Direction Of Maximum Variance Research Articles

Statistical analysis of magnetotail fast flows and related magnetic disturbances

Abstract. This study presents an investigation on the occurrence of fast flows in the magnetotail using the complete available data set of the THEMIS spacecraft for the years 2007 to 2015. The fast flow events (times of enhanced ion velocity) are detected through the use of a velocity criterion, therefore making the resulting database as large as almost 16 000 events. First, basic statistical findings concerning velocity distributions, occurrence rates, group structures are presented. Second, Superposed Epoch Analysis is utilized to account for average profiles of selected plasma quantities. The data reveal representative time series in near and far tail of the Earth with typical timescales of the order of 1–2 min, corresponding to scale sizes of 3 RE. Last, related magnetic field disturbances are analyzed. It is found that the minimum variance direction is essentially confined to a plane almost perpendicular to the main flow direction while, at the same time, the maximum variance direction is aligned with flow and background field directions. The presentation of the database and first statistical findings will prove useful both as input for magneto-hydrodynamical simulations and theoretical considerations of fast flows.

Reformulation in the frequency domain of a critical plane-based multiaxial fatigue criterion

In the present paper, a new computationally-efficient frequency domain formulation of the critical plane-based Carpinteri–Spagnoli (C–S) criterion is proposed to evaluate the fatigue lives of smooth metallic structures subjected to multiaxial random loading. The critical plane orientation is here proposed to depend on the Power Spectral Density (PSD) matrix of the stress tensor. Then, the PSD function of an equivalent normal stress is defined by considering a linear combination of the PSD functions of the normal stress and the projected shear stress along the direction of maximum variance, with such stresses acting on the critical plane. Such an equivalent PSD function allows us to apply the Tovo–Benasciutti method to estimate the fatigue life of the structural components. The present frequency domain formulation of the C–S criterion is applied to some relevant fatigue tests related to smooth specimens under non-proportional bending and torsion random loading.

Estimating the orientation of Stage I crack paths through the direction of maximum variance of the resolved shear stress

Our formalisation of the Shear Stress-Maximum Variance Method takes as a starting point the hypothesis that, in ductile materials subjected to fatigue loading, the crack initiation planes, i.e. the so-called Stage I planes, are those containing the direction experiencing the maximum variance of the resolved shear stress. From a computational point of view, the most remarkable implication of the above assumption is that, as soon as the variance and covariance terms characterising the considered load history are known, the effective time needed to estimate the orientation of the critical plane does not depend on the length of the load history itself. Further, such a computational efficiency is seen to be associated with an high-level of accuracy in estimating fatigue lifetime of both plain and notched engineering components, this holding true under constant as well as under variable amplitude uniaxial/multiaxial fatigue loading. In this scenario, by assuming that the orientation of Stage I planes can directly be determined through the orientation of Stage II crack paths, the present paper investigates whether, independently from the degree of multiaxiality and non-proportionality of the applied loading history, the direction of maximum variance of the resolved shear stress is also capable of accurately estimating the orientation of Stage I crack paths.

PCA based image denoising

Principal component analysis (PCA) is an orthogonal transformation that seeks the directions of maximum variance in the data and is commonly used to reduce the dimensionality of the data. In image denoising, a compromise has to be found between noise reduction and preserving significant image details. PCA is a statistical technique for simplifying a dataset by reducing datasets to lower dimensions. It is a standard technique commonly used for data reduction in statistical pattern recognition and signal processing. This paper proposes a denoising technique by using a new statistical approach, principal component analysis with local pixel grouping (LPG). This procedure is iterated second time to further improve the denoising performance, and the noise level is adaptively adjusted in the second stage.

Evidence of surface wave on the dusk flank of Saturn's magnetopause possibly caused by the Kelvin-Helmholtz instability

Boundary waves on Saturn's dusk side magnetopause were identified from magnetopause crossings made by the Cassini spacecraft. These crossings occurred over a 48 h period from 14 July to 15 July 2007 when Cassini was on the inbound leg of revolution (Rev.) 48, and were located at around 17:00 SLT at low latitudes above the equatorial plane. The boundary normal for each magnetopause crossing was determined from minimum variance analysis (MVA) on the magnetic field within the current sheet interval. MVA was then performed on the normals themselves to identify a maximum variance direction, in an attempt to identify a preferred plane in which the normals varied. The first few crossings of the group exhibited no wave activity. The following 17 crossings showed clear oscillation of consecutive boundary normals about a mean direction in a well defined plane, with a clear periodicity of the order 50 min, suggesting the presence of surface waves with similar characteristics to those observed by Masters et al. (2009). Subsequent normals also exhibited clear consecutive oscillation in a preferred plane, but with no apparent periodicity. The direction along which maximum variance of the normals occurred was assumed to be the direction of wave propagation, and was found to be almost perpendicular to the average magnetospheric field for the duration of the crossings. Both the wave periodicity and propagation direction support the Kelvin-Helmholtz instability as a strong candidate mechanism for generating these waves.

Size, shape, and orientation of magnetosheath mirror mode structures

Information about the size, shape, and orientation of mirror mode structures in the magnetosheath is obtained by combining data from the four Cluster spacecraft, resulting in an estimate of the form of the magnetic field magnitude spatial autocorrelation function. It is shown that the longest direction of the structures is typically inclined at around 30° from the local magnetic field direction, consistent with predictions based on linear growth theory; their smallest size is also consistent with these predictions. The ratio of the longest to shortest scales of the structures is typically 2–6, with some evidence for this value to decrease as the structures are carried downstream. This suggests that the mirror modes are generated as long structures, but gradually become isotropic, possibly as part of their nonlinear growth. In common with earlier work, we find that the direction in which the structures are shortest is typically well aligned with the local magnetopause normal, consistent with compression against this boundary. However, the maximum variance direction is not a good proxy for the long axis of the structures, typically lying around 25° from it.

Magnetosheath fluctuations at Venus for two extreme orientations of the interplanetary magnetic field

Using the magnetosheath crossings of Venus Express on two consecutive days, we investigate magnetic fluctuations in the same locations for two extreme interplanetary magnetic field orientations, i.e., nearly along and nearly perpendicular to the solar wind flow. It is shown that the properties of the fluctuations are drastically different at basically the same location of the spacecraft in the magnetosheath. The strength and properties of the fluctuations are strongly controlled by the types of the upstream bow shock. The magnetic fluctuations behind a quasi‐parallel bow shock are quite strong and turbulent, having a strongly variable angle αeB between maximum variance direction of the fluctuations and the direction of the magnetic field, which may be convected from the upstream waves in the foreshock. The magnetic fluctuations behind a quasi‐perpendicular bow shock are less intensive and wave‐like, showing a less perturbed angle αeB, which are probably generated locally.

A novel engineering method based on the critical plane concept to estimate the lifetime of weldments subjected to variable amplitude multiaxial fatigue loading

ABSTRACTThis paper summarizes an attempt at proposing a new engineering method suitable for estimating the fatigue lifetime of steel‐ and aluminium‐welded connections subjected to variable amplitude multiaxial fatigue loading. In particular, the proposed approach is based on the use of the so‐called Modified Wöhler Curve Method (MWCM), i.e. a bi‐parametrical critical plane approach, whose accuracy has been checked so far solely in addressing the constant amplitude multiaxial fatigue problem. In order to extend the use of our criterion to variable amplitude situations, the critical plane is suggested here as being determined by taking full advantage of the maximum variance concept, that is, such a plane is assumed to be the one containing the direction along which the variance of the resolved shear stress reaches its maximum value. The main advantage of such a strategy is that the cycle counting can directly be performed by considering the shear stress resolved along the maximum variance direction: by so doing, the problem is greatly simplified, allowing those well‐established cycle counting methods specifically devised to address the uniaxial variable amplitude problem to be extended to those situations involving multiaxial fatigue loading. The validity of the proposed methodology was checked by using two different datasets taken from the literature and generated by testing both steel and aluminium tube‐to‐plate welded connections subjected to in‐phase and 90° out‐of‐phase variable amplitude bending and torsion. This new fatigue life assessment technique was seen to be highly accurate allowing the estimates to fall within the calibration scatter bands not only when the constants in the governing equations were calculated by using the experimental uniaxial and torsional fully reversed fatigue curves, but also when they were determined by using the reference curves supplied, for the investigated geometry, by the available standard codes. These results seem to strongly support the idea that, thanks to its peculiar features, our method can be considered as an effective engineering approach capable of performing multiaxial fatigue assessment under variable amplitude loading which fully complies with the recommendations of the available standard codes.

THEMIS observations of duskside compressional Pc5 waves

The five Time History of Events and Macroscale Interactions during Substorms (THEMIS) spacecraft offer new possibilities to analyze ULF waves in the magnetosphere by means of multipoint measurements. During the coast phase, THEMIS observed many compressional oscillations with periods in the Pc5 range and longer. The observed events occur inside a well‐defined spatial domain in the outer equatorial duskside magnetosphere. We analyze these waves using the unique string‐of‐pearls configuration of the THEMIS constellation to evaluate their phase speed and propagation direction. We find that the waves are propagating sunward (westward) and radially outward, orthogonal to the mean magnetic field, with phase speeds around 30 km/s and higher in the spacecraft frame. In the plasma frame the propagation direction is still sunward, with lower speeds (up to 30 km/s for most events). The oscillations exhibit a strong anticorrelation between the magnetic field and the plasma density. On the basis of this, as well as their low propagation speed, orthogonal to the mean magnetic field propagation direction and almost parallel to the magnetic field maximum variance direction, we conclude that the most likely source of these waves is the drift mirror instability.

Uncorrelated linear discriminant analysis (ULDA): A powerful tool for exploration of metabolomics data

The theory together with an algorithm for uncorrelated linear discriminant analysis (ULDA) is introduced and applied to explore metabolomics data. ULDA is a supervised method for feature extraction (FE), discriminant analysis (DA) and biomarker screening based on the Fisher criterion function. While principal component analysis (PCA) searches for directions of maximum variance in the data, ULDA seeks linearly combined variables called uncorrelated discriminant vectors (UDVs). The UDVs maximize the separation among different classes in terms of the Fisher criterion. The performance of ULDA is evaluated and compared with PCA, partial least squares discriminant analysis (PLS-DA) and target projection discriminant analysis (TP-DA) for two datasets, one simulated and one real from a metabolomic study. ULDA showed better discriminatory ability than PCA, PLS-DA and TP-DA. The shortcomings of PCA, PLS-DA and TP-DA are attributed to interference from linear correlations in data. PLS-DA and TP-DA performed successfully for the simulated data, but PLS-DA was slightly inferior to ULDA for the real data. ULDA successfully extracted optimal features for discriminant analysis and revealed potential biomarkers. Furthermore, by means of cross-validation, the classification model obtained by ULDA showed better predictive ability than PCA, PLS-DA and TP-DA. In conclusion, ULDA is a powerful tool for revealing discriminatory information in metabolomics data.

Advanced method to derive the IMF direction near Mars from cycloidal proton distributions

In a previous paper, we showed a method for deriving the interplanetary magnetic field (IMF) orientation from the velocity distribution of ring-like distributed ions as measured by the Ion Mass Analyser (IMA) on board Mars Express (MEX). This method has been improved so that one can derive the IMF orientation from a very limited portion of the ring distributions, i.e., only the highest energy portion of the ring distribution. This method uses the maximum variance direction L instead of the minimum variance direction N, which are derived from manually selected ring data. Because IMA's count rate for a semi-persistent ring distribution is nearly proportional to energy squire, L is most likely aligned to the tangential direction of the ring distribution at its highest energy, and this tangential direction is parallel or anti-parallel to the electric field. A vector product of L and the solar wind direction ( X) gives the IMF orientation projected to the Y– Z plane. The tilt angle of IMF toward the X direction from the Y– Z plane is the same as the angle between the X direction and the ring plane, and is obtained from two methods when the initial speed of the ring ions is estimated to be much smaller than the solar wind speed: (1) angle between the velocity of ring's maximum energy portion and the solar wind vector, and (2) energy ratio between the solar wind and the maximum energy of the ring. The present method is applied to the IMA data from 3 June 2005 (0605–0640 UT) when the Mars Global Surveyor (MGS) magnetometer data are available. Using these data, we also tried to determine the sign of the IMF direction by estimating the evolution direction of the ring ions.

Problems and Techniques

This issues features papers about mathematical genetics, data compression, and data assimilation in oceanography. 1. Computer algebra can help to predict the genetic makeup of a population. This is what we can conclude from the first paper, “Buchberger's Algorithm and the Two-Locus, Two-Allele Model.” With reproduction rates and probability of chromosome recombination as parameters, and one variable for each chromosome type, one can set up a system of polynomial equations. The solution value for a variable indicates how often this chromosome type occurs in the population. The question is now, over all parameter choices, how many different equilibrium solutions can there be? In other words, in the long term, can a chromosome type occur with any possible frequency? Arthur Copeland's answer is no: the number of solutions is almost always finite (and conjectured to be 15) in the special case when a population contains at most four different chromosome types. He arrives at his answer by means of affine varieties, ideals, and Gröbner bases. He also shows that the equilibrium solutions vary smoothly with the parameters. 2. The second paper, “A Direct Formulation for Sparse PCA Using Semidefinite Programming,” by Alexandre d'Aspremont, Laurent El Ghaoui, Michael I. Jordan, and Gert R. G. Lanckriet, is concerned with principal component analysis (PCA), a technique for compressing data so that as little information as possible is lost. PCA accomplishes this by finding directions of maximum variance in the data. In data analysis of gene expressions, for instance, different elements of a direction vector correspond to different genes. However, interpreting the meaning of a direction is much easier if only a few genes contribute to the direction. Such a direction vector has many zero elements and is called sparse. The authors propose a semidefinite programming method that maximizes variance as well as sparsity. The method is computationally efficient and robust. 3. The topic of the third paper, “A Reduced-Order Kalman Filter for Data Assimilation in Physical Oceanography,” by D. Rozier, F. Birol, E. Cosme, P. Brasseur, J. M. Brankart, and J. Verron, is a technique used by oceanographers to increase the accuracy of numerical models for predicting ocean circulation. Data assimilation combines the output of a simulation step with observed data and error statistics and then returns this “corrected” output to the simulation as the basis for the next step. Data assimilation based on Kalman filters assumes that measurement and model errors are unbiased and Gaussian. This is expensive, however, because an error covariance matrix must be constructed at every step. A reduced-order Kalman filter (called SEEK) is a cheaper version that reduces (or compresses) the size of the covariance matrix—the same key idea as in the paper described above. The authors give a very readable account of the issues associated with so-called ocean global circulation models: choice of vertical coordinate system (depending on the ocean region: shallow coastal shelf, steeply sloping ocean floors, stratified regions); limitations of models for ocean circulation simulations; sequential data assimilation techniques; adaptation of Kalman filters to geophysical applications; and modes of data acquisition (via satellites, ships, floats, mooring networks). The effectiveness of the SEEK filter is illustrated with numerical simulations of the Gulf Stream and the Bay of Biscay.

Cluster observations of flux rope structures in the near-tail

Abstract. An investigation of the 2003 Cluster tail season has revealed small flux ropes in the near-tail plasma sheet of Earth. These flux ropes manifest themselves as a bipolar magnetic field signature (usually predominantly in the Z-component) associated with a strong transient peak in one or more of the other components (usually the Y-component). These signatures are interpreted as the passage of a cylindrical magnetic structure with a strong axial magnetic field over the spacecraft position. On the 2 October 2003 all four Cluster spacecraft observed a flux rope in the plasma sheet at X (GSM) ~-17 RE. The flux rope was travelling Earthward and duskward at ~160 kms-1, as determined from multi-spacecraft timing. This is consistent with the observed south-then-north bipolar BZ signature and corresponds to a size of ~0.3 RE (a lower estimate, measuring between the inflection points of the bipolar signature). The axis direction, determined from multi-spacecraft timing and the direction of the strong core field, was close to the intermediate variance direction of the magnetic field. The current inside the flux rope, determined from the curlometer technique, was predominantly parallel to the magnetic field. However, throughout the flux rope, but more significant in the outer sections, a non-zero component of current perpendicular to the magnetic field existed. This shows that the flux rope was not in a "constant α" force-free configuration, i.e. the magnetic force, J×B was also non-zero. In the variance frame of the magnetic field, the components of J×B suggest that the magnetic pressure force was acting to expand the flux rope, i.e. directed away from the centre of the flux rope, whereas the smaller magnetic tension force was acting to compress the flux rope. The plasma pressure is reduced inside the flux rope. A simple estimate of the total force acting on the flux rope from the magnetic forces and surrounding plasma suggests that the flux rope was experiencing an expansive total force. On 13 August 2003 all four Cluster spacecraft observed a flux rope at X (GSM) ~-18 RE. This flux rope was travelling tailward at 200 kms-1, consistent with the observed north-then-south bipolar BZ signature. The bipolar signature corresponds to a size of ~0.3 RE (lower estimate). In this case, the axis, determined from multi-spacecraft timing and the direction of the strong core field, was directed close to the maximum variance direction of the magnetic field. The current had components both parallel and perpendicular to the magnetic field, and J×B was again larger in the outer sections of the flux rope than in the centre. This flux rope was also under expansive magnetic pressure forces from J×B, i.e. directed away from the centre of the flux rope, and had a reduced plasma pressure inside the flux rope. A simple total force calculation suggests that this flux rope was experiencing a large expansive total force. The observations of a larger J×B signature in the outer sections of the flux ropes when compared to the centre may be explained if the flux ropes are observed at an intermediate stage of their evolution after creation by reconnection at multiple X lines near the Cluster apogee. It is suggested that these flux ropes are in the process of relaxing towards the force-free like configuration often observed further down the tail. The centre of the flux ropes may contain older reconnected flux at a later evolutionary stage and may therefore be more force-free.

High speed anisotropic etching of Pyrex ® for microsystems applications

We report high speed etching of glass (Pyrex ® 7740) substrates using an inductively coupled plasma (ICP) reactive ion etching (RIE) process employing sulfur hexafluoride/argon (SF 6/Ar) based chemistry. Electroplated Ni over a patterned Cr/Au seed layer was used as the hard mask for etching. Detailed process characterization was performed by varying the process parameters which include substrate temperature, ICP power, substrate power, operating pressure, distance of substrate holder from ICP source and composition and flow rates of the etching gases. An rms surface roughness of 1.97 nm at a high etch rate of 0.536 μm/min was achieved by process optimization. We used least square fit to find the direction of maximum variance in the process parametric space and to reduce the dimensionality of the process parametric space. The etch rate was then linearly related to a new variable termed as the etch rate number.

Motion and orientation of magnetic field dips and peaks in the terrestrial magnetosheath

Using data from the four Cluster spacecraft, estimates of the motion of isolated magnetic field dips and peaks in the magnetosheath are presented. Their motion and orientation are deduced using four‐spacecraft timing, with typical spacecraft separations of ∼100 km. A set of 39 clear isolated dips and 12 peaks, spread over 4 days, is considered. The calculated velocities in the plasma frame are scattered around zero, with a width of ∼30 km/s and an average absolute value of 21 km/s. This places an upper limit on the speed of these structures, and within timing errors these values are consistent with the structures being stationary in the plasma frame. Two events had estimated speeds of over half the local Alfvén speed; it is shown that these two estimates were unreliable as a result of uncertainties in timings. The inferred propagation speeds were typically slower than local wave speeds (95% less than half the local Alfvén speed), even allowing for their orientations (85% less than the slow mode speed and 56% less than half of it). These results are consistent with the structures being stationary in the plasma frame, as expected for mirror modes, rather than being propagating waves or solitons. In addition, they are not consistent with structures standing in the flow. However, it is not possible to eliminate the possibility that a small number of the structures may have been slowly propagating. The normals of the structures deduced from the timings suggest that they are cylinders rather than sheets and that the maximum variance direction is a reliable estimate of their axis orientation, with a precision of ∼20° for maximum/intermediate eigenvalue ratios of 10 or more.

Reconstruction of two-dimensional magnetopause structures from Cluster observations: verification of method

Abstract. A recently developed technique for reconstructing approximately two-dimensional (∂/∂z≈0), time-stationary magnetic field structures in space is applied to two magnetopause traversals on the dawnside flank by the four Cluster spacecraft, when the spacecraft separation was about 2000km. The method consists of solving the Grad-Shafranov equation for magnetohydrostatic structures, using plasma and magnetic field data measured along a single spacecraft trajectory as spatial initial values. We assess the usefulness of this single-spacecraft-based technique by comparing the magnetic field maps produced from one spacecraft with the field vectors that other spacecraft actually observed. For an optimally selected invariant (z)-axis, the correlation between the field components predicted from the reconstructed map and the corresponding measured components reaches more than 0.97. This result indicates that the reconstruction technique predicts conditions at the other spacecraft locations quite well. The optimal invariant axis is relatively close to the intermediate variance direction, computed from minimum variance analysis of the measured magnetic field, and is generally well determined with respect to rotations about the maximum variance direction but less well with respect to rotations about the minimum variance direction. In one of the events, field maps recovered individually for two of the spacecraft, which crossed the magnetopause with an interval of a few tens of seconds, show substantial differences in configuration. By comparing these field maps, time evolution of the magnetopause structures, such as the formation of magnetic islands, motion of the structures, and thickening of the magnetopause current layer, is discussed. Key words. Magnetospheric physics (Magnetopause, cusp, and boundary layers) – Space plasma physics (Experimental and mathematical techniques, Magnetic reconnection)

A computational feature binding model of human texture perception

We present a computational model for human texture perception which assigns functional principles to the Gestalt laws of similarity and proximity. Motivated by early vision mechanisms, in the first stage, local texture features are extracted by utilizing multi-scale filtering and nonlinear spatial pooling. In the second stage, features are grouped according to the spatial feature binding model of the competitive layer model (CLM; Wersing et al. 2001). The CLM uses cooperative and competitive interactions in a recurrent network, where binding is expressed by the layer-wise coactivation of feature-representing neurons. The Gestalt law of similarity is expressed by a non-Euclidean distance measure in the abstract feature space with proximity being taken into account by a spatial component. To choose the stimulus dimensions which allow the most salient similarity-based texture segmentation, the feature similarity metrics is reduced to the directions of maximum variance. We show that our combined texture feature extraction and binding model performs segmentation in strong conformity with human perception. The examples range from classical microtextures and Brodatz textures to other classical Gestalt stimuli, which offer a new perspective on the role of texture for more abstract similarity grouping.

Cluster magnetic field observations in the magnetosheath: four-point measurements of mirror structures

Abstract. The Cluster spacecraft have returned the first simultaneous four-point measurements of the magnetosheath. We present an analysis of data recorded on 10 November 2000, when the four spacecrafts observed an interval of strong mirrorlike activity. Correlation analysis between spacecraft pairs is used to examine the scale size of the mirror structures in three dimensions. Two examples are presented which suggest that the scale size of mirror structures is ~ 1500–3000 km along the flow direction, and shortest along the magnetopause normal (< 600 km), which, in this case, is approximately perpendicular to both the mean magnetic field and the magnetosheath flow vector. Variations on scales of ~ 750–1000 km are found along the maximum variance direction. The level of correlation in this direction, however, and the time lag observed, are found to be variable. These first results suggest that variations occur on scales of the order of the spacecraft separation ( ~ 1000 km) in at least two directions, but analysis of further examples and a statistical survey of structures observed with different magnetic field orientations and tetrahedral configurations will enable us to describe more fully the size and orientation of mirror structures.Key words. Magnetosphenic physics (magnetosheath; plasma waves and instabilities)

A fast nearest-neighbor algorithm based on a principal axis search tree

A new fast nearest-neighbor algorithm is described that uses principal component analysis to build an efficient search tree. At each node in the tree, the data set is partitioned along the direction of maximum variance. The search algorithm efficiently uses a depth-first search and a new elimination criterion. The new algorithm was compared to 16 other fast nearest-neighbor algorithms on three types of common benchmark data sets including problems from time series prediction and image vector quantization. This comparative study illustrates the strengths and weaknesses of all of the leading algorithms. The new algorithm performed very well on all of the data sets and was consistently ranked among the top three algorithms.

Evolution of mirror structures in the magnetosheath of Saturn from the bow shock to the magnetopause

Mirror modes have been systematically observed by Voyagers 1 and 2 in wide portions of Jupiter's and Saturn's magnetosheaths. In particular, in one crossing of Saturn's subsolar magnetosheath, mirror waves are present almost continuously from the bow shock to the magnetopause. Therefore in this crossing, taking advantage also of relatively steady interplanetary conditions, we can track the evolution of mirror structures from a quasi‐perpendicular bow shock to a low‐shear magnetopause. We find that these structures evolve from quasi‐sinusoidal waves to nonperiodic structures, consisting of both magnetic field enhancements and wells, and, finally, to dips in the plasma depletion layer (PDL) close to the magnetopause. Both the amplitude and wavelength of the fluctuations tend to increase with increasing distance from the bow shock, except in the PDL, where they decrease toward the magnetopause. The waves are always compressional, and the direction of maximum variance forms an angle of ∼30° with B in the outer magnetosheath and a smaller angle in the inner magnetosheath. A comparison with the predictions of a nonlinear theory of the mirror instability shows some discrepancies, indicating that further theoretical studies are necessary.