Elastic constants estimation

140 160 180 200 220 240

0 0.05 0.1

Frequency (kHz)

Amplitude (a.u.)

Figure 3.4: Portion of a measured spectrum (plain line), spectrum reconstructed after the linear prediction filter method (dashed line), and spectrum reconstructed after the non-linear optimization method (dot line, almost indistinguishable from the measured spectrum). Resonant frequencies obtained from linear prediction (•) and non-linear optimization (+). Measurement on a epoxy/glass composite sample.

with more degrees-of-freedom than necessary. In that case, the fitting algorithm is able to discard some components, giving them a very low amplitude, a very lowQ, or a very high or low frequency so that they do not add any significant contribution to the model.

3.5 Elastic constants estimation

In RUS, the elastic constant are generally obtained through an optimization process, in which the stiffness constants are adjusted until the best match between measured and predicted frequencies has been obtained. For that purpose, a cost function of the type

F =X

n

f_n^meas.−fn^pred.(Cij) f_n^meas.

!2

(3.11) is defined and minimized. This process requires that each measured frequency is cor-rectly paired with its corresponding predicted frequency. The pairing is not straight-forward because some predicted modes can be missed during the measurement (this is particularly true for highly attenuating materials, such as bone (Bernard et al., 2013) and because vibration modes predicted for the initial guess of the stiffness constants can appear in a different order than in the measurement.

We here describe a probabilistic formulation of the estimation of the elastic constants, in a Bayesian framework, which provides a quantitative criterion for the

automated pairing of the predicted and measured resonant modes. This Bayesian formulation of the problem requires the definition and use of prior information on the elastic constants.

3.5.1 Bayesian formulation of the RUS inverse problem

We denote m a random vector in them dimensional space of the parameters (the stiffness constants) and p_M(m) the prior distribution of the parameters. The ob-served data (the measured frequencies) are also represented by a probability distri-bution p_D(d), whered is a vector of thep dimensional data space. The functional relation between the data and the parameters is noted d =g(m) and will be sup-posed to be exact (all the errors are included in the distribution of the frequencies).

The posterior distribution of the parameters for given data d (the solution of the inverse problem) is given by the Bayes’ rule. Following Tarantola(2005), it is ex-pressed by

σ(m) = pD(g(m))pM(m)

Rp_D(g(m⁰))p_M(m⁰)dm⁰. (3.12) Let us assume that the probability distributions on the parameters as on the data are multivariate normal distributions. The distribution over the data space is characterized by a mean d_obs and a covariance matrix C_d and is given by:

pD(g(m))∝exp

We further assume that the measured frequencies are mutually independent, i.e. C_d is diagonal with entriesC_dⁱ. The prior distribution on the parameters is characterized by a mean mpriorand a covariance matrix Cm and is given by:

pM(m)∝exp

Under the multivariate normal assumption, Eq. (3.12) can be written

σ(m)∝exp(−S(m)), (3.15)

WhereS is themisfit function:

2S(m) = (g(m)−d_obs)^tC_d⁻¹(g(m)−d_obs) + (m−m_prior)^tC_m⁻¹(m−m_prior).

(3.16) Therefore, finding the point of maximum a posteriori probability (MAP) is identical to finding the point that minimize the misfit function S. Under an assumption of small non-linearity, the MAP estimate can be obtained recursively using a quasi-Newton method (Tarantola,2005):

m_n+1=m_n+ (G^t_nC_d⁻¹G_n+C_m⁻¹)⁻¹(G^t_nC_d⁻¹(d_n−d_obs) +C_m⁻¹(m_n−m_prior)), (3.17)

3.5. Elastic constants estimation 51

where G is the matrix of the partial derivatives of the frequencies with respect to each elastic constant evaluated at step nfrom Eq. (3.3). Parameters were adjusted using Eq. (4.15) until there was no more significant changes in the value of the misfit function (10⁻⁵ relative change). Once the final point was obtained, the posterior covariance matrix of the parametersC˜_m was calculated (Tarantola,2005):

C˜m≈(G^tC_d⁻¹G+Cm−1)⁻¹. (3.18)

The diagonal entries of this matrix are the posterior variances of each parameter while the off-diagonal entries represent correlations between parameters.

At this point, we have a solution to the RUS inverse problem that allows one to properly include prior information on the stiffness constants and uncertainties on the measured frequencies. This solution is more general than the least-square solution usually defined in RUS, Eq. (3.11), which can be obtained here as a special case (assuming infinite prior dispersion of the parameters and diagonal covariance matrix C_d with variances proportional to the squared frequencies). However, it still requires the correct pairing of the measured and predicted frequencies. The objective of the next section is to define a criterion for an automated pairing of the frequencies.

3.5.2 Automated pairing of the resonant modes

Suppose that we have a set of p measured frequencies dⁱ_obs (i= 1, ..., p) and a set of q predicted frequencies g^j(m) (j= 1, ..., q,q ≥p). Stating that we do not know the correct pairing of the modes is equivalent to stating that we do not know which componentg^j(m)of the vector g(m)to pair with the i-th measured frequency dⁱ_obs to calculate the density pD(g(m)) in Eq. (3.13). To find the correct pairing, we will use the approach of Bayesian model selection and hence make the assumption that the correct pairing of the frequencies is the one which maximize the evidence (Wasserman,2000), i.e. maximize the integral R

pD(g(m⁰))pM(m⁰)dm⁰ (denomina-tor in (3.12)). Therefore, one solution could be to calculate the evidence for each possible pairing combination of frequencies. However for about p = 20 measured frequencies that can reasonably be supposed to correspond to one of theq = 30−40 first predicted frequencies, there is an extremely large number of combinations and it is not possible to evaluate all of them in a reasonable amount of time. Then, we consider each frequency independently and compute

P_i^j =

Rpⁱ_D(g^j(m))pM(m)dm P

j⁰

Rpⁱ_D(g^j0(m))pM(m)dm (3.19)

for each possible combination of i and j (it gives a reasonable number p×q of

We then look for the maximal individual normalized evidence max_i,j(P_i^j) and as-sociate the corresponding measured and predicted frequencies. For this couple of frequencies we compute the MAP estimate from (4.15) and the posterior covari-ance matrix from Eq. (3.18). This posterior distribution becomes the new prior distribution, for which the evidences Eq. (3.19) are recomputed. A second pair of frequencies is associated. The process is repeated for p steps, until each measured frequency has been associated to one of the predicted frequencies (Fig.3.5).

The solution of the inverse problem is the MAP estimate obtained when all the measured frequencies have been taken into account. Uncertainty on this solution is obtained from the final posterior covariance matrix.

3.5.3 Implementation

Integrals in Eq. (3.19) were calculated using Monte-Carlo simulation. We gener-ated a number N of samples m1, ...,m_N of the prior distribution and estimate the integrals as We found that for N = 500the variance on the estimated P_i^j was sufficiently small for the whole process to be repeatable. The total number of computations of the resonant frequencies (evaluations of g(m)) is then equal to 500×p. Samples were generated in a unit normal distribution and transformed into samples of the desired normal distribution pM(m) using the Cholesky decomposition of the covariance matrix (Kroese et al., 2011). We assumed that this distribution was multivariate normal because it allows iterative computing of the MAP estimate. However, this is not strictly consistent with physical boundaries on the stiffness constants. Indeed, any multivariate normal distribution attributes a non-zero probability to elastic tensors that are not positive-definite. Therefore, when generating samples of this distribution we tested each sample and rejected all the samples for which the elastic tensor was not positive-definite.

There is an infinite number of theoretical vibration modes for a solid of finite shape. However, the Rayleigh-Ritz approximation method that is used to compute the frequencies (section 3.2) only considers the ∼ 10³ first modes (852 for a RP sample with a polynomial order N = 10). Moreover, the experimental spectra were measured in a limited frequency band containing the ∼30 first modes. Then,

3.5. Elastic constants estimation 53

Figure 3.5: Diagram of the algorithm for the automated pairing of the resonant frequencies and resolution of the inverse problem. Starts at step n = 0 and stops whenn=p. At the end of the last step each measured frequencies is paired with one of the predicted frequencies and the solution of the inverse problem is the posterior distribution. See text of section3.5for details.

the number q of predicted modes that are considered in the inverse problem can be reasonably fixed to about 50 without risking to not consider a mode that was actually observed.

The distribution p_D(d) should reflect the knowledge of the experimental data.

Because we assumed that the frequencies were mutually independent, we only have to specify the mean dobs and the diagonal terms of the covariance matrix. The mean values of the frequencies measured over several measurement (section3.4) were naturally chosen as the central values of the distributions. Variance terms must take into account measurement uncertainties and potential model errors (such as errors due to imperfect sample geometry), which sum under the Gaussian assumption (Tarantola, 2005). Uncertainties on the measurement are given by the standard deviation over the measured frequencies, but modeling errors are more difficult to evaluate a priori. A solution is to start with a reasonable guess for model errors, and then to evaluate them a posteriori from the value of the misfit function. Based on previous experiment in RUS measurements as on literature review, we started by postulating variances equal to0.5%of each frequency (measurement and model error combined). This value corresponds to usual misfit obtained in RUS. The uncertainty estimation was then refined at the end of the inverse problem, using the method which consists in adjusting the assumed error so that the final misfit value 2S is equal to the number of data points p (wich is approximately the expected value of a p degrees-of-freedomχ² distribution) (Press et al.,1992).