The goal of this calibration step is to obtain a complex-valued range profile for a channel, representing a measure proportional to the scene reflectivity as a function of range. This includes compensating for cable delays and the effects of mutual antenna coupling. Cable attenuation does not need to be compensated for to produce range profiles and is, therefore, neglected.

#### 3.1. Signal Model for a Single Channel

The frequency-domain signal for a particular transmit-receive channel measured by the VNA may be modelled as

where

$j=\sqrt{-1}$, and

${T}_{Cable}$ is the one-way cable delay. The frequency extends across the signal bandwidth

B from

$f={f}_{c}-B/2$ to

${f}_{c}+B/2$ in steps of

${\delta}_{f}$. The reflectivity

$\xi \left(f\right)$ may further be decomposed into two parts:

where

${\xi}^{Scene}\left(f\right)$ is the scene reflectivity and

${\xi}^{Coupling}\left(f\right)$ is the component due to mutual antenna coupling.

The cable delay term can easily be removed by estimating ${T}_{Cable}$ and multiplying ${S}^{Meas}\left(f\right)$ by ${e}^{j4\pi f{T}_{Cable}}$. The estimate for ${T}_{Cable}$ can be done using manufacturer data or by using dips in the voltage standing wave ratio. The cable delay term is henceforth assumed to be compensated for, resulting in

The range profile

$x\left(R\right)$ is obtained by taking the inverse discrete Fourier transform (iDFT) of (

3):

where

R is the one-way range from the equivalent monostatic antenna phase centre, which lies halfway between the transmitting and receiving antenna.

R extends from 0 m to the unambiguous range

${R}_{u}={c}_{0}/\left(2{\delta}_{f}\right)$ in steps of

$\delta R={c}_{0}/\left(2{f}_{s}\right)$. The implicit time-domain sampling rate

${f}_{s}$ was set to

${f}_{s}=10B$. The oversampling factor of 10 results in good linear interpolation results in tomogram formation. The resulting iDFT length is

${N}_{DFT}={f}_{s}/{\delta}_{f}+1$.

${x}^{Scene}\left(R\right)$ is the scene reflection and

${x}^{Coupling}\left(R\right)$ is the mutual coupling component. The symbol ⊛ denotes circular convolution [

17]. The function

$\mathrm{sinc}\left(a\right)=\mathrm{sin}\left(a\right)/a$, where

$a=2\pi BR/{c}_{0}$ arises due to the finite frequency extent of the signal

${S}^{Meas}\left(f\right)$. The range profile, therefore, consists of the sum of two components: the scene reflectivity circularly convoluted with

$\mathrm{sinc}\left(2\pi BR/{c}_{0}\right)$, and the mutual coupling component circularly convoluted with

$\mathrm{sinc}\left(2\pi BR/{c}_{0}\right)$.

#### 3.2. When Mutual Coupling Is a Problem

The mutual coupling occurs near the antennas ($R\approx 0$ m), which is several resolution cells away from the observed scene. This means that the resolution is sufficient to separate the mutual coupling response from the forest response. But when convoluted with $\mathrm{sinc}\left(2\pi BR/{c}_{0}\right)$, the mutual coupling energy spreads over the scene reflectivity due to the side-lobes of the sinc function. The severity of this interference depends on the combination of four factors:

The mutual coupling amplitude. A large $|{x}^{Coupling}\left(R\right)|$ relative to the scene reflectivity amplitude $|{x}^{Scene}\left(R\right)|$ causes strong interference by mutual coupling. This is the case for BorealScat’s P to L-band observations.

Signal bandwidth. If B is small, high side-lobes of $\mathrm{sinc}\left(2\pi BR/{c}_{0}\right)$ spread out in range, increasing the interference by mutual coupling. This is the case for BorealScat’s P-band observations.

Antenna-scene separation. The side-lobe amplitude of $\mathrm{sinc}\left(2\pi BR/{c}_{0}\right)$ decreases with increasing R. A small antenna-scene separation, which is true for most ground-based experiments, increases the interference by mutual coupling.

Unambiguous range. Due to the circular convolution in (

4), the mutual coupling peak is repeated at

$R\approx {R}_{u}$. The side-lobes from this peak may interfere with scatterers near

${R}_{u}$, such as BorealScat’s trihedral corner reflector.

BorealScat’s P-band measurements fall in all four categories, and are, therefore, severely affected by mutual coupling. If the mutual coupling interference approaches the strength of the trihedral corner reflector’s response, the reflector cannot be used for calibrating phase differences between channels to construct focused tomographic images (

Section 4). Side-lobes also cause artefacts in the tomographic images that distort the scene reflectivity. The mutual coupling component must, therefore, be suppressed in order to be able to calibrate the radar instrument and form tomographic images.

#### 3.3. Mutual Coupling Side-Lobe Suppression

Side-lobes can be suppressed at the cost of worsening the range resolution by choosing a frequency-domain window function which strongly tapers the bandwidth-limited frequency-domain signal ${S}^{Meas}\left(f\right)$. Such a window function will reduce the available number of looks, which is already low due to the small bandwidth at P-band. The resolution-side-lobe trade-off is due to the time-frequency uncertainty principle and cannot be overcome without introducing new information.

In this study, a new method for mutual coupling suppression in VNA radar measurements was developed. The method is based on the assumption that the mutual coupling component consists of scattering mechanisms, such as a direct transmit–receiving path between antennas and reflections off the supporting metallic structure. Such mechanisms can be assumed to be equivalent to a finite set of independent point scatterers near an ideal monostatic antenna. If the observed reflectivity

$\xi \left(f\right)$ can be decomposed into equivalent reflections from point scatterers, the component due to scatterers near

$R=0$ m can be separated and removed, suppressing the mutual coupling component along with its side-lobes. This concept is illustrated in

Figure 6.

In the frequency domain, this decomposition is a sum of

${N}_{s}$ complex exponentials:

where

${\alpha}_{n}$ and

${\phi}_{n}$ are the scattering amplitude and phase for point scatterer

n, and

${R}_{n}$ is the one-way antenna-scatterer range. The residual component

$r\left(f\right)$ encompasses small variations of

${S}^{Meas}\left(f\right)$ which are not well modelled by the sum of

${N}_{s}$ complex exponentials. If the number of scatterers

${N}_{s}$ is known, the problem is to estimate

${\alpha}_{n}$,

${\phi}_{n}$ and

${R}_{n}$. This is a parametric line spectrum estimation problem, for which several methods exist [

18,

19]. The root MUSIC algorithm [

20] was chosen for estimating

${R}_{n}$ due to its high computational efficiency compared to nonlinear least squares estimation. After

${R}_{n}$ has been estimated for all

n, the complex scattering amplitudes

${\alpha}_{n}{e}^{j{\phi}_{n}}$ can be estimated using linear least squares.

The number of scatterers

${N}_{s}$ determines what portion of

${S}^{Meas}\left(f\right)$ is well modelled by the summation term in (

5); i.e., the

signal subspace in MUSIC terminology. The remaining component of

${S}^{Meas}\left(f\right)$ is what defines

$r\left(f\right)$; i.e., the

noise subspace.

${N}_{s}$ was tuned such that the range profile corresponding to the summation term in (

5) closely resembles the range profile corresponding to

${S}^{Meas}\left(f\right)$, which only needs to be done once per frequency band.

The estimated scattering amplitudes

${\alpha}_{n}$ are only accurate for dominant scatterers, which in this case are the equivalent point scatterers due to mutual coupling. The forest region should not be represented in this manner as the line spectrum model estimates are radiometrically inaccurate for these relatively weak scatterers. The estimated mutual coupling component,

${\widehat{\xi}}^{Coupling}\left(f\right)$, was, therefore, extracted by selecting all the point scatterers near the antenna (

${R}_{n}\le 24$ m), where mutual coupling scattering mechanisms take place and where no forest is present:

The estimated frequency-domain signal with mutual coupling suppressed,

${S}^{MCS}\left(f\right)$, is then obtained by subtracting

${\widehat{\xi}}^{Coupling}\left(f\right)$ from the measured signal:

This mutual coupling-suppressed signal is still of finite length, and so the scene reflectivity is still circularly convolved with a sinc function. This has the effect of spreading the forest reflection in range. A moderate tapering window function ${W}_{R}\left(f\right)$, the Hamming window, was, therefore, applied to suppress these side-lobes at the cost of a slightly worsened resolution (a factor of 1.36 increase).

#### 3.4. Range Profiles

Range profiles, representing the complex reflectivity as a function of range, are computed as

Figure 7 shows an example of a measured magnitude-squared range profile, the estimated coupling component and the coupling-suppressed signal. This example uses a P-band, VV-polarised measurement, which is the type of BorealScat measurement that is most severely affected by mutual coupling. By applying the proposed method, the mutual coupling peak at

$R\approx 0$ m is significantly suppressed (>40 dB) along with its side-lobes, revealing the trihedral corner reflector peak. The region of the range profile containing the forest response is also significantly altered, showing how severe the interference is between the forest reflection and mutual coupling if not accounted for. The trihedral reflector response in the range profile can then be used for calibrating magnitude and phase errors between channels.