A principal component approach to space-based gravitational wave astronomy

The current approach to data analysis for the Laser Interferometry Space Antenna (LISA) depends on the time delay interferometry observables (TDI) which have to be generated before any weak signal detection can be performed. These are linear combinations of the raw data with appropriate time shifts that lead to the cancellation of the laser frequency noises. This is possible because of the multiple occurrences of the same noises in the different raw data. Originally, these observables were manually generated starting with LISA as a simple stationary array and then adjusted to incorporate the antenna's motions. However, none of the observables survived the flexing of the arms in that they did not lead to cancellation with the same structure. The principal component approach is another way of handling these noises that was presented by Romano and Woan which simplified the data analysis by removing the need to create them before the analysis. This method also depends on the multiple occurrences of the same noises but, instead of using them for cancellation, it takes advantage of the correlations that they produce between the different readings. These correlations can be expressed in a noise (data) covariance matrix which occurs in the Bayesian likelihood function when the noises are assumed be Gaussian. Romano and Woan showed that performing an eigendecomposition of this matrix produced two distinct sets of eigenvalues that can be distinguished by the absence of laser frequency noise from one set. The transformation of the raw data using the corresponding eigenvectors also produced data that was free from the laser frequency noises. This result led to the idea that the principal components may actually be time delay interferometry observables since they produced the same outcome, that is, data that are free from laser frequency noise. The aims here were (i) to investigate the connection between the principal components and these observables, (ii) to prove that the data analysis using them is equivalent to that using the traditional observables and (ii) to determine how this method adapts to real LISA especially the flexing of the antenna. For testing the connection between the principal components and the TDI observables a 10x 10 covariance matrix containing integer values was used in order to obtain an algebraic solution for the eigendecomposition. The matrix was generated using fixed unequal arm lengths and stationary noises with equal variances for each noise type. Results confirm that all four Sagnac observables can be generated from the eigenvectors of the principal components. The observables obtained from this method however, are tied to the length of the data and are not general expressions like the traditional observables, for example, the Sagnac observables for two different time stamps were generated from different sets of eigenvectors. It was also possible to generate the frequency domain optimal AET observables from the principal components obtained from the power spectral density matrix. These results indicate that this method is another way of producing the observables therefore analysis using principal components should give the same results as that using the traditional observables. This was proven by fact that the same relative likelihoods (within 0.3%) were obtained from the Bayesian estimates of the signal amplitude of a simple sinusoidal gravitational wave using the principal components and the optimal AET observables. This method fails if the eigenvalues that are free from laser frequency noises are not generated. These are obtained from the covariance matrix and the properties of LISA that are required for its computation are the phase-locking, arm lengths and noise variances. Preliminary results of the effects of these properties on the principal components indicate that only the absence of phase-locking prevented their production. The flexing of the antenna results in time varying arm lengths which will appear in the covariance matrix and, from our toy model investigations, this did not prevent the occurrence of the principal components. The difficulty with flexing, and also non-stationary noises, is that the Toeplitz structure of the matrix will be destroyed which will affect any computation methods that take advantage of this structure. In terms of separating the two sets of data for the analysis, this was not necessary because the laser frequency noises are very large compared to the photodetector noises which resulted in a significant reduction in the data containing them after the matrix inversion. In the frequency domain the power spectral density matrices were block diagonals which simplified the computation of the eigenvalues by allowing them to be done separately for each block. The results in general showed a lack of principal components in the absence of phase-locking except for the zero bin. The major difference with the power spectral density matrix is that the time varying arm lengths and non-stationarity do not show up because of the summation in the Fourier transform.

Small strain elastic behaviour of unsaturated soil investigated by bender/extender element testing

The aim of this project was to investigate very small strain elastic behaviour of soils under unsaturated conditions, using bender/extender element (BEE) testing. The behaviour of soils at very small strains has been widely studied under saturated conditions, whereas much less work has been performed on very small strain behaviour under unsaturated conditions. A suction-controlled double wall triaxial apparatus for unsaturated soil testing was modified to incorporate three pairs of BEEs transmitting both shear and compression waves with vertical and horizontal directions of wave transmission and wave polarisation. Various different techniques for measuring wave travel time were investigated in both the time domain and the frequency domain and it was concluded that, at least for the current experimental testing programme, peak-to-first-peak in the time domain was the most reliable technique for determining wave travel time. An experimental test programme was performed on samples of compacted speswhite kaolin clay. Two different forms of compaction were employed (i.e. isotropic and anisotropic). Compacted kaolin soil samples were subjected to constant suction loading and unloading stages at three different values of suction, covering both unsaturated conditions (s= 50kPa and s= 300kPa) and saturated conditions (s=0). Loading and unloading stages were performed at three different values of stress ratio (η=0, η=1 and η=-1 ). In some tests a wetting-drying cycle was performed before or within the loading stage, with the wetting-drying cycles including both wetting-induced swelling and wetting-induced collapse compression. BEE tests were performed at regular intervals throughout all test stages, to measure shear wave velocity Vs and compression wave velocity Vp and hence to determine values of shear modulus G and constrained modulus M. The experimental test programme was designed to investigate how very small strain shear modulus G and constrained modulus M varied with unsaturated state variables, including how anisotropy of these parameters developed either with stress state (stress-induced anisotropy) or with previous straining (strain-induced anisotropy). A new expression has been proposed for the very small strain shear modulus G of an isotropic soil under saturated and unsaturated conditions. This expression relates the variation of G to only mean Bishop’s stress p* and specific volume v, and it converges to a well-established expression for saturated soils as degree of saturation approaches 1. The proposed expression for G is able to predict the variation of G under saturated and unsaturated conditions at least as well as existing expressions from the literature and it is considerably simpler (employing fewer state variables and fewer soil constants). In addition, unlike existing expressions from the literature, the values of soil constants in the proposed new expression can be determined from a saturated test. It appeared that, in the current project at least, any strain-induced anisotropy of very small strain elastic behaviour was relatively modest, with the possible exception of loading in triaxial extension. It was therefore difficult to draw any firm conclusion about evolution of strain-induced anisotropy and whether it depended upon the same aspects of soil fabric as evolution of anisotropy of large strain plastic behaviour. Stress-induced anisotropy of very small strain elastic behaviour was apparent in the experimental test programme. An attempt was made to extend the proposed expression for G to include the effect of stress-induced anisotropy. Interpretation of the experimental results indicated that the value of shear modulus was affected by the values of all three principal Bishop’s stresses (in the direction of wave transmission, the direction of wave polarisation and the third mutually perpendicular direction). However, prediction of stress-induced anisotropy was only partially successful, and it was concluded that the effect of Lode angle was also significant.