Dante Fratta

Discrete Signals and Inverse Problems: An Introduction for Engineers and Scientists

Preface. Brief Comments on Notation. 1. Introduction. 1.1 Signals, Systems, and Problems. 1.2 Signals and Signal Processing -- Application Examples. 1.3 Inverse Problems -- Application Examples. 1.4 History -- Discrete Mathematical Representation. 1.5 Summary. Solved Problems. Additional Problems. 2. Mathematical Concepts. 2.1 Complex Numbers and Exponential Functions. 2.2 Matrix Algebra. 2.3 Derivatives -- Constrained Optimization 2.4 Summary. Further Reading. Solved Problems. Additional Problems. 3. Signals and Systems. 3.1 Signals: Types and Characteristics. 3.2 Implications of Digitization -- Aliasing 3.3 Elemental Signals and Other Important Signals. 3.4 Signal Analysis with Elemental Signals. 3.5 Systems: Characteristics and Properties. 3.6 Combination of Systems 3.7 Summary. Further Reading. Solved Problems. Additional Problems. 4. Time Domain Analyses of Signals and Systems. 4.1 Signals and Noise. 4.2 Cross- and Autocorrelation: Identifying Similarities. 4.3 The Impulse Response -- System Identification. 4.4 Convolution: Computing the Output Signal. 4.5 Time Domain Operations in Matrix Form. 4.6 Summary. Further Reading. Solved Problems. Additional Problems. 5. Frequency Domain Analysis of Signals (Discrete Fourier Transform). 5.1 Orthogonal Functions -- Fourier Series. 5.2 Discrete Fourier Analysis and Synthesis. 5.3 Characteristics of the Discrete Fourier Transform. 5.4 Computation in Matrix Form. 5.5 Truncation, Leakage, and Windows. 5.6 Padding. 5.7 Plots. 5.8 The Two-dimensional Discrete Fourier Transform. 5.9 Procedure for Signal Recording. 5.10 Summary. Further Reading and References. Solved Problems. Additional Problems. 6. Frequency Domain Analysis of Systems. 6.1 Sinusoids and Systems - Eigenfunctions. 6.2 Frequency Response. 6.3 Convolution. 6.4 Cross-spectral and Autospectral Densities. 6.5 Filters in the Frequency Domain -- Noise Control. 6.6 Determining H with Noiseless Signals (Phase Unwrapping). 6.7 Determining H with Noisy Signals (Coherence). 6.8 Summary. Further Reading and References. Solved Problems. Additional Problems. 7. Time Variation and Nonlinearity. 7.1 Nonstationary Signals: Implications. 7.2 Nonstationary Signals: Instantaneous Parameters. 7.3 Nonstationary Signals: Time Windows. 7.4 Nonstationary Signals: Frequency Windows. 7.5 Nonstationary Signals: Wavelet Analysis. 7.6 Nonlinear Systems: Detecting Nonlinearity. 7.7 Nonlinear Systems: Response to Different Excitations. 7.8 Time-varying Systems. 7.9 Summary. Further Reading and References Solved Problems. Additional Problems. 8. Concepts in Discrete Inverse Problems. 8.1 Inverse Problems -- Discrete Formulation. 8.3 Data-driven Solution -- Error Norms. 8.4 Model Selection -- Ockhams Razor. 8.5 Information. 8.6 Data and Model Errors. 8.7 Nonconvex Error Surfaces. 8.8 Discussion on Inverse Problems. 8.9 Summary. Further Reading and References. Solved Problems. Additional Problems. 9. Solution by Matrix Inversion. 9.1 Pseudoinverse. 9.2 Classification of Inverse Problems. 9.3 Least Squares Solution (LSS). 9.4 Regularized Least Squares Solution (RLSS). 9.5 Incorporating Additional Information. 9.6 Solution Based on Singular Value Decomposition. 9.7 Nonlinearity. 9.8 Statistical Concepts -- Error Propagation. 9.9 Experimental Design for Inverse Problems. 9.10 Methodology for the Solution of Inverse Problems. 9.11 Summary. Further Reading. Solved Problems. Additional Problems. 10. Other Inversion Methods. 10.1 Transformed Problem Representation. 10.2 Iterative Solution of System of Equations. 10.3 Solution by Successive Forward Simulations. 10.4 Techniques from the Field of Artificial Intelligence. 10.5 Summary. Further Reading. Solved Problems. Additional Problems. 11. Strategy for Inverse Problem Solving. 11.1 Step 1: Analyze the Problem. 11.2 Step 2: Pay Close Attention to Experimental Design. 11.3 Step 3: Gather High-quality Data. 11.4 Step 4: Pre-process the Data. 11.5 Step 5: Select an Adequate Physical Model. 11.6 Step 6: Explore Different Inversion Methods. 11.7 Step 7: Analyze the Final Solution. 11.8 Summary. Solved Problems. Additional Problems. Index.

Combined TDR and P-Wave Velocity Measurements for the Determination of In Situ Soil Density--Experimental Study

This paper summarizes a new nondestructive approach for the evaluation of soil density and water content. This new measurement methodology involves evaluating the dielectric permittivity and the P-wave velocity in soils as the water content is increased. These values are then related to the volumetric water content, porosity, and skeleton shear stiffness, which are needed to back-calculate the density and water content of the tested soil specimens. Experimental laboratory results are briefly summarized. These test results show a potential for developing a new device. Electronic equipment and sensors for the proposed device include a TDR system, miniature piezoelectric accelerometers, signal conditioner system, and oscilloscope for data acquisition.

Mechanistic Corrections for Determining the Resilient Modulus of Base Course Materials Based on Elastic Wave Measurements

The mechanical performance of pavement systems depends on the stiffness of subsurface soil and aggregate materials. The moduli of base course, subbase, and subgrade soils included in pavement systems need to be characterized for their use in the new empirical-mechanistic design procedure (NCHRP 1-37A). Typically, the resilient modulus test is used in the design of base and subbase layers under repetitive loads. Unfortunately, resilient modulus tests are expensive and cannot be applied to materials that contain particles larger than 25 mm (for 125-mm diameter specimens) without scalping the large grains. This paper examines a new methodology for estimating resilient modulus based on the propagation of elastic waves. The method is based on using a mechanistic approach that relates the P-wave velocity-based modulus to the resilient modulus through corrections for stress, void ratio, strain, and Poissons ratio effects. Results of this study indicate that resilient moduli are approximately 30% of Youngs moduli based on seismic measurements. The technique is then applied to specimens with large-grain particles. Results show that the methodology can be applied to large-grained materials and their resilient modulus can be estimated with reasonable accuracy based on seismic techniques. An approach is proposed to apply the technique to field determinations of modulus.

Effect of Fine Particle Migration on the Small-Strain Stiffness of Unsaturated Soils

The paper presents the results of an experimental investigation of fine particle migration from pore body to the pore throat and toward the contact between particles and its effect on skeleton stiffness of granular materials. We hypothesize that the suspended colloids in the pore fluid migrate and deposit on the contact surface between the skeleton-forming particles and change the magnitude of the soil stiffness. Three specimens were prepared using uniform spherical glass particles that were saturated with deionized water and kaolinite or silt-base slurries. The specimens were drained by evaporation which retained the fines in the soil while increasing the matric suction. Changes in soil dynamic stiffness were evaluated using piezoelectric transducers while the migration of fines and the changes of the properties of the pore fluid were monitored using synchrotron X-ray microcomputed tomography (SMCT) on identical specimens. The wave propagation experiments show that the stiffness of the tested specimens i...

A Suction-Control Apparatus for the Measurement of P and S-wave Velocity in Soils

This paper presents a new apparatus for the measurement of P and S-wave velocities in unsaturated soil specimens under controlled net stress and matric suction conditions. The system consists of a triaxial cell modified to enable an independent control of the pore air and water pressures as well as the confining pressure, and to accommodate P and S-wave sources and receivers. The system also includes three servo-controlled pressure pumps, and a computer control system that drives the pressures and acquires volume changes. Three sets of experiments were conducted to verify the system performance. Specific matric suction values were applied while monitoring the stabilization process of capillary pressures using the P and S-wave velocity measurements. Both P and S-wave velocities increase due to the rise in interparticle forces resulting from the increase in matric suction.

Freeze-Thaw Effects on Stiffness of Unbound Recycled Road Base

A major concern for using recycled pavement material as an unbound base or subbase layer is the effect of changing seasons on the properties of the recycled material. Three sources of recycled concrete aggregate (RCA) and recycled asphalt pavement (RAP), and one conventional base aggregate, were used to investigate the effect of freeze-thaw cycles on the stiffness of unbound road base/subbase layers. Effects of freeze-thaw cycling on the mechanical behavior of three gradations (coarse, medium, fine) of recycled materials were systematically evaluated to determine how climatic factors and aging affect the resilient modulus. Sealed specimens were exposed to 5, 10, and 20 sets of freeze-thaw cycles. Resilient modulus tests were conducted according to NCHRP 1-28A after the final freeze-thaw cycle. Freeze-thaw cycling caused a decrease in the stiffness (i.e., the summary resilient modulus) of RAP samples and Class 5 aggregate due to the effect of the water retained in the pores. An increase in the stiffness of RCA was observed over 20 freeze-thaw cycles and is attributed to self-cementitious behavior of crushed concrete. Seismic modulus testing was used to investigate the continuous rate of change (daily) of the stiffness for RCA and Class 5 aggregate. The seismic modulus test confirmed the trends observed in resilient modulus testing and served as a non-destructive method for tracking changes in stiffness over time and freeze-thaw cycling.

An integrated acoustic and electromagnetic wave-based technique to estimate subbottom sediment properties in a freshwater environment

An integrated acoustic and electromagnetic (EM) wave-based technique was developed to estimate sediment porosity and top-layer thickness in shallow freshwater environments. The integrated methodology reduces the limitation of each of the individual techniques and combines the data for a more robust inversion solution. The acoustic and EM-wave reflection coefficients are determined based on the ratios of reflected signal strengths from sediments and a reference aluminium plate. An iterative algorithm that uses reflection coefficients to optimize the sediment porosity was developed. Once the optimal sediment porosity is obtained, the acoustic and EM wave speeds and then the top-layer thickness were evaluated. In comparison with ground truth data, the measured and estimated sediment porosity and top-layer thickness show differences less than 8.6%. The new integrated method provides an efficient and accurate methodology to obtain sediment properties under different sediment conditions.

Dimensionless Limits for the Collection and Interpretation of Wave Propagation Data in Soils

The use of bender elements to generate and receive shear waves in soils has become a very popular technique in geotechnical engineering studies. However as with any other wave propagation technique, the interpretation of bender element-collected data is controlled by wave characteristics, boundary conditions, and properties of the medium. This paper presents experimental data and simple closed-form solutions in order to investigate and to evaluate the effects due to the near field and boundary conditions in different types of specimen geometries and boundary conditions. Results yield dimensionless limits that must be taken into account to properly monitor soil parameters and to avoid misleading results in the interpretation of wave propagation data from the bender elements.

The Use of Low-Cost MEMS Accelerometers for the Near-Surface Monitoring of Geotechnical Engineering Systems

The urban underground environment is under continuous development as the infrastructure ages and new technologies are added to the existing networks. Design, construction, installation, and repair in this ever-changing environment can benefit significantly from high-resolution imaging to acquire adequate subsurface characterization. This paper documents the development of MEMS accelerometers as a low-cost alternative sensor to geophones and piezocrystal accelerometers for monitoring wave propagation in soils. The experimental study includes an evaluation of limitations and the development of a simple packaging system for the broad application of MEMS in the geoenvironment. The calibrated accelerometers are used to monitor elastic wave propagation for the evaluation of changes in the effective state of stress within a 1g model of a braced excavation. The experimental program captures the effects of load and deformation as elastic waves propagate behind a sheet-pile system. Results hint at the development of long-term non-destructive monitoring systems for the diagnostics and control of geotechnical processes.

Evaluation of the Zone of Influence and Stiffness Improvement from Geogrid Reinforcement in Granular Materials

The longevity of a pavement system is closely related to the amount of deformation of both the asphalt surface and the underlying layers. Placing a geogrid in the granular base decreases the amount of rutting at the surface by providing some strength to the base and increasing stiffness and thus limiting elastic deformations and loads transmitted to the sub-grade. However, the zone of influence of the geogrid layer on surrounding soil particles and the increase in modulus in this zone are not well known. A testing scheme aimed at quantifying both the zone of influence and the increase in the modulus caused by the presence of a geogrid in granular materials was developed. Both P-wave velocity and the shear strain induced by loading a 150-mm diameter plate were examined. P-wave velocity results indicated a change in modulus across the geogrid from a minimum of 1.35 (at 75-mm geogrid depth) to a maximum of 2.66 (at 100-mm geogrid depth) over modulus expected in unreinforced soils. Expected internal soil rotations were modeled with PLAXIS and compared with laboratory tests. These analyses showed that the shearing of soil was confined to a zone above the geogrid. Rotation tests showed a zone of influence of 30 to 40 mm on both sides of the geogrid reinforcement; however, the zone of influence depends on the position of the geogrid, with a geogrid 100 mm in depth seeming most able to constrain subsurface soils and distribute the shear stresses caused by the 150-mm-diameter loading plate.