferometric phase	$$\Phi^{\mathrm{WB}} = 2 \frac{\omega_0}{c} \mathbf L_1 \cdot \mathbf b$$	$\Phi^{\mathrm{UNB}} = 2 s_d T_\phi (f_1^d - f_2^d)$
Flattened phase	$$2 \frac{ \omega_0 }{c } \frac{ \mathbf b_1^\perp \cdot \boldsymbol\ell }{ R_1 }$$	$$-2 \frac{ \omega_0 }{ c } \frac{ s_d T_\phi }{ R_1 } \mathbf v_1^\perp \cdot \boldsymbol\ell$$
Baseline degrees of freedom	Two positions ($\mathbf b$)	Two velocities ($\mathbf v$)
Key approximation	$\|\mathbf b\| \ll R_1$	Small baseline and velocity difference versus range

Doppler-SAR’s main distinguishing features are the substitution of velocity baselines for position baselines, and the extraction of elevation from the interferometric Doppler-rate structure, exploiting the fine Doppler resolution naturally afforded by ultra-narrowband CW operation (Yazici et al., 2017).

6. Numerical Validation and Simulation Results

Validation comprises simulated experiments on a $128 \times 128$ m scene (1 m pixels), with a single target at ($-20$ m, $-31$ m, $50$ m). For wideband SAR (antennas at 3 km and 4 km altitude, $100$ m/s velocity, $100$ MHz bandwidth, $8$ GHz center), reconstructed lay-over positions were ($-41$, $-31$, $0$) and ($-48$, $-31$, $0$) for antennas 1 and 2, and interferometric elevation recovery was exact (50 m).

For Doppler-SAR (antennas at 2 km and 4 km, velocities $100$ m/s and $400$ m/s, single-frequency $8$ GHz CW, window $T_\phi = 0.01$ s), reconstructed lay-over positions were ($-34$, $-31$, $0$) and ($-48$, $-31$, $0$) and elevation retrieval was exact (50 m). Both methods exhibited meter-level horizontal accuracy and exact height mapping in noiseless conditions (Yazici et al., 2017).

7. Practical Considerations and System Advantages

Doppler-SAR interferometry offers significant operational and implementation advantages:

• Ultra-narrowband hardware is low-cost, lightweight, and requires reduced power.
• Fine Doppler resolution provides high angular discrimination at long ranges despite coarse range resolution.
• Sensor design is drastically simplified, requiring no wideband pulse generators or high-speed ADCs.
• Passivity: The modality is compatible with sources of opportunity (FM radio, digital TV), enabling passive radar architectures.
• Suitability for small platforms: The method is well-matched to the constraints of micro-satellites, small UAVs, or any low-payload application.
• Spectral footprint: The technique is environmentally benign due to extremely low spectral occupancy.

In sum, Doppler-SAR interferometry constitutes an alternative for topographic mapping that exchanges direct range measurement for Doppler-driven spatial inference via velocity baselines, providing a viable path for resource-constrained, passive, or environmentally stringent radar deployments (Yazici et al., 2017).