Spherical Harmonic Approach Research Articles

A new spherical harmonic approach to residual terrain modeling: a case study in the central European Alps

For decades, the residual terrain model (RTM) concept (Forsberg and Tscherning in J Geophys Res Solid Earth 86(B9):7843–7854, https://doi.org/10.1029/JB086iB09p07843, 1981) has been widely used in regional quasigeoid modeling. In the commonly used remove-compute-restore (RCR) framework, RTM provides a topographic reduction commensurate with the spectral resolution of global geopotential models. This is usually achieved by utilizing a long-wavelength (smooth) topography model known as reference topography. For computation points in valleys this neccessitates a harmonic correction (HC) which has been treated in several publications, but mainly with focus on gravity. The HC for the height anomaly only recently attracted more attention, and so far its relevance has yet to be shown also empirically in a regional case study. In this paper, the residual spherical-harmonic topographic potential (RSHTP) approach is introduced as a new technique and compared with the classic RTM. Both techniques are applied to a test region in the central European Alps including validation of the quasigeoid solutions against ground-truthing data. Hence, the practical feasibility and benefits for quasigeoid computations with the RCR technique are demonstrated. Most notably, the RSHTP avoids explicit HC in the first place, and spectral consistency of the residual topographic potential with global geopotential models is inherently achieved. Although one could conclude that thereby the problem of the HC is finally solved, there remain practical reasons for the classic RTM reduction with HC. In this regard, both intra-method comparison and ground-truthing with GNSS/leveling data confirms that the classic RTM (Forsberg and Tscherning 1981; Forsberg in A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modeling. Report 355, Department of Geodetic Sciences and Surveying, Ohio State University, Columbus, Ohio, USA, https://earthsciences.osu.edu/sites/earthsciences.osu.edu/files/report-355.pdf, 1984) provides reasonable results also for a high-resolution (degree 2160) RTM, yet neglecting the HC for the height anomaly leads to a systematic bias in deep valleys of up to 10–20 cm.

Fourier-Hankel-Abel Nyquist-limited tomography: A spherical harmonic basis function approach to tomographic velocity-map image reconstruction.

A new method for the fully generalized reconstruction of three-dimensional (3D) photoproduct distributions from velocity-map imaging (VMI) projection data is presented. This approach, dubbed Fourier-Hankel-Abel Nyquist-limited TOMography (FHANTOM), builds on recent previous work in tomographic image reconstruction [C. Sparling and D. Townsend, J. Chem. Phys. 157, 114201 (2022)] and takes advantage of the fact that the distributions produced in typical VMI experiments can be simply described as a sum over a small number of spherical harmonic functions. Knowing the solution is constrained in this way dramatically simplifies the reconstruction process and leads to a considerable reduction in the number of projections required for robust tomographic analysis. Our new method significantly extends basis set expansion approaches previously developed for the reconstruction of photoproduct distributions possessing an axis of cylindrical symmetry. FHANTOM, however, can be applied generally to any distribution-cylindrically symmetric or otherwise-that can be suitably described by an expansion in spherical harmonics. Using both simulated and real experimental data, this new approach is tested and benchmarked against other tomographic reconstruction strategies. In particular, the reconstruction of photoelectron angular distributions recorded in a strong-field ionization regime-marked by their extensive expansion in terms of spherical harmonics-serves as a key test of the FHANTOM methodology. With the increasing use of exotic optical polarization geometries in photoionization experiments, it is anticipated that FHANTOM and related reconstruction techniques will provide an easily accessible and relatively low-cost alternative to more advanced 3D-VMI spectrometers.

Statistical anisotropy in galaxy ellipticity correlations

As well as the galaxy number density and peculiar velocity, the galaxy intrinsic alignment can be used to test the cosmic isotropy. We study distinctive impacts of the isotropy breaking on the configuration-space two-point correlation functions (2PCFs) composed of the spin-2 galaxy ellipticity field. For this purpose, we build a formalism for general types of the isotropy-violating 2PCFs and a methodology to efficiently compute them by generalizing the polypolar spherical harmonic decomposition approach to the spin-weighted version. As a demonstration, we analyze the 2PCFs when the matter power spectrum has a well-known g *-type isotropy-breaking term (induced by, e.g., dark vector fields). We then confirm that some anisotropic distortions indeed appear in the 2PCFs and their shapes rely on a preferred direction causing the isotropy violation, d̂. Such a feature can be a distinctive indicator for testing the cosmic isotropy. Comparing the isotropy-violating 2PCFs computed with and without the plane parallel (PP) approximation, we find that, depending on d̂, the PP approximation is no longer valid when an opening angle between the directions towards target galaxies is 𝒪(1°) for the density-ellipticity and velocity-ellipticity cross correlations and around 10° for the ellipticity auto correlation. This suggests that an accurate test for the cosmic isotropy requires the formulation of the 2PCF without relying on the PP approximation.

Experiences with the RTM Method in Local Quasi-Geoid Modeling

In local quasi-geoid modeling, the residual terrain modeling (RTM) method is often used to remove short-wavelength gravity field signals from the measured gravity on the ground in order to obtain a regularized and smooth gravity field that is suited for field interpolation and modeling. Accurate computation of RTM corrections plays a crucial role in computing an accurate local quasi-geoid, and it requires a set of fine-tuned parameters, including the combination of DEMs with different resolutions for suitably representing the real topography, the choice of integration radius for properly defining the extent of the computation zone, and the determination of reference topography to properly describe the RTM-reduced Earth’s surface. To our knowledge, this has not been systematically documented, despite its obvious importance. This study aims to systematically investigate the impact of these factors on RTM correction computation and, consequently, on local quasi-geoid modeling to provide practical guidelines for real-world applications. The tesseroid-based gravity forward modeling technique is employed to investigate the following issues existing in the practical use of the RTM method: ① Can the combination of a high-resolution DEM and a DEM with a lower resolution replace the single use of the high-resolution DEM for RTM correction computation while maintaining accuracy and improving efficiency? If it does, how do I properly choose the resolution of this coarse DEM as well as the integration radius r1 for the inner zone and r2 for the outer zone? ② How large would the differences between the RTM corrections computed by three types of reference topographies, which are obtained from the direct averaging (DA) approach, the moving averaging (MA) approach, and the spherical harmonic (SH) approach, be, and how large would their impact on quasi-geoid modeling be? To obtain objective findings, two research regions were selected for this investigation. One is the Colorado test area (USA) with rugged terrain, and the other is the Auvergne test area (France) with moderate terrain. The main numerical findings are: (1) the combination of the 3” resolution DEM (inner zone) and the 30″ resolution DEM (outer zone) is sufficient for accurate and efficient RTM correction computation; (2) if the resolution of the reference topography is 5′ or slightly lower, all three types of reference topographies are able to obtain local quasi-geoid models at a similar accuracy level, while the values of r1 and r2 are preferred to be at least 20 km and 111 km, respectively; (3) if the reference topography has a resolution of 30′ or lower, the MA or SH reference topography is recommended, especially for the latter one, and the values of r1 and r2 are suggested to be at least 20 km and 222 km, respectively. The above numerical findings can be taken as a reference for local quasi-geoid determination in areas with different topographic regimes than the two selected test areas.

Robust Object Classification Approach Using Spherical Harmonics

Point clouds produced by either 3D scanners or multi-view images are often imperfect and contain noise or outliers. This paper presents an end-to-end robust spherical harmonics approach to classifying 3D objects. The proposed framework first uses the voxel grid of concentric spheres to learn features over the unit ball. We then limit the spherical harmonics order level to suppress the effect of noise and outliers. In addition, the entire classification operation is performed in the Fourier domain. As a result, our proposed model learned features that are less sensitive to data perturbations and corruptions. We tested our proposed model against several types of data perturbations and corruptions, such as noise and outliers. Our results show that the proposed model has fewer parameters, competes with state-of-art networks in terms of robustness to data inaccuracies, and is faster than other robust methods. Our implementation code is also publicly available at <uri>https://github.com/AymanMukh/R-SCNN</uri>

Strain and Grain Size Determination of CeO2 and TiO2 Nanoparticles: Comparing Integral Breadth Methods versus Rietveld, μ-Raman, and TEM

Various crystallite size estimation methods were used to analyze X-ray diffractograms of spherical cerium dioxide and titanium dioxide anatase nanoparticles aiming to evaluate their reliability and limitations. The microstructural parameters were estimated from several integral breadth methods such as Scherrer, Monshi, Williamson–Hall, and their variants: (i) uniform deformation model, (ii) uniform strain deformation model, and (iii) uniform deformation energy density model. We also employed the size–strain plot and Halder–Wagner method. For this purpose, an instrumental resolution function of an Al2O3 standard was used to subtract the instrumental broadening to estimate the crystallite sizes and strain, and the linear regression analysis was used to compare all the models based on the coefficient of determination. The Rietveld whole powder pattern decomposition method was introduced for comparison purposes, being the best candidate to fit the X-ray diffraction data of metal-oxide nanoparticles. Refined microstructural parameters were obtained using the anisotropic spherical harmonic size approach and correlated with the above estimation methods and transmission electron microscopy images. In addition, μ-Raman spectra were recorded for each material, estimating the mean crystallite size for comparison by means of a phonon confinement model.

Mapping the gravitational-wave sky with LISA: a Bayesian spherical harmonic approach

ABSTRACT The millihertz gravitational-wave frequency band is expected to contain a rich symphony of signals with sources ranging from Galactic white dwarf binaries to extreme mass ratio inspirals. Many of these gravitational-wave signals will not be individually resolvable. Instead, they will incoherently add to produce stochastic gravitational-wave confusion noise whose frequency content will be governed by the dynamics of the sources. The angular structure of the power of the confusion noise will be modulated by the distribution of the sources across the sky. Measurement of this structure can yield important information about the distribution of sources on Galactic and extragalactic scales, their astrophysics and their evolution over cosmic time-scales. Moreover, since the confusion noise is part of the noise budget of Laser Interferometer Space Antenna (LISA), mapping it will also be essential for studying resolvable signals. In this paper, we present a Bayesian algorithm to probe the angular distribution of the stochastic gravitational-wave confusion noise with LISA using a spherical harmonic basis. We develop a technique based on Clebsch–Gordan coefficients to mathematically constrain the spherical harmonics to yield a non-negative distribution, making them optimal for expanding the gravitational-wave power and amenable to Bayesian inference. We demonstrate these techniques using a series of simulations and analyses, including recovery of simulated distributed and localized sources of gravitational-wave power. We also apply this method to map the gravitational-wave foreground from Galactic white dwarfs using a simplified model of the Galactic white dwarf distribution.

On some features of the eigenvalue problem for the [formula omitted] approximation of the neutron transport equation

In this work some characteristics of the eigenvalue problem for the neutron transport equations are considered. Various formulations are examined, discussing some theoretical and practical aspects. The standard multiplication eigenvalue that is particularly relevant for nuclear reactor physics applications is analysed, together with the time eigenvalue, including also the contribution of delayed neutrons. In addition, the less common collision and density eigenvalues are also discussed, highlighting interesting physical features. A semianalytical approach is developed allowing to evidence some interesting structures of the eigenvalue spectra.The study is carried out within the spherical harmonics approach. For the plane one dimensional geometry, the mathematical relationship between even and odd-order approximations for the homogeneous form of the equations for the eigenvalue formulation is investigated. It is shown that the even-order system of equations can be re-cast in the form of the contiguous lower odd-order one. Numerical results are obtained in the two-group energy model for various configurations for which a reference is available, providing also results for high-order approximations. The study includes a presentation and discussion of the spectra patterns for the various eigenvalue formulations.

Rietveld Refinement, μ-Raman, X-ray Photoelectron, and Mössbauer Studies of Metal Oxide-Nanoparticles Growth on Multiwall Carbon Nanotubes and Graphene Oxide

Applying a modified coprecipitation method, maghemite and anatase nanoparticles embedded in graphene oxide and multiwall carbon nanotube frameworks were prepared, and a detailed structural characterization is presented. Transmission electron images have revealed that the multiwall carbon nanotubes and graphene oxide act as substrates to reduce the nanoparticle agglomeration with narrow sizes of ca. 9–20 nm, in agreement with the results of the Rietveld refinement, which have also indicated their crystallite apparent size and shapes using the spherical harmonics approach. In structural studies of maghemite nanoparticles by Raman spectroscopy, it was found that the effect of optical density and laser power intensity plays a significant role. When no optical filter was located between the powder sample and the laser source, a transformation from the γ-Fe2O3 to the α-Fe2O3 phase was observed, as demonstrated by the disappearance of the characteristic broad Raman peak (A1g) of the γ-Fe2O3 phase when increasing the laser power. X-ray photoelectron spectroscopy has also brought insights into the functionalization mechanism, suggesting that the one-pot reduction of the graphene oxide is favored by the alkaline γ-Fe2O3 nanoparticle growth. The temperature dependence of the 57Fe Mössbauer spectra has indicated that the effective anisotropy constant of Fe oxide-based nanoparticles is similar to that of bulk maghemite, and magnetic relaxation of Fe3+ spins depends on particle sizes.

Eigenvalue Formulations for the PN Approximation to the Neutron Transport Equation

The study of the eigenvalues of the neutron transport operator yields an important insight into the physical features of the neutronic phenomena taking place in a nuclear reactor. Although the multiplication eigenvalue is the most popular because of its implication in the engineering design of multiplying structures, alternative interesting formulations are possible. In this paper the interest is focused on the multiplication, collision and time eigenvalues. The transport model is considered in the spherical harmonics approximation and the study is restricted to the one-dimensional plane geometry in the monokinetic case. The spectra of the different eigenvalues are investigated using a numerical code, validating its performance against the results available in the literature. The observation of the convergence trends allows to establish the performance of even- and odd-order approximations. It is shown that in general even-order approximations yield slightly less accurate results, nevertheless they appear to converge to the reference values. The effect of the choice of the boundary conditions according to the methodologies proposed by either Mark or Marshak is also investigated. The analysis of all the results presented allows to characterize the convergence properties of the spherical harmonics approach to neutron transport. The spectrum of the time eigenvalues retains a very rich physical meaning, as they are the actual time constants of the time-dependent solution of the transport problem. Therefore, in the last part of the paper the behavior of the pattern of the spectrum of the time eigenvalues when changing the scattering ratio and the order of the approximation is examined.

Modeling Australian TEC Maps Using Long-Term Observations of Australian Regional GPS Network by Artificial Neural Network-Aided Spherical Cap Harmonic Analysis Approach

The global ionosphere map (GIM) is not capable of serving precise positioning and navigation for single frequency receivers in Australia due to sparse International GNSS Service (IGS) stations located in the vast land. This study proposes an approach to represent Australian total electron content (TEC) using the spherical cap harmonic analysis (SCHA) and artificial neural network (ANN). The new Australian TEC maps are released with an interval of 15 min for longitude and latitude in 0.5° × 0.5°. The validation results show that the Australian Ionospheric Maps (AIMs) well represent the hourly and seasonally ionospheric electrodynamic features over the Australian continent; the accuracy of the AIMs improves remarkably compared to the GIM and the model built only by the SCHA. The residual of the AIM is inversely proportional to the level of solar radiation. During the equinoxes and solstices in a solar minimum year, the residuals are 2.16, 1.57, 1.68, and 1.98 total electron content units (TECUs, 1 TECU = 1016 electron/m2), respectively. Furthermore, the AIM has a strong capability in capturing the adequate electrodynamic evolutions of the traveling ionospheric disturbances under severe geomagnetic storms. The results demonstrate that the ANN-aided SCHA method is an effective approach for mapping and investigating the TEC maps over Australia.

Temporal Gravity Signals in Reprocessed GOCE Gravitational Gradients

The reprocessing of the satellite gravitational gradiometry (SGG) data from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite mission in 2018/2019 considerably reduced the low-frequency noise in the data, leading to reduced noise amplitudes in derived gravity field models at large spatial scales, at which temporal variations of the Earth’s gravity field have their highest amplitudes. This is the motivation to test the reprocessed GOCE SGG data for their ability to resolve time-variable gravity signals. For the gravity field processing, we apply and compare a spherical harmonics (SH) approach and a mass concentration (mascon) approach. Although their global signal-to-noise ratio is &lt;1, SH GOCE SGG-only models resolve the strong regional signals of glacier melting in Greenland and Antarctica, and the 2011 moment magnitude 9.0 earthquake in Japan, providing an estimation of gravity variations independent of Gravity Recovery and Climate Experiment (GRACE) data. The benefit of combined GRACE/GOCE SGG models is evaluated based on the ice mass trend signals in Greenland and Antarctica. While no signal contribution from GOCE SGG data additional to the GRACE models could be observed, we show that the incorporation of GOCE SGG data numerically stabilizes the related normal equation systems.

Effects of bathymetry complexity on tsunami propagation: a spherical harmonics approach

SUMMARYThis paper explores the effects on tsunami simulations of the level of detail of the bathymetric grid in use. For this purpose, we expand available bathymetric data sets of the Pacific Basin in spherical harmonics. For realistic scenarios of tsunamis generated by earthquake dislocations, we conclude that an expansion to a maximum degree lmax = 40, corresponding to wavelengths of 1000 km, is sufficient to reproduce the main features of the tsunami wavefield synthesized in deep water, that is, without considering final shoaling and interaction with coastal features.

Dynamic spherical harmonics approach for shape classification of migrating cells

Cell migration involves dynamic changes in cell shape. Intricate patterns of cell shape can be analyzed and classified using advanced shape descriptors, including spherical harmonics (SPHARM). Though SPHARM have been used to analyze and classify migrating cells, such classification did not exploit SPHARM spectra in their dynamics. Here, we examine whether additional information from dynamic SPHARM improves classification of cell migration patterns. We combine the static and dynamic SPHARM approach with a support-vector-machine classifier and compare their classification accuracies. We demonstrate that the dynamic SPHARM analysis classifies cell migration patterns more accurately than the static one for both synthetic and experimental data. Furthermore, by comparing the computed accuracies with that of a naive classifier, we can identify the experimental conditions and model parameters that significantly affect cell shape. This capability should – in the future – help to pinpoint factors that play an essential role in cell migration.

X-ray diffraction assessment of complex sample states on the basis of “combined analysis”

The multi-reflection grazing-incidence X-ray diffraction technique and the analysis of the thereby recorded datasets are extended in this paper on hexagonal lattice types with distinct elastic anisotropy. The evaluation of the complete stress tensor is based on X-ray diffraction pattern decomposition and considers for texture by the spherical harmonics approach. This procedure, following the directives of “combined analysis”, enables the extraction of the mean macroscopic strain/stress tensor weighted by texture assuming a quasi-isotropic state with respect to sample elasticity. The procedure has been applied to study plastic deformation and the resulting evolution of triaxial stress states in zinc green-compacts after uniaxial compression. Due to the application of asymmetric diffraction and associated with the chosen incidence angle the method focuses on the near-surface region.

GOCE-Derived Coseismic Gravity Gradient Changes Caused by the 2011 Tohoku-Oki Earthquake

In contrast to most of the coseismic gravity change studies, which are generally based on data from the Gravity field Recovery and Climate Experiment (GRACE) satellite mission, we use observations from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) Satellite Gravity Gradient (SGG) mission to estimate the coseismic gravity and gravity gradient changes caused by the 2011 Tohoku-Oki Mw 9.0 earthquake. We first construct two global gravity field models up to degree and order 220, before and after the earthquake, based on the least-squares method, with a bandpass Auto Regression Moving Average (ARMA) filter applied to the SGG data along the orbit. In addition, to reduce the influences of colored noise in the SGG data and the polar gap problem on the recovered model, we propose a tailored spherical harmonic (TSH) approach, which only uses the spherical harmonic (SH) coefficients with the degree range 30–95 to compute the coseismic gravity changes in the spatial domain. Then, both the results from the GOCE observations and the GRACE temporal gravity field models (with the same TSH degrees and orders) are simultaneously compared with the forward-modeled signals that are estimated based on the fault slip model of the earthquake event. Although there are considerable misfits between GOCE-derived and modeled gravity gradient changes (ΔVxx, ΔVyy, ΔVzz, and ΔVxz), we find analogous spatial patterns and a significant change (greater than 3σ) in gravity gradients before and after the earthquake. Moreover, we estimate the radial gravity gradient changes from the GOCE-derived monthly time-variable gravity field models before and after the earthquake, whose amplitudes are at a level over three times that of their corresponding uncertainties, and are thus significant. Additionally, the results show that the recovered coseismic gravity signals in the west-to-east direction from GOCE are closer to the modeled signals than those from GRACE in the TSH degree range 30–95. This indicates that the GOCE-derived gravity models might be used as additional observations to infer/explain some time-variable geophysical signals of interest.

A comparison of different GRACE solutions in terrestrial water storage trend estimation over Tibetan Plateau

The Gravity Recovery and Climate Experiment (GRACE) terrestrial water storage anomalies (TWSA) estimations provide valuable information for the monitoring of land water resources changes. Multiple parameters and strategies for inversion of the water storage changes have been explored. The explorations on differences between GRACE solutions in local regions and basins are fundamental and important. This study focuses on comparisons of TWSA trends between different GRACE solutions over Tibetan Plateau (TP), both storage and flux among solutions were compared. Results show that great discrepancies exist in TWSA between GRACE solutions derived from the standard spherical harmonic approach (SSH) and the mascon approach. Three SSH-based GRACE solutions (JPL, CSR, and GFZ) detect no significant TWSA changes for the whole area of Tibetan Plateau, whereas JPL mascon solution (JPL-M) and CSR mascon solution (CSR-M) gave decreasing trends of 3.10 km3/yr and 3.77 km3/yr, respectively. This difference also exists in the Yangtze River-Yellow River basin (YYR basin) in the TP. Although five solutions derived consistent TWSA trends in northwest river basin (NWR basin) and southwest river basin (SWR basin) in the TP, the variations between different solutions are 2.88 km3/yr and 4.75 km3/yr for NWR and SWR basin respectively, which could not be neglected. The JPL-M solution, as a result, would overestimate both TWSA decreasing and increasing trends comparing with other GRACE solutions. The results of this study are expected to provide references for the studies of water resource dynamics over Tibetan Plateau and the surrounding areas based on GRACE TWSA products.

A New Hybrid Method for Estimating Hydrologically Induced Vertical Deformation From GRACE and a Hydrological Model: An Example From Central North America

AbstractHydrologically induced deformation of Earth's surface can be measured with high precision geodetic techniques, which in turn can be used to study the underlying hydrologic process. For geodetic study of other Earth processes such as tectonic and volcanic deformation, or coastal subsidence and its relation to relative sea level rise and flood risk, hydrological loading may be a source of systematic error, requiring accurate correction. Accurate estimation of the hydrologic loading deformation may require consideration of local as well as regional loading effects. We present a new hybrid approach to this problem, providing a mathematical basis for combining local (near field) and regional to global (far field) loading data with different accuracies and spatial resolutions. We use a high‐resolution hydrological model (WGHM) for the near field and GRACE data for the far field. The near field is defined as a spherical cap and its contribution is calculated using numerical evaluation of Green's functions. The far field covers the entire Earth, excluding only the near‐field cap. The far‐field contribution is calculated using a modified spherical harmonic approach. We test our method with a large GPS data set from central North America. Our new hybrid approach improves fits to GPS‐measured vertical displacements, with 25% and 35% average improvement relative to GRACE‐only or WGHM‐only spherical harmonic solutions. Our hybrid approach can be applied to a wide variety of environmental surface loading problems.

Mathematical and Numerical Validation of the Simplified Spherical Harmonics Approach for Time-Dependent Anisotropic-Scattering Transport Problems in Homogeneous Media

ABSTRACTIn this work, we demonstrate theoretically and numerically the equivalence between the SPN and PN approximation for time-dependent problems in homogeneous media with anisotropic scattering. Our theory extends the solid harmonics derivation, which was used by Ackroyd et al. to derive the steady-state SPN equations, to transient problems. The derivation expands the angular flux in ordinary surface harmonics but uses harmonic polynomials to generate additional surface spherical harmonic terms to be used in Galerkin projection. Also, we use the line source problem and McClarren's “box” problem to demonstrate such equivalence numerically. Both problems were initially proposed for isotropic scattering, but here we add higher order scattering moments to them. Numerical results show that the difference between the SPN and PN scalar flux solution is at the roundoff level.

Cortical complexity in bipolar disorder applying a spherical harmonics approach

Recent studies using surface-based morphometry of structural magnetic resonance imaging data have suggested that some changes in bipolar disorder (BP) might be neurodevelopmental in origin. We applied a novel analysis of cortical complexity based on fractal dimensions in high-resolution structural MRI scans of 18 bipolar disorder patients and 26 healthy controls. Our region-of-interest based analysis revealed increases in fractal dimensions (in patients relative to controls) in left lateral orbitofrontal cortex and right precuneus, and decreases in right caudal middle frontal, entorhinal cortex, and right pars orbitalis, and left fusiform and posterior cingulate cortices. While our analysis is preliminary, it suggests that early neurodevelopmental pathologies might contribute to bipolar disorder, possibly through genetic mechanisms.