Error-propagation in weakly nonlinear inverse problems

In applications of inversion methods to real data, nonlinear inverse problems are often simplied to more easily solvable linearized inverse problems. By doing so one introduces an error made by the linearization. Nonlinear inverse methods are more accurate because the methods that are used are more correct from a physical point of view. However, if data are used that have a statistical error, nonlinear inversion methods lead to a bias in the retrieved model parameters, caused the by nonlinear propagation of errors. If the bias in the estimated model parameters is larger than the linearization error, a linearized inverse problem leads to better estimation of the model parameter. In this paper the error-propagation is investigated for inversion methods that account the nonlinearity quadratically

Genomic analysis of diet composition finds novel loci and associations with health and lifestyle

We conducted genome-wide association studies (GWAS) of relative intake from the macronutrients fat, protein, carbohydrates, and sugar in over 235,000 individuals of European ancestries. We identified 21 unique, approximately independent lead SNPs. Fourteen lead SNPs are uniquely associated with one macronutrient at genome-wide significance (P < 5 × 10−8), while five of the 21 lead SNPs reach suggestive significance (P < 1 × 10−5) for at least one other macronutrient. While the phenotypes are genetically correlated, each phenotype carries a partially unique genetic architecture. Relative protein intake exhibits the strongest relationships with poor health, including positive genetic associations with obesity, type 2 diabetes, and heart disease (rg ≈ 0.15–0.5). In contrast, relative carbohydrate and sugar intake have negative genetic correlations with waist circumference, waist-hip ratio, and neighborhood deprivation (|rg| ≈ 0.1–0.3) and positive genetic correlations with physical activity (rg ≈ 0.1 and 0.2). Relative fat intake has no consistent pattern of genetic correlations with poor health but has a negative genetic correlation with educational attainment (rg ≈−0.1). Although our analyses do not allow us to draw causal conclusions, we find no evidence of negative health consequences associated with relative carbohydrate, sugar, or fat intake. However, our results are consistent with the hypothesis that relative protein intake plays a role in the etiology of metabolic dysfunction

Genomic analysis of diet composition finds novel loci and associations with health and lifestyle

We conducted genome-wide association studies (GWAS) of relative intake from the macronutrients fat, protein, carbohydrates, and sugar in over 235,000 individuals of European ancestries. We identified 21 unique, approximately independent lead SNPs. Fourteen lead SNPs are uniquely associated with one macronutrient at genome-wide significance (P < 5 x 10(-8)), while five of the 21 lead SNPs reach suggestive significance (P < 1 x 10(-5)) for at least one other macronutrient. While the phenotypes are genetically correlated, each phenotype carries a partially unique genetic architecture. Relative protein intake exhibits the strongest relationships with poor health, including positive genetic associations with obesity, type 2 diabetes, and heart disease (r(g) approximate to 0.15-0.5). In contrast, relative carbohydrate and sugar intake have negative genetic correlations with waist circumference, waist-hip ratio, and neighborhood deprivation (|r(g)| approximate to 0.1-0.3) and positive genetic correlations with physical activity (r(g) approximate to 0.1 and 0.2). Relative fat intake has no consistent pattern of genetic correlations with poor health but has a negative genetic correlation with educational attainment (r(g) approximate to-0.1). Although our analyses do not allow us to draw causal conclusions, we find no evidence of negative health consequences associated with relative carbohydrate, sugar, or fat intake. However, our results are consistent with the hypothesis that relative protein intake plays a role in the etiology of metabolic dysfunction.Public Health and primary carePrevention, Population and Disease management (PrePoD

Surface wave scattering theory : with applications to forward and inverse problems in seismology

Scattering of surface waves in a three dimensional layered elastic medium with embedded heterogeneities is described in this thesis with the Born approximation. The dyadic decomposition of the surface wave Green's function provides the crucial element for an efficient application of Born theory to surface wave scattering. This is because the dyadic Green's function allows for an efficient bookkeeping of the different processes that contribute to the scattered surface wave: excitation, propagation, scattering (conversion), and oscillation. One can argue that the most crucial (and surprisingly also the simplest) expression in this thesis is equation (3) of chapter 2. The resulting surface wave scattering theory for buried heterogeneities in a flat geometry (chapter 2), can easily be extended to incorporate the effects of surface topography (chapter 3), and a spherical geometry (chapters 6 and 7). In practice, the Born approximation imposes a lower limit on the periods that can be analyzed. This limit depends both on the properties of the heterogeneity and on the source receiver separation. An analysis of the surface wave coda recorded in stations of the NARS array shows that the surface wave coda level differs substantially for different regions. For paths through eastern and middle Europe, the Born approximation breakS down for periods shorter than 30 s., while for paths through the western Mediterranean periods as short as 20 s. can be analyzed with linear theory (chapter 8). In exploration seismics, linear theory is usually used to establish a relation between the heterogeneity and the reflected waves, as well as for the inversion of these reflection data. It is therefore not surprising that the surface wave coda can in principle be used to map the heterogeneity in the Earth, with an inversion scheme which is reminiscent to Kirchoff migration as used in exploration seismics (chapter 2). In a simple field experiment the feasibility of such an inversion scheme is established (chapter 4). It is also possible to formulate the waveform inversion of surface wave data as a (huge) matrix problem. The least squares solution of these matrix equations can iteratively be constructed. These reconstructed models have the same characteristics as the models found with a simple holographic inversion (chapter 8). Inversion of the surface wave coda recorded in stations of the NARS array produce chaotic models of scatterers which are difficult to interpret unambiguously. Apart from a lack of enough data to perform a good imaging, this inversion is hampered by an appreciable noise component in the surface wave coda. This noise level might be acceptable if the data set were redundant, so that this noise component can be averaged out. However, the 42 available seismograms lead to an underdetermined system of linear equations, which make it likely that the noise in the surface wave coda introduces artifacts in the reconstructed model (chapter 9). Born theory for surface waves describes the distortion of the wavefield due to the heterogeneity of the medium. This distortion consists of true surface wave scattering due to abrupt lateral inhomogeneities, as well as a distortion of the direct surface wave due to smooth variations of the heterogeneity. Up to first order, ray geometrical effects follow from linear scattering theory (chapter 5). Furthermore, the scattering coefficient for forward scattering of unconverted waves is proportional to the phase velocity perturbation of these waves (chapter 3). This makes it possible to reconstruct phase velocity fields for surface waves using a large scale linear waveform inversion of the direct surface wave (chapter 8). This inversion is applied to the direct surface wave train recorded in stations of the NARS array. This results in detailed reconstructions of the phase velocity of the fundamental Rayleigh mode. In this inversion, a variance reduction of approximately 40% is achieved. By combining this information for different frequencies, detailed models of the S-velocity under Europe and the Mediterranean are reconstructed (chapter 9). With the present data set, the resolution of this model differs considerably from region to region. The only way to overcome this restriction is to use more data, which can be realized by employing dense networks of digital seismic stations. There is still a considerable amount of research to be performed on scattering theory of elastic waves. Apart from the restriction of linearity, the theory presented in this thesis is only valid in the far field. This means that the inhomogeneity should be several wavelengths removed from the source and the receiver (and their antipodes). In practice, this is a troublesome limitation, because seismic stations are often located on top of heterogeneities, and earthquakes usually occur in heterogeneous areas such as subduction zones. The interaction terms are valid both in the far field and in the near field (chapter 7), so that in order to resolve the far field restriction, the propagator terms need to be investigated. Future theoretical research should also address the problem of conversions between surface waves and body waves. This issue is related to the near field problem, because in the near field the concepts of "surface waves" and "body waves" are poorly defined. It would be interesting to use portable seismic stations for local investigations by recording scattered surface waves in the vicinity of strong lateral variations in the crust and upper mantle. In this way, it should be possible to probe tectonic features such as subduction zones using scattered surface waves. The waveform inversions of the direct surface waves, as presented in this thesis, can be applied to other regions of the Earth with a good coverage with digital seismic stations (e.g. Japan, the continental US), and possibly for lower frequencies on a global scale. In this way, large scale waveform inversions for both the phase and amplitude of surface wave data may dramatically increase our knowledge of the Earth's interio

High-frequency precursors to P-wave arrivals in New Zealand : implications for slab structure

This report revisits the very early high-frequency slab phases from earthquakes in the Kermadec slab (between −25°S and −37°S) that arrive as a precursor to the P wave onset at stations in New Zealand. The analysis of short-period digital records for station SNZO (South Karori New Zealand) for the time period between 1980 and 1986 involved 4 times as many data as in previous studies. We confirm most of the earlier observations, in particular the relationship between source region and observed waveform. Numerical waveform modeling using two-dimensional slab models confirm that the high-frequency precursor can be explained by P wave propagation in a thin (8–10 km) high-velocity layer (HVL). The sensitivity of the high-frequency phase to source position in the slab is investigated, and we demonstrate that the slab phase does not survive interruptions or strong lateral variations of the HVL. Assuming the shape of the slab from tomographic images and continuity of the HVL, we infer that the large negative travel time residuals can be explained if P wave propagation is 4 to 5% faster than in the ambient mantle for depths to 300 km and about 2% faster for depths between 300 and 600 km. Rather than resorting to regional differences in structure and composition of the HVL, we argue that specific source-receiver configurations and the shape of the slab explain the observed regional variation in the character of the high-frequency slab phases

High-frequency precursors to P-wave arrivals in New Zealand : implications for slab structure

This report revisits the very early high-frequency slab phases from earthquakes in the Kermadec slab (between −25°S and −37°S) that arrive as a precursor to the P wave onset at stations in New Zealand. The analysis of short-period digital records for station SNZO (South Karori New Zealand) for the time period between 1980 and 1986 involved 4 times as many data as in previous studies. We confirm most of the earlier observations, in particular the relationship between source region and observed waveform. Numerical waveform modeling using two-dimensional slab models confirm that the high-frequency precursor can be explained by P wave propagation in a thin (8–10 km) high-velocity layer (HVL). The sensitivity of the high-frequency phase to source position in the slab is investigated, and we demonstrate that the slab phase does not survive interruptions or strong lateral variations of the HVL. Assuming the shape of the slab from tomographic images and continuity of the HVL, we infer that the large negative travel time residuals can be explained if P wave propagation is 4 to 5% faster than in the ambient mantle for depths to 300 km and about 2% faster for depths between 300 and 600 km. Rather than resorting to regional differences in structure and composition of the HVL, we argue that specific source-receiver configurations and the shape of the slab explain the observed regional variation in the character of the high-frequency slab phases

Error-propagation in weakly nonlinear inverse problems

In applications of inversion methods to real data, nonlinear inverse problems are often simplied to more easily solvable linearized inverse problems. By doing so one introduces an error made by the linearization. Nonlinear inverse methods are more accurate because the methods that are used are more correct from a physical point of view. However, if data are used that have a statistical error, nonlinear inversion methods lead to a bias in the retrieved model parameters, caused the by nonlinear propagation of errors. If the bias in the estimated model parameters is larger than the linearization error, a linearized inverse problem leads to better estimation of the model parameter. In this paper the error-propagation is investigated for inversion methods that account the nonlinearity quadratically

Error-Propagation in Weakly Nonlinear Inverse Problems

In applications of inversion methods to real data, nonlinear inverse problems are often simplified to more easily solvable linearized inverse problems. By doing so one introduces an error made by the linearization. Nonlinear inverse methods are more accurate because the methods that are used are more correct from a physical point of view. However, if data are used that have a statistical error, nonlinear inversion methods lead to a bias in the retrieved model parameters, caused the by nonlinear propagation of errors. If the bias in the estimated model parameters is larger than the linearization error, a linearized inverse problem leads to better estimation of the model parameter. In this paper the error-propagation is investigated for inversion methods that account the nonlinearity quadratically. 1 Introduction Inverse problems are widely used in many fields of science to relate measured data to physically relevant model parameters. In applications of inversion methods to real data, in..