Direct calculation of electromagnetic apparent resistivity in whole time domain based on piecewise fitting of least squares polynomials

Real-time downhole time-domain electromagnetic imaging technology requires a fast calculation of apparent resistivity in downhole.As the relationship between induction voltage and apparent resistivity is not a monotonic function, the apparent resistivity cannot be calculated directly from induction voltage.The existing methods mainly consist of late apparent resistivity approximation method, kernel function step-by-step approxi-mation method and kernel function translation method.The late apparent resistance approximation method pro-duces some calculation errors in the early and middle stages.The kernel function approximation method needs to calculate the approximation resistivity step by step for many times, which takes a long time for calculation.The Kernel function translation method needs interpolation to calculate the time corresponding to the measured intercept of each time channel on the theoretical intercept curve, then the apparent resistivity is calculated according to the stretching characteristic of the induced voltage attenuation curve, it is more complex.In order to shorten the calculation time, according to the feature that the kernel function expression remains unchanged in the calculation of apparent resistivity, the authors propose a direct calculation method of electromagnetic apparent resistivity in the whole time domain based on least square polynomial fitting.The inverse function of the left and right branches of the kernel function is fitted by the higher order least square polynomial, and the direct calculation formula of apparent resistivity polynomial fitting is derived.In order to reduce the span of the fitting algorithm and reconcile the decimal problem caused by the multiple orders of magnitude, avoid too large a dynamic range to fit directly, a double logarithmic transformation is used to reduce the dynamic range, achieve the purpose of increasing fitting accuracy and speed.A homogeneous half-space and layered earth model is established, and the results of the kernel function dichotomy step-by-step approximation method and the logarithmic transformation least squares pol-ynomial fitting method are compared and analyzed by using the 12-order polynomial fitting method.Finally, the correctness of the method is verified by the field test of the known water-filled goaf.The results show that the method based on the least squares polynomial piecewise fitting can calculate the apparent resistivity correctly, which is basically consistent with the model and the actual situation.In comparison with the kernel function dichotomy step-by-step approximation method, the method proposed shortens the calculation time and increases the calculation speed for nearly five times by using logarithmic transformation.