Internal structure imaging of Didymos with JuRa using homogeneous asteroid models

1 Introduction

The unique abilities of low frequency spaceborne radars to probe internal structure imaging of planetary bodies at high resolution have resulted in their use on missions dedicated to kilometric-size planetary bodies. Radar instruments on these missions include in particular the upcoming JuRa instrument onboard the HERA mission which aims at imaging the binary S-type asteroid 65803 Didymos [1]. One of the major challenge when exploiting the data to reconstruct the internal structure of kilometric-size planetary bodies lies in the relatively large size of the planetary body with regard to the radar carrier signal wavelength. Given the 60MHz radar carrier frequency, Didymos' 800m diameter and Dimorphos' 160m diameter 3D domains require too much computational resources with current available hardware to perform multiple iterations of a gradient descent algorithm on all radar orbits. This problem is common in spaceborne subsurface radar sounding where the radar signal carrier frequency, driven by geophysical and technical considerations, often results in domains too large to accurately simulate radio wave propagation [2]. In order to overcome this limitation, we propose to investigate the possibility to linearize or use approximate forward operator in gradient descent algorithms. This work aims at developing robust analysis methods to process JuRa data in order to image the interior of the Didymos binary asteroid system.

2 Radar interior imaging

Internal structure imaging from spaceborne radar consists in finding the effective permittivity contrast m={χ(x)}_{x ∈ ℜ³} of the asteroid from the observed scattered field at the radar antenna for multiple locations x_l on the orbit and for angular frequency ω in the radar signal frequency band. It can be understood as a minimization problem of a chosen cost function S (e.g. least squares, Wasserstein distance) with regard to the effective permittivity contrast [3]. Such minimization problem is often solved using gradient algorithms

m_n+1 = m_n - W_n [∂f(m)/∂m]^∗_n [∂S/∂f(m)]_n

where the forward operator f(m) corresponds to Maxwell's equation along with the far-field antenna gain. W_n is a positive-definite operator (e.g. Hessian of the cost function in Newton method), [∂S/∂f(m)]_n  is the usual adjoint source (e.g. weighted residuals in least squares) and [∂f(m)/∂m]^∗_n is the conjugate transpose of the Frechet derivative [3]. JuRa is a monostatic radar, accordingly the kernel of the Frechet derivative can be expressed as a function of the the total electric field E_l,n(x,ω) associated to m_n induced by the radar source. In order to perform internal structure imaging with JuRa, it is thus only necessary to compute at each step n the electric field for each position x_l on the orbit and each angular frequency ω in the frequency band.

Fig.1: (a) Free space background model, (b) homogeneous asteroid background model, (c) true model.

Computing the electric field in Didymos and Dimorphos on all radar orbits for multiple steps n corresponds to the usual full waveform inversion algorithm [4]. Such a brute force approach requires considerable computational resources. One straightforward way to decrease the computational load is to linearize the forward operator around a fixed background effective permittivity contrast m_b

f(m) ≈ f(m_b) + [∂f(m)/∂m]_b (m- m_b)

For example, it is possible to use a free space background effective permittivity contrast, i.e. m_b=0 (Fig. 1a), which yields the popular and low computation back-propagation and pseudo-inverse methods. However, the linearization of the forward operator is only valid as long as residuals between the true and background effective permittivity contrast remain sufficiently small. The real part of the effective permittivity of Didymos and Dimorphos is estimated to vary between ε_r ∈ [4 ,7] [2] and has been showed to be too large for the linearization around a free space background to perform well due to poor focusing [5].

Homogeneous asteroid model

Fig. 2: Dimorphos homogeneous asteroid background model E_l,n(x,ω) for a given position and angular frequency along the antenna polarization component.

As an alternative, we propose to use as a background effective permittivity contrast homogeneous asteroid models with an interior effective permittivity set to an a priori estimated average value (Fig. 1b). Homogeneous asteroid models uses the shape models of Didymos and Dimorphos and strongly reduces residuals between the true and background effective permittivity contrast. Unfortunately, unlike the free space background effective permittivity contrast where E_l,n(x,ω) admits a closed form, computing E_l,n(x,ω) still requires to simulate the electric field in Didymos and Dimorphos, albeit only once. Considering Dimorphos’s largest dimension at central frequency, the domain size varies between 70 to 90 wavelengths with the expected range of ε_r. We investigate the possibility to use different numerical schemes to compute the electric field in the asteroid domain and their interior imaging performances including Discrete Dipole Approximation (DDA) and Physical Optics (PO). DDA is an accurate method which naturally handles scattering from far field sources and provides iterative accuracy improvement of E_l,n(x,ω) [6]. Our current DDA GPU implementation on an Nvidia H100 80GB requires ∼10min to accurately compute the electric field inside Dimorphos for a given position x_l and a given angular frequency ω (Fig. 2) which remains prohibitive. In order to process JuRa data volume sufficiently rapidly, a compromise needs to be found between accuracy and computational load.

Fig. 3: Comparison between simulated E_l,n(x,ω) using PO (left) and DDA (right) inside the highlighted area in Fig. 2.

PO is a surface integral method [7] which could be used to estimate E_l,n(x,ω) (Fig. 3). Additional investigation venues include: (i) reducing the number of iterations in the DDA linear solver or (ii) using geometric optics to compute E_l,n(x,ω) and physical optics to compute f(m_b).

 

References

[1] Michel P et al. 2022 The planetary science journal 3 160
[2] Hérique A et al. 2018 Advances in Space Research 62 2141–2162
[3] Tarantola A 2005 Inverse problem theory and methods for model parameter estimation (siam)
[4] Deng et al. (2022) IEEE Trans. Antennas & Propagation, 70(12) 11934–11945
[5] Dufaure A et al.2023 Astronomy & Astrophysics 674 A72
[6] Yurkin M A 2023 Discrete dipole approximation Light, Plasmonics and Particles (Elsevier) pp 167–198
[7] Berquin Y et al. 2015 Radio Science 50 1097–1109