Spherical Harmonic

Key Takeaways

• The International Geomagnetic Reference Field (IGRF) represents the Earth's main geomagnetic field as a truncated spherical harmonic expansion with degree and order up to 13 (13 x 15 = 195 Gauss coefficients), updated every 5 years by the International Association of Geomagnetism and Aeronomy (IAGA); in oil and gas directional drilling, MWD magnetic tools measure the three components of the Earth's magnetic field (Bx, By, Bz in the tool coordinate frame) and use the IGRF model to compare the measured field with the model prediction to compute borehole azimuth, inclination, and tool face; the accuracy of the azimuth determination depends critically on the accuracy of the IGRF model at the survey location and time, and the IGRF model error (which is larger in regions with strong local crustal magnetic anomalies or near the geomagnetic poles) is the primary uncertainty in MWD magnetic survey azimuth; for wells in areas with significant local crustal magnetic anomalies (volcanic rocks, magnetized basement outcrops, or mineral deposits), the IGRF model must be supplemented by a local geomagnetic field model derived from ground or aeromagnetic survey data to achieve the azimuth accuracy required for precision directional drilling.
• Normal modes of the Earth (free oscillations) are the whole-Earth resonance frequencies that are excited by large earthquakes (moment magnitude above approximately 7.5 to 8.0) and that can be observed for hours to days after the event on long-period seismographs as the Earth "rings" at its natural frequencies; the normal modes are indexed as spheroidal modes (nSl, which involve radial and horizontal motion of the Earth's surface with a spherical harmonic degree l spatial pattern) and toroidal modes (nTl, which involve horizontal shear motion of the Earth's surface with a twisting pattern); the frequencies of the normal modes (which range from approximately 0.3 millihertz for the gravest modes with the simplest spatial pattern to tens of millihertz for higher overtones) are sensitive to the density, rigidity, and compressibility of the Earth's interior, making normal mode analysis one of the key constraints on Earth structure models used in global seismology; in the petroleum geoscience context, knowledge of Earth's normal modes is important for understanding the long-period background noise (the "Earth hum") observed on broadband seismometers that limits the sensitivity of ultra-long-period surface wave and ambient noise methods used in regional and global seismic tomography.
• Spherical harmonic analysis of the gravity field provides the fundamental model of the Earth's shape and density structure: the geoid (the equipotential surface of Earth's gravity field that defines mean sea level) is described by its spherical harmonic expansion, with the lower-degree harmonics (l = 2 to 10) representing large-scale density variations in the deep mantle and the higher-degree harmonics (l = 10 to several hundred) representing crustal density variations and topographic effects; modern high-degree gravity field models (EGM2008, EIGEN-6C, the GRACE and GRACE-FO satellite gravity models) include spherical harmonic coefficients up to degree and order 2190 (EGM2008), corresponding to a spatial resolution of approximately 10 kilometers; in oil and gas gravity surveys, the regional gravity field (obtained by removing the lower-degree spherical harmonic contribution, the "normal gravity field," from the observed gravity) reveals the residual Bouguer anomaly field caused by subsurface density contrasts in the exploration target zone, with the depth and density contrast of the anomalous mass controlling the amplitude and wavelength of the Bouguer anomaly; spherical harmonic gravity models also provide the reference for geodetic datum definitions (WGS84, GRS80) used in GPS positioning and well location reporting.
• Seismic tomography reconstructs the three-dimensional velocity structure of the Earth's interior by inverting large datasets of seismic travel time residuals (the difference between the observed traveltime and the prediction of a reference 1D Earth model) from earthquakes recorded by global seismograph networks; the velocity perturbation field (the three-dimensional variation of seismic velocity relative to the reference model) is parameterized as a linear combination of spherical harmonic basis functions in the horizontal direction and depth-dependent functions in the vertical direction, and the amplitudes of each basis function (the model parameters) are found by least-squares inversion of the travel time residuals; higher-degree spherical harmonics are needed to resolve finer-scale features of the mantle velocity structure (subducting slabs, plumes, heterogeneous lower mantle), but more model parameters require more data and more computation; global seismic tomography has imaged subducting oceanic slabs (which appear as high-velocity anomalies in the mantle), plumes of hot upwelling material (which appear as low-velocity anomalies), and the thermal and compositional structure of the mantle at wavelengths of 1,000 to 10,000 kilometers that are relevant for understanding the long-term tectonic framework of sedimentary basins.
• Practical applications of spherical harmonics in petroleum engineering beyond geomagnetics include the representation of acoustic emission (microseismic) radiation patterns from hydraulic fractures: the seismic moment tensor (which describes the forces acting at a microseismic source) can be decomposed into its spherical harmonic components (isotropic component representing volume change, double-couple component representing shear on a fault, and compensated linear vector dipole component representing tensile crack opening), enabling the type of fracture mechanism to be inferred from the observed radiation pattern; the spherical harmonic representation of the radiation pattern allows automated inversion of microseismic data for moment tensor components that distinguish shear fracturing (double-couple) from tensile hydraulic fracture opening (isotropic + CLVD), providing information on the hydraulic fracture geometry and stimulated rock volume that is valuable for completion design optimization and well spacing decisions in unconventional plays.