Patent ID: 7130758

Claim:
A method for obtaining closed form expressions for subsurface temperature depth distribution along with its error bounds, the method comprising: defining at least two different types of boundary conditions involving at least three different heat sources; selecting computer inputs from said defined boundary conditions; using said inputs in a computerized stochastic heat conduction equation incorporating random thermal conductivity to obtain a mean and variance in temperature fields for said input, said equation being: ⅆ ⅆ z ⁢ { ( K _ + K ′ ⁡ ( z ) ) ⁢ ⅆ T ⅆ z } = - A ⁡ ( z ) ( 1 ) where T is the temperature (° C.), A(z) is the radiogenic heat source (μW/m 3 ), K(z)={overscore (K)}+K′(z) is the thermal conductivity (W/m° C.) which is expressed as a sum of a deterministic component {overscore (K)} and a random component K′(z) is the random component with mean zero and a Gaussian colored noise correlation structure represented by E ( K ′( z ))=0 (2) E ( K ′( z 1 ) K ′( z 2 ))=σ {overscore (K)} 2 =σ {overscore (K)} 2 e −ρ|Z −z | (3) where σ {overscore (K)} 2 is the variance in thermal conductivity (W/m° C.) ρ is the correlation decay parameter m −1 (or 1/ρ is the correlation length scale) and z 1 and z 2 are the depths (m); and outputting said mean and variance in temperature fields.