Systems and methods to determine elastic properties of materials

A method for determining a value of one or more elastic properties of a material, the method comprising:

• (a) exerting a driving force on the material by an actuator at a driving point;
• (b) producing input signals comprising a first input signal representative of the driving force exerted on the material, a second input signal representative of a driving-point velocity corresponding to the driving force, and a third input signal representative of a driving-point displacement corresponding to the driving force
• (c) generating a basis function signal for each basis function term of a differential equation model of the motion of the driving-point, wherein at least one basis function term comprises a function of the second input signal, and at least one basis function term comprises a function of the third input signal;
• and
• (d) determining coefficients of a linear combination of the basis function signals which fits the first input signal, wherein at least one of the determined coefficients is representative of the value of one or more elastic properties of the material.

A system for determining a value of one or more elastic properties of an in situ material at or near a surface of the earth, the system using input signals comprising a first input signal representative of a driving force exerted on the in situ material by an actuator at a driving-point;

• a second input signal representative of a driving-point velocity corresponding to the driving force;
• and a third input signal representative of a driving-point displacement corresponding to the driving force, the system comprising:
• (a) a generator for generating a basis function signal for each basis function term of a differential equation model of the motion of the driving-point, wherein the differential equation model comprises a polynomial function of the driving-point displacement and a polynomial function of the driving-point velocity, and wherein generating a basis function signal comprises generating a basis function value at each of a set of distinct times;
• and
• (b) a processor which uses a fitting process for determining coefficients of a linear combination of the basis function signals which fits the first input signal, wherein the basis function signals and the first input signal comprise an overdetermined system of linear equations, and wherein the fitting process comprises solving the overdetermined system of linear equations to determine a value of each coefficient, and wherein each determined coefficient corresponding to a displacement term is then representative of the value of a stiffness property of the in situ material, and each determined coefficient corresponding to a velocity term is then representative of the value of a viscosity property of the in situ material.