Radio Science

Data set generation and inversion simulation of radio waves propagating through a two-dimensional comet nucleus (CONSERT experiment)



[1] To prepare the Comet Nucleus Sounding Experiment using Radio wave Transmission during the Rosetta mission, we study the electromagnetic wave propagation through a comet nucleus model and tomographic inversion in a two-dimensional setting. For the propagation, the Ray Tracing Method (RTM) is validated with respect to the Pseudo-Spectral Time Domain (PSTD) method. For the inverse problem, a Tikhonov-like inverse RTM method based on weak permittivity assumptions is used, with synthetic data derived from the PSTD algorithm. Reconstruction results show that the Consert data will permit a reliable tomography of the comet nucleus. Surface data will enhance the quality of the imaging.

1. Introduction

[2] The ROSETTA cometary mission is a cornerstone mission of the European Space Agency (ESA) dedicated to the study of comet 46P/Wirtanen in 2011. ROSETTA will carry a Surface Science Package called RoLand which will land on the nucleus to perform in-situ investigations. The CONSERT experiment (COmet Nucleus Sounding Experiment by Radiowave Transmission) is one of these investigations which involves both the orbiter and the lander. It aims at imaging the internal structure of the comet nucleus by analysing the time-delay perturbations affecting radiowaves propagating through the nucleus. The principle of this experiment is detailed by Kofman et al. [1998] and Barbin et al. [1999].

[3] Dedicated data processing techniques are currently under investigations. Hérique et al. [1999] proved that the mean nucleus permittivity can be derived from CONSERT measurements and proposed a statistical characterisation of comet nucleus in two-dimension. Barriot et al. [1999] demonstrated that we can map large scale heterogeneities in the frame of the small perturbation theory by using Tikhonov-like techniques.

[4] To develop these inverse methods, one needs to generate synthetic data which mimic real measurements as closely as possible. One of the simplest ways to simulate wave propagation is the Ray Tracing Method (RTM) which is generally used in seismic tomography [Snieder and Sambridge, 1991]. CONSERT is a more complex case because of the following:

  1. The medium in which waves propagate is a completely bounded body, unlike in geophysical prospecting. The propagation in such a medium is affected by abrupt interface reflexions and refractions, trajectory deviations, diffractions and interferences. All these phenomena can have a great impact on the measurements collected by the orbiter.
  2. The CONSERT instrument is designed to detect very low level signals corresponding to indirect paths (internal reflections).
  3. In the CONSERT experiment, we are not only interested in measuring the propagation delay between the lander and the orbiter, but also in measuring the signal power which is providing by itself additional information on the crossed medium. The initial circular polarization of the emitted waves, combined with the 3-D complex shape of the body, greatly complexifies the modeling of the received signal power.

[5] All of these constraints implied to develop a Ray Tracing Method tailored to the CONSERT experiment. An evident first step was to develop a 2-D algorithm to validate the use of the RTM and to study its performances and limitations before using it for the 3-D case. The validation of the RTM method was done using the Pseudo Spectral Time Domaine Method (PSTD), based on the resolution of the Maxwell equations. This method is more accurate than the RTM method but numerically untractable up to now for the 3-D case.

[6] In this paper we present these two methods and compare their performances in a 2-D case. We also use the derived synthetic data to test a 2-D inversion scheme (imaging), based on Thikonov-like least-squares techniques, with statistical a priori assumptions about the structure of the inhomogeities and about the surface permittivity distribution.

2. Pseudo-Spectral Time Domain Method (PSTD)

2.1. Introducing PSTD

[7] Solving Maxwell equations has been widely used to simulate radio wave propagation since Yee [1966] described the Finite Difference Time Domain (FDTD) method in the sixties. The FDTD method is based on a central differential operator applied to both time and space coordinates. It results in a second order accuracy with respect to the time and space discretisation. FDTD remains popular among scientists although it suffers from anisotropic numerical dispersion. Phase velocity in vacuum is minimum along the x and y axis and maximum along the diagonals. It leads to phase errors that accumulate with time. Reducing this anisotropy requires finer grids. To achieve good accuracy, problems of moderate size (100 λ) require at least 10 or 20 nodes per wavelength. Our simulations are 500 λ large since the wavelength λ is thought to be 2.4 m in the nucleus (relative permittivity of 2) and the latter to be 1200 m across [Lamy et al., 1998]. To avoid computing on too many nodes, in order to keep anisotropy low, higher order solutions were developed such as 4th order finite difference method.

[8] Another class of methods, called spectral methods appeared in the seventies. Spectral methods were developed to solve partial differential equations in fields ranging from fluid dynamics, weather forecasting, to wave propagation [Fornberg, 1998]. In contrast to the FDTD method, the spectral methods require only a few nodes (in theory two) per wavelength to achieve infinitely good accuracy. In spectral methods, functions are approximated by polynomials of special functions.

[9] Using trigonometric functions leads to the so-called pseudospectral method [Orszag, 1972]. Fornberg [1990] shows that pseudospectral methods can be viewed as finite difference methods of infinite order. A difference however is that the electro-magnetic components are all computed on the same nodes in the PSTD method (a regular grid to be compared to the staggered grid used in the FDTD method). When applied to the resolution of Maxwell equations, the pseudospectral method takes the name of PSTD by reference to the FDTD method.

2.2. Principles of PSTD

[10] We choose to solve te Maxwell equations in the case of 2-D Tranverse Magnetic (TM) mode:

equation image

where Ez is a component of the electric field; Hx and Hy are the components of the magnetic field; Jz is a current source; and μ and ϵ are respectively the permeability and permittivity of the medium.

[11] The PSTD method consists of (1) computing space derivative approximations using a spectral method on the right hand side of equation (1), (2) estimating the time derivatives in the left hand side of equation (1) from step one, and (3) integrating in time. Finite difference can be used, or more sophisticated approaches.

2.2.1. Estimation of the Space Derivatives

[12] The pseudospectral method approximates the electromagnetic fields with polynomials of trigonometric functions. It relies then on the discrete Fourier transform: derivation of a N-point function in the Fourier domain reduces to N single multiplications. Let us note f(x) a function of x

equation image

where equation imagex and equation imagex−1 denote respectively the discrete Fourier transform and its inverse and kx the wavenumbers.

[13] The introduction of Fast Fourier Transform (FFT) made this operation attractive. To save more time and memory, the use of hermitian symmetry is highly recommended, so that only N/2 Fourier coefficients need to be computed.

2.2.2. Absorbing Boundary Conditions

[14] Using the discrete Fourier transform implies periodicity of the computational domain: a pulse outgoing from one edge of the computational domain will enter it on the opposite edge. To ensure simulations on an infinite domain, we adopted Bérenger's Perfectly Matched Layers (PLM) as Absorbing Boundary Condition to massively absorb outgoing waves. Bérenger [1994] described the PLM technique for the FDTD algorithm. Implementation to the PSTD algorithm is straightforward.

[15] PML technique lies on the splitting of the electrical field component Ez (for 2-D TM case) in two sub components Ezx and Ezy. To absorb waves in the boundary layers, electrical and magnetical losses are introduced, by the mean of electrical and magnetical conductivities, respectively (σx, σy) and (ρx, ρy). The resulting Maxwell equations are

equation image

[16] A PML medium is then defined by the 6 following parameters: (ϵ, μ, σx, σy, ρx, ρy). Vacuum is a particular PML medium the parameters of which are (ϵ0, μ0, 0, 0, 0, 0). Bérenger noticed that the impedance of a PML medium is that of vacuum if ϵ = ϵ0, μ = μ0 and

equation image

He demonstrated the transmission properties through an interface between two PML media of parameters (ϵ, μ, σx1, σy1, ρx1, ρy1) and (ϵ, μ, σx2, σy2, ρx2, ρy2) verifying equation (4). If this interface is normal to the x-axis, waves crossing it are completely transmitted, whatever their frequencies and incidence angles, if

equation image

The same property is derived for an interface normal to y axis: y has then to be replaced by x in equation (5).

[17] Bérenger [1994] then proposed to belt the computational domain with such PMLs. For efficiency reasons, we proceeded the way described in Figure 1 using discrete Fourier transform properties. To avoid numerical noise while waves enter the PMLs, we used a quadratic law for the conductivities:

equation image

where 2δ is the PML width and r is the distance to the computational domain. The maximum conductivity σM is reached in the middle of the PML at r = δ.

Figure 1.

Implementation of Absorbing Boundary Conditions using PMLs.

[18] Absorbing Boundary Conditions are matched to vacuum, so that outgoing waves are massively absorbed without any spurious reflections. That way, the computational domain seems to be infinite.

2.2.3. Condition of Validity

[19] The use of discrete Fourier transform implies some constraints on space discretisation. To avoid aliasing and then large errors when computing space derivatives within the Fourier space, Shannon theorem claims that the space steps Δx and Δy have to be shorter than half the shortest wavelength λmin

equation image

[20] Our simulations run with Δx = Δy. Theoretically, equation image, it means two nodes per shortest wavelength, should define a fine enough grid.

[21] But two difficulties arise. The first one is in defining λmin. Its expression is

equation image

So computing λmin requires to know the maximal permittivity of the medium ϵMax and the maximal frequency of the emitted signal νMax. The maximum permittivity ϵMax is easy to get whereas defining νMax is much more complicated. Signal shape is a trade-off between pulse duration and frequency broadening, so that signal cannot be bounded in frequency.

[22] The second difficulty concerns the enlargement of the spectrum when a pulse crosses an interface. The explanation of this significant enlargement is in the product between 1/ϵ and the space derivatives of the magnetic field equation image in equation (3). This product acts as a convolution in the wavenumber domain. This leads to unphysical Gibbs oscillations which spread through the whole computational domain and spoil the simulation. Gibbs phenomenon has to be rejected below the upper limit of the acceptable numerical noise, by oversampling the propagating signal and avoiding strong permittivity gradients. Oversampling means that cells have to be smaller than the Nyquist wavelength. We experimentally choose Δx = Δy = λmin/5, to cope with our models which contain steep interfaces between media of permittivity ϵr = 1 (vacuum) and ϵr = 2 (comet nucleus).

2.2.4. Time Stepping

[23] Once the four space derivatives in Maxwell equations (equation (3)) are estimated, time has to be stepped forward. Time derivative are estimated and then integrated using values of the fields at previous time steps. We adopted the method after Kinnmark and Gray [1984] which integrates sets of differential equations of the form

equation image

where ut is a vector of time and space functions and D is a constant matrix with respect to time. The numerical Maxwell equations can be expressed that way if

equation image


equation image

where equation imagex and equation imagey are the derivative operators with respect to x and y. Kinnmark and Gray [1984] proposed a one step numerical integration of the form

equation image

where Δt is the time step and αg are scalars. From a practical point of view, powers of D are not computed. It is better to use the iterative equivalent (equation 12) of equation (11). Time integration is then performed in G substeps.

equation image

In our simulations, we set G = 4, as we noticed that pulses are abnormally attenuated for G = 3 compared to propagation in vacuum. The coefficients αg are then

equation image

2.2.5. Numerical Stability

[24] To ensure algorithm stability, time steps have to be small enough. Considering the space and time eigenvalue problem and introducing results from Kinnmark and Gray [1984], we can derive the stability requirement

equation image

However, this stability criterium only holds for homogeneous media. Deriving such a criterium for heterogeneous media is not obvious. For this reason, we require equation (14) to be verified in each homogeneous part of the computational domain. Nevertheless this might not be sufficient, so we took a much smaller time step

equation image

2.2.6. Excitation

[25] As in the FDTD method, wave propagation can be initiated by using nonzero initial condition or a current source. In the first case, one initializes the three fields Ez, Hx and Hy to coherent values to cause the electromagnetic field to propagate. However in our simulations, the source lies on the comet nucleus surface, so that interaction between waves and the surface cannot be a priori easily described. Thus a current source Jz is introduced in Maxwell equations and the fields are initialized to zero. We then observe, Figure 2, that the radiation diagram is not uniform due to the nucleus. To improve coupling, it means the energy transmitted inside the nucleus, the source is put right under the surface of the nucleus.

Figure 2.

Radiation diagram due to the current source in the presence of the nucleus surface. Nucleus permittivity ϵr is 2. Frequency Contents of the Current Source

[26] The signal for our simulations has properties close to that of the signal used by CONSERT, namely a 9 MHz bandwidth signal modulated at 90 MHz. For simplicity, we choose a 90 MHz modulated Gaussian pulse of standard deviation σt = 1/2πσν with σν = 4.5 MHz. The maximal frequency νMax was then said to be 100 MHz, meaning that λmin = 2.1 m. Spatial Shape of the Current Source

[27] Because of the use of discrete Fourier transform to compute space derivatives, the spatial shape of the current source Jz has to respect the Shannon-Nyquist criterion. Thus, we choose a 2-D Gaussian shape of standard deviation σs for the spatial extension of the source. If we set σs = 2.85Δx, this form rejects the numerical noise at least 140 dB below the emitted signal. The counterpart is that the source is slightly over one wavelength large (2σs limit). The full expression of the current source is then

equation image

where n0 is the time step at which the maximum of the pulse is emitted.

3. Ray Tracing Method (RTM)

[28] The Ray Tracing Method (RTM) is a practical tool to model large scale wave propagation problems in “smooth” media without having to track the whole wave-field evolution. It founds its major area of application in high frequency geophysical tomography, X-ray medical imaging and optics. It is a unique way to model a high frequency propagation in the case of bounded 3-D nonhomogeneous media with complex boundaries and over long propagation times.

[29] In this section we present this technique, and apply it to a 2-D bounded medium in order to show its capabilities and limitations comparing to the PSTD method. Starting from the TM mode Maxwell equations (Equation (1)), we have

equation image

By looking to an harmonic process, i.e. with

equation image

we obtain the Helmholtz equation:

equation image

which simplifies outside the source to

equation image

equation image being the wavenumber and equation image(equation image) being the local relative dielectric permittivity of the medium.

[30] In the case of a smooth medium (i.e., in a mathematical sense, with a sufficiently differentiable permittivity function with scale variations larger than a few wavelengths λ), we can expect that the solution of the Helmholtz equation (equation (20)) will look-like a plane wave affected by a phase shift. This wave can be written as:

equation image

where A(equation image, k) and φ(equation image) are respectively the local amplitude and the local phase of the wave. Assuming the asymptotic expansion [Keller, 1962]:

equation image

we arrive at (by substituting equations (22) and (21) in (20) and ordering the result with respect to the inverse powers of k)

equation image

called the eikonal equation, and

equation image

called the transport equation.

[31] If we can write equation image as:

equation image

ϵ0 being a constant background permittivity, we have:

equation image

where s is the rectilinear distance from the source. The term:

equation image

is the phase perturbation affecting the wave. We will use these two last equations later. The characteristics of the eikonal equation are the ray equations:

equation image

where equation image(s) is the direction vector of the ray path at a given point equation image and n = equation image is the refractive index. We can add to the ray differential equations (Equation (28)) an additional equation to compute the travel time T along the ray path:

equation image

These equations characterize completely the ray path in a continuous medium. When a discontinuity, in the form of a smooth shape interface, is encountered, the paths of the reflected and the refracted rays can be calculated by following Descartes-Snell's laws.

3.1. Ray Path and Phase Determination

[32] The simplest form of ray tracing is shooting (initial value problem), in which the initial position equation imageS and the initial direction equation imageS = equation image(equation imageS) of the ray are imposed. Another form of ray tracing is the “two point ray tracing”, where the source position equation imageS and the receiver position equation imageR are known and we have to find the rays connecting these two points. To solve such a problem we can use the shooting technique by starting with an initial position equation imageS and a shooting direction equation imageS, and recording the final position equation imageF. By analyzing the evolution of the couple (equation imageF, dequation imageF/dequation imageS) we can adjust the shooting direction until the final condition equation imageF = equation imageR is met. Such a shooting algorithm can diverge in the case of highly nonhomogeneous media for which the final conditions (equation imageF, equation imageF) are very sensitive to the variations of the initial values (equation imageS, equation imageS). In this paper we will use the first form which is efficient for a small shooting step γ.

3.2. Amplitude Determination

[33] The two different effects that can be calculated using the RTM in a nondispersive medium are as follows:

  • The geometric spreading (which consists on the geometric expansion of the wavefront with the distance from the source), by solving the ray equations (equation (28)) for a thin beam with an aperture angle δ (Figure 3). From the transport equation (equation (24)) evaluated at two arbitrary positions equation image0 and equation image along the central ray, we have:
    equation image
    where dS0 and dS are the cross-section surfaces at respectively equation image0 and equation image. This follows from the conservation of the power along the beam. The power per surface unit at position equation image is then given by:
    equation image
    where P0 = A02(equation image0)n(equation image0)dS0 is the emitted power by the source along the beam. In 2-D power calculation, dS0 and dS are lineic quantities. Difficulties arise in special points where dS = 0. These locations, where the power amplitude is “infinite”, are called caustics. Generally, we will simply reject the rays ending at caustics or we will determine their powers by continuity from adjacent “well defined” rays. For an anisotropic source (Figure 2) the emitted power is function of the emission direction. P(equation image) is then written as
    equation image
    where P(θ, δ) is the power emitted in the angular direction θ by a beam with an angular aperture size δ.
  • The reflection/transmission coefficients, that account for the splitting of energy at the interfaces when the incoming wave separates into refracted and transmitted outcoming waves. The outgoing power is
    equation image
    where Pin(equation image) is the power carried by the incident ray, Pout(equation image) is the power carried by the reflected or transmitted ray and T is the reflection or transmission coefficient.
  • The loss in the medium, which attenuates the wave amplitude during propagation from the source S to the intercept point R. It depends on the medium composition and on its inhomogeneities:
    equation image
    where α(equation image(s)) is the local attenuation coefficient of the medium.
Figure 3.

Beam used in amplitude determination.

3.3. Polarization Determination

[34] The polarization determination is of interest only in the case of 3-D propagation or in the case of a complex source polarization (circular or elliptic). To model it we have to consider, when we reach the interface, the appropriate transmission/reflection ratio for each polarization direction, so that we can determine the polarization state at the receiver position. In our 2-D case the polarization remain constant during the wave propagation corresponding to a TM mode (electric field perpendicular to the propagating plane).

4. Results of Wave Propagation Simulation and Methods Comparison

4.1. Nucleus Model

[35] There are many possible nucleus models that can be used for this study [Whipple, 1950; Weissman, 1962; Möhlmann, 1996; Weidenschilling, 1997]. As the aim of this paper is not to discuss which one is the most plausible, we will consider only one of these possible models: a 2-D comet nucleus model that is a conglomerate of homogeneous subnuclei (cometesimals). This model, already presented by Benna and Barriot [2002] allows us to easily compute analytically the local permittivity. The permittivity distribution model inside the comet nucleus is then given by:

equation image

where ϵ0 is the background permittivity, N is the number of cometesimals into the nucleus, (Λi, xi, yi, bi) being respectively the central permittivity perturbation, the cartesian coordinates of the cometesimal centre, and a characteristic length assumed to be the radius of the ith cometesimal. The term ϵ0 can be seen as the mean permittivity value of the nucleus, although it is not strictly the mean value on a mathematical sense. The comet is bounded by a geometrically smooth shape interface which makes the ray tracing technique valid and prevents surface diffraction phenomena. Figure 9a shows this model with its permittivity distribution. The characteristic length for this model is (∇n/n)−1 ≥ 500 m. This allows us to use the geometric optics approximation (λ = 2.4 m) and so we can apply the RTM. The wave source (lander) is positioned on the surface, and the receiver positions (orbiter) are represented by a circular orbit around this nucleus model.

4.2. Rays Shots

[36] We developed for this simulation a ray tracing algorithm based on the previous exposed shooting technique (section 3.1):

  1. Each ray is characterized by the initial shooting direction equation imageI. The shooting angle sampling which defines the rays density is γ.
  2. When an interface is encountered, if a refraction occurs, we propagate simultaneously the refracted ray and the partially reflected component, and so on, taking into account the power attenuation of each partially reflected ray.
  3. If the total length for a given ray exceeds a defined limit lmax or a refraction number limit Refmax, we will consider that the attenuation due to several interface reflections and to media losses makes the power carried by the ray out of the detection range of the receiver, and reject the ray. In our simulation lmax = 15000 m, corresponding to attenuation values greater than 200 dB. The geometrical spreading will also often increase this attenuation.

[37] If the ray is considered as valid, we start its amplitude computation: We shoot a second ray from the same source in a direction very close to the direction of the original ray in order to evaluate the geometric ray divergence. The received power is the combination of effects of spreading, of interface reflections and of losses in the medium (already discussed in the section 3.2) and can be written as:

equation image

where Ti are the reflection or transmission coefficients corresponding to the N interface encounters undertook by the ray. The final amplitude attenuation in dB is then given by:

equation image

where Attenu(r) is the local attenuation of the medium in dB, and dS is the cross-section distance of the beam when it intercepts the orbit, allowing us to compute the geometric spreading. D(θ, δ) is the source directivity:

equation image

If we end up with a caustic (dS = 0), we simply reject the ray. This does not affect the final result, since we shoot highly concentrated rays.

4.3. Comparison

[38] We run the PSTD and the RTM algorithms on the model described in section 4.1. The PSTD and RTM algorithms used the parameters described respectively in Tables 1 and 2.

Table 1. Parameters of the PSTD Algorithm to Obtain a 120 dB Dynamic
   Size of computational domainNx = Ny = 3072
   Space samplingequation image
   Time samplingequation image
Absorbing Boundary Condition
   Apparent transmission coefficientT0 = −120 dB
   Half-width of PMLsδ = 40Δx
   Sizeσs = 2.85Δx = 1.18 m
   Delay of the maximum of the pulset0 = 627Δt
Table 2. Parameters of the RTM Algorithm
   Ray trajectory integration stepds = 1 m
   Angular shooting stepγ = 0.05°
   beam angle sizeδ = 0.005°
   Max. ray lengthlmax = 15000 m
   Max. ReflectionsRefmax = 8
   Background permittivityϵ0 = 2
   Local attenuationAttenu(r) = 0 dB/m

[39] To compare the two algorithms, we need to extract the travel time and intensity of the signals on the orbit from the outputs of the PSTD algorithm. As the medium is not dispersive, the travel time is defined by the time elapsed between the emission and the reception of the maximum of the Gaussian pulse. Its amplitude is given by the maximum of the pulse. Figure 4 displays the travel time of waves through the model (Figure 9a) using the PSTD algorithm (Figure 4a) and the RTM algorithm (Figure 4b). Along the abscissae we have the angular position on the orbit, with color reflecting the power density in dB. We notice that the PSTD algorithm gives much more details than the RTM by taking into account the diffraction phenomena. However, the diffraction encountered here is a numerical artifact: the PSTD algorithm requires the discretisation of the propagation medium. So the surface is like stairs and is responsible for diffraction, whereas for the RTM, the surface is continuous. Diffraction can be removed by smoothing lightly the interface. Besides diffraction, the main patterns are identical on both graphs. The main pulses have the same power and are at the same location. To identify the main patterns, we refer to Figure 5 which displays the intensity of the electric field in the model versus time. We then notice that wave front 1 corresponds to direct propagation between the source and orbit in vacuum. Wave fronts 2 and 3 correspond to main waves which only undergo one refraction at the nucleus-vacuum interface. The others wavefronts correspond to pulses which reflected either completely one or several times on the interface (4, 5, 7 and 8) or partially (6).

Figure 4.

Comparison between the RTM and PSTD algorithms. Figures (a) and (b) show the complete signals received along the orbit computed respectively from the PSTD and RTM algorithms. Details of the 2 and 3 main wave-fronts are shown in figures (c)–(d) and (e)–(f). They correspond to a direct pulse which propagated through the nucleus and undertook only a refraction.

Figure 5.

Propagation of the electric field in the comet nucleus model (time sequence). Wave-fronts are labeled for identification on Figure 4.

Figure 5.


Figure 5.


Figure 5.


Figure 5.


Figure 5.


Figure 5.


Figure 5.


[40] Let us have a closer look on the two direct pulses (2 and 3). We see on Figure 4e that the RTM algorithm gives continuous arrival times whereas the PSTD algorithm gives discrete arrival times. This is normal as the PSTD algorithm uses discrete time steps. We notice on both Figure 4c and Figure 4e that there is a good agreement between the two methods with respect to the arrival time. On each graph, both curves follow the same variations and the differences are within one time step (11 ns). These differences can be explained by errors on the detection of the maxima. Concerning amplitudes the agreement is also satisfactory, according to the previous remarks. However, the RTM does not describe interferences in opposition to the PSTD algorithm. That's why amplitude variations are important on the border of wave-front 2 for the PSTD algorithm outputs (Upper graph in Figure 4d. Other points correspond to a wave front arriving later). In areas where interference does not occur, agreement between both methods is really good (Figure 4f). Note that these interferences are physical and are not due to the discretisation of the surface. They occur because two or more rays converge on the same orbit point.

4.4. Conclusion

[41] The PSTD algorithm permits to model the full solution of the Maxwell equations. Unlike the RTM algorithm, it can handle diffraction, caustics and fully describe interference phenomena. As expected, the RTM algorithm is efficient, provided that the model variations are smooth (i.e., the heterogeneities scales are large with respect to the wavelength). Thus, the RTM will be implemented to study the propagation in 3-D which is needed for CONSERT, as the PSTD algorithm is too CPU and memory demanding to be extended to the three-dimensional case.

5. Data Inversion

[42] The inverse imaging problem was already discussed by Barriot et al. [1999] in the case of a 2-D model and using data generated from RTM method. In this paper we will only discuss the use of the phase information (time propagation). We consider as data the phase perturbation φ1(s) = φ(s) − φ0(s) due to the permittivity perturbation ϵ1(equation image(s)) = equation image(equation image(s)) − ϵ0 along the ray, where φ0 is the phase generated by the background permittivity ϵ0. The forward problem can then be written in a discrete form (Figure 6) by using equation (26) as:

equation image

where N is the number of cells of the discretized model, φ1j is the phase perturbation of the jth ray, ϵ1i is the permittivity perturbation of the ith cell of the discrete model, and Δji is the part of jth ray path contained in the ith cell (Figure 6). For the M available data points {φ1j}1≤jM we obtain the following matrix form:

equation image

where d = [φ1j1≤jM is the vector of data points, p = [ϵ1i1≤iN is the permittivity perturbation vector and G = [Δij1≤iN, 1≤jM is a linear matrix operator. This matrix form expresses the linearized forward problem. The image reconstruction (inverse problem) consists in finding a structure equation image which verifies equation (40).

Figure 6.

Discrete propagating ray.

[43] The resolution of this inverse problem is based on the Tikhonov-Arsenine formula [Tarantola and Valette, 1982; Tarantola, 1987]:

equation image

where Cd = σdI is the sum of data error variance (supposed to be noncorrelated) and of the forward modeling error variance. Cp is the variance-covariance matrix of the model, containing all a priori information we have about the nucleus interior. In their paper, Benna and Barriot [2002] analyzed the structure of this matrix, how to construct it and finally what is its impact on the reconstructed image.

5.1. Data Extraction

[44] During data gathering, the Consert instrument will measure the total travel times and amplitudes of propagating wave pulses. The correspondence between the phase φj and the travel time Tj of the jth measure is given by:

equation image

ν being the radiowave frequency. The propagation delay is counted from the lander to the orbiter.

[45] We need to extract from this total phase, the part due to the propagation into the nucleus φcj = φj − φvj, which contains the useful information. The free space phase φvj is due to the propagation of the ray in the vacuum from the nucleus surface to the orbiter, so we have:

equation image

where Lvj is the total ray length in the vacuum for the jth measure, and ϵv = 1 is the dielectric permittivity of the vacuum. The quantity Lvj is function of:

  1. The location of the emerging point of the considered ray j on the comet surface, which is itself a function of the trajectory of this ray into the nucleus and then so a function of the heterogeneity distribution throughout the nucleus.
  2. The local refraction index on the surface which determines the emerging direction of a ray leaving the comet and so the length of this ray from the nucleus to the orbit intercept point. A small deviation of this emerging direction produces a large deviation along the orbit and so a large variation of the length Lvj. This effect is shown in Figures 7a and 7b where we report respectively the differences between the trajectory lengths in a homogeneous and inhomogeneous model and the distance between the two orbit intercept points for a given shooting angle.
Figure 7.

Ray path length differences between anhomogeneous and an inhomogeneous model for a givenshooting angle: (a) the total ray length difference betweenthe lander and the intercept point in orbit, (b) the rayintercept position dispersion between the two models.

[46] Since we will not be able to measure the arrival direction of the ray along the orbit, we will not be able to estimate correctly Lvj. A good solution to this problem consists in back-propagating the ray from the real orbit to the nucleus surface or to a virtual orbit closer to the nucleus surface. This method will reduce the effect of errors due to the estimation of Lvj (ideally this error will vanish if we will be able to back-propagate down to the surface of the nucleus). Hérique et al. [1999] showed that a viable back-propagating method exists for 2-D circular nucleus shape models, provided that we have a complete or near complete data coverage over the orbit.

[47] With a more complex shape, we can use a more general but less accurate method based on the assumption that the heterogeneities of the nucleus are week enough. In this case, the average length of a thin ray pencil is roughly equal to the ray length computed in the case of a homogeneous model which propagates and arrives over the same “receiving” region.

[48] The data extraction strategy is then the following: for each ray j computed from RTM method for an homogeneous model and defined by its intercept position P(j) along the orbit, we average the N measurements φk located in a given reception window W = [−Δ, Δ] centered around P(j). In our case, P(j) is the angular position of the ray intercept point along the circular orbit. This averaged measurement includes an averaged phase perturbation equation image1j which is close to the real perturbation φ1j. The window W should be large enough to contain all measurements coming from the same region of the straight ray j. In our case, we choose Δ to be equal to the maximum sampling angle tolerated for the experiment [see Kofman, 1995]. We remove from this averaged total phase the homogeneous component φ0j. This can be written as:

equation image

where δk,j = P(j) − P(k). Figure 8a shows the extracted phase perturbation compared to the real one.

Figure 8.

Data extraction: (a) the extracted phase perturbation using equation (44) (cross) compared to the real phase perturbation of the ray equation (27) (dot), caustic regions are filtered, (b) internal illumination of the nucleus (cells not crossed by waves are in black).

5.2. Inversion Results

[49] A first inversion was done using PSTD data and the extraction process detailed above (section 5.1). We generated the a priori covariance matrix of the expected permittivity from an analytical model [see Benna and Barriot, 2002]. This matrix contains the a priori correlation between all cells in the discretized nucleus. Figure 8b presents cells crossed by waves for which we can use both phase information and a priori information for inversion. The permittivity of remaining cells (black cells) is determined using only the a priori information which define their correlations with the illuminated region. Figure 9c presents the inversion result. Figure 9d is the error between the real discretized model (Figure 9b) and this reconstructed model. Another reconstruction was made by adding the surface permittivity as an a priori information: we supposed in this case that the permittivity of surface cells was known with an error bar vector Cs. We injected this information to make a new forward problem by writing:

equation image


equation image

s being a vector containing the permittivity perturbation values of surface cells. The inversion equation (Equation (41)) remains the same and we have just to replace G by the new operator G′ and d by the new data vector d′. The data variance-covariance matrix Cd merges the phase variances and the surface permittivity variances:

equation image
Figure 9.

Original model and inversion results: (a) structure of the original nucleus model (ϵ0 = 2); (b) corresponding discretized nucleus with 25 m × 25 m cells, (*) represents the lander position; (c) reconstruction using the phase perturbation; (d) error between the first reconstruction (c) and the original discretized model (b); (e) reconstruction using the phase perturbation and a priori values for the surface permittivity perturbations; (f) error between the second reconstruction (e) and the original discretized model (b).

[50] Figure 9e presents the result of this new reconstruction. We can notice that the addition of the information concerning the permittivity surface decreases the reconstruction error (Figure 9f) not only in surface regions but also in the deep interior region: the permittivity perturbations were all well positioned with the correct signs. This surface information could come from other ROSETTA Experiments which will quantify the surface mineralogy and surface composition.

6. Overall Conclusion

[51] In this paper we achieved a complete 2-D wave propagation simulation (forward problem) of the Consert experiment and synthetic data acquisition using two methods: The Pseudo Spectral Time Domaine Method (PSTD) and the Ray Tracing Method (RTM). These data were also inverted using a Thikonov-like scheme. In the forward problem we showed the following:

  1. The PSTD method is very efficient for the simulation of complex high frequency wave propagations over a large scale medium. This method is today numerically intractable for a full scale 3-D Consert simulation.
  2. The RTM method is a good alternative to the PSTD method, being able to simulate wave propagation with a good accuracy. Diffraction phenomena could be added in the frame of the Geometrical Theory of Diffraction which is similar to the Geometrical Optic Theory used in this paper [Keller, 1962]. This improvement will extend the ability of our algorithm to model real diffractions due to the small irregularities of the nucleus surface. The RTM technique could be used to select the best landing site for the lander as well as orbit parameters in order to maximise the scientific return.

[52] In the preparation to the backward problem, we achieved a data filtering algorithm to determine the phase perturbation along the orbit that is an alternative to the back-propagation method of Hérique [1999] in the case of weak permittivity perturbations. We demonstrated, in the backward (inverse) problem, that adding surface permittivity values will significantly improve the imaging. These values could be derived from the global chemical surface mapping provided by other orbiter instruments and from the lander analysis, by using permittivity evaluation equations [Sihvola and Kong, 1988]. Nevertheless, we will always need to use either a priori information such as a global covariance matrix of the permittivity in order to regularize this ill-posed inverse problem.

[53] In a future work, we will go ahead and generalize the RTM method for a 3-D nucleus shape. Therefore, we will also need to take into account the real circular polarization of the emitted waves in the modeling of the power of received signals.


[54] All the computations using PSTD and RTM codes presented in this paper were respectively performed at the Service Commun de Calcul Intensif de l'Observatoire de Grenoble (SCCI) and at the Centre de Calcul Intensif du Centre National d'Etudes Spatiales (CNES). This research was founded by the Centre National de la Recherche Scientifique and the CNES. The Ph.D. grant of M. Benna is provided by the Tunisian government. The Ph.D. grant of A. Piot is provided by the French government.