Rectangular Harmonic Analysis Research Articles

Rectangular harmonic analysis for gravity field representation by fusing ground gravity and airborne gravity gradient tensor data: A case study in Vinton salt dome, USA

Gravity and Gravity Gradient Tensor (GGT) surveys are commonly used to investigate mineral deposits, oil and gas reservoirs, etc. However, the different kinds of discrete observations with unknown noises are usually irregularly and unevenly spaced. Therefore, the practical observations need to be self-consistently and reliably fused and then the regularly distributed grid data of desired physical quantity can be predicted by the data fusion method. Rectangular Harmonic Analysis (RHA) is one of efficient and easy-to-implement methods for data processing, gravity field representation and so on. Combining ground gravity and airborne GGT data to determine RHA coefficients is beneficial for improving the reliability of complete wavelength components of the gravity field. In this study, we employ the modified RHA method to recover the gravity field by fusing ground gravity and airborne GGT data sets. Through a series of tests with synthetic data, we discuss the option of maximum RHA order, data sampling spaces and performances by using different simulated data sets assemble to represent the gravity field under different practical conditions. The synthetic tests show that the RHA can effectively extract the coherent signals as far as possible from the different data sets and guarantees better fitness between the predicted gravity and GGT data to observed data. Finally, we apply the RHA method to practical ground gravity and airborne GGT data sets over the Vinton salt dome, Louisiana, USA, to represent and predict the gravity and GGT data sets at different altitudes over the study region. As an example of geological application, the predicted GGT data sets are employed to interpret the lithological boundaries with density contrasts beneath the study area.

Geomagnetic reference map super-resolution using convolutional neural network

Geomagnetic matching navigation technology, as an important assisted navigation method, has gradually been wildly used in the navigation field, in which the quality of geomagnetic reference map is of great significance on improving the guidance and positioning accuracy. However, the existing geomagnetic reference map reconstruction methods can hardly satisfy the actual needs. Considering the popularity of deep learning methods in the image super-resolution (SR) field, a geomagnetic reference map SR method using a convolution neural network is proposed. First, rectangular harmonic analysis theory is used to construct the training dataset, which is based on data from the United States Geological Survey and the International Geomagnetic Reference Field. Then, a multi-channel convolutional neural network, which integrates all the three independent components in the geomagnetic reference field, is proposed of reference map SR reconstruction. Experimental results show the effectiveness of our proposed method over other state-of-the-art methods.

A Quick-Look Software for In Situ Magnetic Field Modeling from Onboard Unmanned Aircraft Vehicles (UAVs) Measurements

UAVs represent a tremendous opportunity to perform geophysical and repeated experiments, particularly in volcanic contexts. Their ability to be deployed rapidly and fly at various altitudes and the fact that they are easy to operate despite complex field conditions make them attractive for magnetic surveys. Detailed maps of the magnetic field in turn bring key constraints on the rocks’ composition, thermal anomalies, intrusive systems, and crustal contrast evolution. Yet, raw magnetic field measurements require careful processing to minimize directional, positional, and crossover errors. Moreover, stitching together adjacent or overlapping surveys acquired at different times and altitudes is not a trivial task. Therefore, it is challenging in remote areas to directly evaluate the consistency of a survey and to ascertain the success of the field mission. In this paper, we present a fast algorithm allowing for a quick-look modeling of scalar magnetic intensity measurements. The approach relies on rectangular harmonic analysis (RHA). The field measurements are automatically corrected for a global main field. Then, they are projected along this main field and modeled in terms of RHA functions. The software can exploit the quality indices provided with data and a procedure is applied to mitigate the effect of outliers. Maps for the scalar and the vector anomaly fields are readily built on an interpolated regular grid leveled at a constant altitude. In order to assess the modeling and the inversion procedures, analyses are carried out with synthetic measurements derived from a high-resolution global lithospheric magnetic field model estimated on the French aeromagnetic grid and at UAV locations with some added nonrandom noise. These analyses indicate that RHA is efficient for first-order and direct mapping of the crustal magnetic field structures measured by UAVs but that it could be applied on airborne and marine magnetic intensity data covering dense and large geographical extensions.

Comparison of spherical cap and rectangular harmonic analysis of airborne vector gravity data for high-resolution (1.5 km) local geopotential field models over Tanzania

SUMMARYThis study examines local geopotential field modelling over a mountainous region in Tanzania using vector airborne gravity data. We use the adjusted spherical cap and rectangular harmonic analyses. Both methods are based on expansion of gravitational potential into a series of orthogonal harmonic basis functions of local support in such a way that the expansion coefficients are determined by gravity observations. All three components of gravity vector are simultaneously inverted to derive the geopotential coefficients. In order to evaluate the accuracy of the local models, independent checkpoints are selected within the study region and around its boundary and the computed gravity vectors are compared with the independent gravity observations. The results show an excellent agreement with root mean square error (RMSE) of < 1.6 mGal over the study area. On the contrary, the RMSEs of global geopotential models against the checkpoints data are 7 mGal for the models up to the maximum degree of 2190 (a resolution of ∼9.1 km) and 5 mGal to 5399 (∼3.7 km). Our local models are significantly more accurate than the state-of-the-art global models and fully exploit the airborne vector data with the measurement error of ∼1 mGal. We also present the regional models constrained only by radial (vertical) or by lateral (horizontal) gravity observations. Those models are considerably less accurate than the one from 3-D gravity data inversion. Lastly, the regional models are validated against topography data. It is found that the gravity–topography correlation is 0.8–0.9 at 100 km, 0.5 at 20 km and higher than the correlations of the global models at all frequencies. The gravity–topography admittances estimated from our regional models indicate ∼130 mGal km−1 and imply the effective density of 2500 kg m−3 for topographical mass.

Super-Resolution Geomagnetic Reference Map Reconstruction Based on Dictionary Learning and Sparse Representation

Geomagnetic matching navigation plays a vital role in the navigation and guidance field, and the construction of geomagnetic reference maps plays a significant role in geomagnetic matching navigation. This paper addresses the problem of generating high-resolution geomagnetic reference maps with limited measured data. Most existing methods can hardly satisfy the accuracy requirements. We base our study on the theory of image super-resolution reconstruction and approach this problem from the perspective of dictionary learning and sparse representation. We propose a method of sparse dictionary initialization based on prior information from rectangular harmonic analysis, and then the K-SVD algorithm is applied to the dictionary training process to improve the performance of the sparse dictionary. Three components of the geomagnetic field are considered for multi-channel sparse representation to enhance the quality of the constructed maps. Experimental results show that our proposed method outperforms other geomagnetic reference map construction methods in the reconstructed precision as well as the robustness to noise.

Regional gravity field modeling based on rectangular harmonic analysis

Regional gravity field modeling with high-precision and high-resolution is one of the most important scientific objectives in geodesy, and can provide fundamental information for geophysics, geodynamics, seismology, and mineral exploration. Rectangular harmonic analysis (RHA) is proposed for regional gravity field modeling in this paper. By solving the Laplace’s equation of gravitational potential in local Cartesian coordinate system, the rectangular harmonic expansions of disturbing potential, gravity anomaly, gravity disturbance, geoid undulation and deflection of the vertical are derived, and so are the formula for signal degree variance and error degree variance of the rectangular harmonic coefficients (RHC). We also present the mathematical model and detailed algorithm for the solution of RHC using RHA from gravity observations. In order to reduce the edge effects caused by periodic continuation in RHA, we propose the strategy of extending the size of computation domain. The RHA-based modeling method is validated by conducting numerical experiments based on simulated ground and airborne gravity data that are generated from geopotential model EGM2008 and contaminated by Gauss white noise with standard deviation of 2 mGal. The accuracy of the 2.5′×2.5′ geoid undulations computed from ground and airborne gravity data is 1 and 1.4 cm, respectively. The standard error of the gravity disturbances that downward continued from the flight height of 4 km to the geoid is only 3.1 mGal. Numerical results confirm that RHA is able to provide a reliable and accurate regional gravity field model, which may be a new option for the representation of the fine structure of regional gravity field.

Gravity and geoid model in South Korea and its vicinity by spherical cap harmonic analysis

Long wavelength gravity anomalies derived from satellite data have limitations of spatial resolution and accuracy. To complement these weak points, combined datasets merged with shipborne/land gravity data and satellite data are generally compiled using global spherical harmonic, rectangular harmonic, or spherical cap harmonic analysis etc. However, each method also has a different resolution and accuracy. In this study, gravity anomalies and geoid undulations of South Korea and its vicinity are calculated using the spherical cap harmonic method, and the results are compared with those of GGM02C and EGM2008. Absolute and relative geoid height differences between GPS/leveling data and the results of various other methods show that the method used in this study has the lowest RMS error (0.271 m vs 0.344–0.416 m) and standard deviation (0.236 m vs 0.309–0.362 m). This implies that the spherical cap harmonic method has better resolution and accuracy than the other methods. The short wavelength gravity anomalies derived from the spherical cap harmonic coefficients are used to estimate the sediment thickness of the Ulleung Basin, which is poorly known. The sediment thickness of the Ulleung Basin is about 5 km at the center of the basin, and about 12 km at the southern margin.

Rectangular harmonic analysis revisited

Alldredge's method of rectangular harmonic analysis has been reexamined. After correction of errors, it is found to give improbable values between the data points and wild values outside them. A much more realistic model has been obtained by (1) determining only the most significant coefficients (those that exceed their standard deviations, obtained by an iterative process), (2) introducing new parameters to allow for a linear trend across the region, and (3) increasing the scaling factors so that the sinusoids start and finish outside the region. Modification 1 is the most important for improving the interpolative qualities of the model. Modifications 2 and 3 reduce, but do not entirely eliminate, the wild values near the edges.

Extraction of magnetic anomalies of crustal origin from Magsat data over the area of the Japanese Islands

Three component data along the Magsat paths in the area of 110°E–170°E and 8°N–68°N were fitted by polynomial series of the coordinates of the satellite position along the path. The coefficients decrease sharply up to the sixth degree and become almost constant. The fields represented by the sixth degree polynomials were subtracted from the data to remove the core and the external fields. For the altitude correction a rectangular harmonic analysis was applied to the residuals thus obtained by employing local Cartesian coordinates. The residual fields were expanded into double Fourier series of the maximum order of n = m = 3 over areas of 4000×4000 km and 2000×2000 km, and anomaly fields were computed for the satellite altitude. It was revealed that (1) an intense positive anomaly of the vertical component ran along the Kuril trench, (2) the Japan Sea was covered with a negative vertical anomaly in the northeastern part, and with a positive in the southwestern part, (3) there was a pair of positive and negative vertical anomalies covering the area from northern Korea to the East China Sea, suggesting the existence of an extremely localized magnetization like a dipole near the surface in the southwestern tip of the Korean Peninsula.

Delineation of long-wavelength magnetic anomalies over central India by rectangular harmonic analysis

The method of rectangular harmonic analysis is applied to the geomagnetic field data from central India to isolate long wavelength magnetic anomalies associated with largescale crustal structures. The long-wavelength anomalies have accounted for approximately 20 % of the spatial variability of the residual magnetic field over the International Geomagnetic Reference Field. On the magnetic anomaly map, reflecting the surface expression of longwavelength anomalies, the Tapi-Narmada-Son zone is characterized by a feeble positive anomaly bounded by a strong negative anomaly. The anomaly pattern is believed to be caused by the large-scale undulation in Moho and related variations in the thickness of the lower (basaltic) crust. The other two prominent anomalies, the magnetic low striking northwest and the magnetic high trending east-northeast, appear to be related to the deep structural feature of the Godavari graben and the eastern Rajasthan lineament respectively.

Rectangular harmonic analysis applied to the geomagnetic field

Spherical harmonic analysis of the earth's magnetic field is limited in the resolution that can be obtained. This limitation is caused by inadequacies of computers and of available data sets. The fundamental wavelength in spherical harmonic analysis is the circumference of the earth. To resolve wavelengths as short as 100 km would require a spherical harmonic analysis carried out to a degree and order 400 involving 160,800 coefficients. This is impractical even with modern computers. This limitation of spherical harmonic analysis can be overcome by using rectangular harmonic analysis in successively smaller areas so that the data are more fully utilized. Rectangular harmonic analysis is illustrated for data from Europe and then again for a subset of the data from a smaller area of Europe. The magnetic field at 15 observatories for the smaller area can be computed to an rms residual of only 7 nT for all three components using two sets of rectangular harmonic coefficients and the AWC/75 world chart model. Rectangular harmonic analysis and spherical harmonic analysis are complementary.