Mode Rayleigh Waves Research Articles

Isolating retrograde and prograde Rayleigh-wave modes using a polarity mute

Estimates of S-wave velocity with depth from Rayleigh-wave dispersion data are limited by the accuracy of fundamental and/or higher mode signal identification. In many scenarios, the fundamental mode propagates in retrograde motion, whereas higher modes propagate in prograde motion. This difference in particle motion (or polarity) can be used by joint analysis of vertical and horizontal inline recordings. We have developed a novel method that isolates modes by separating prograde and retrograde motions; we call this a polarity mute. Applying this polarity mute prior to traditional multichannel analysis of surface wave (MASW) analysis improves phase velocity estimation for fundamental and higher mode dispersion. This approach, in turn, should lead to improvement of S-wave velocity estimates with depth. With two simple models and a field example, we have highlighted the complexity of the Rayleigh-wave particle motions and determined improved MASW dispersion images using the polarity mute. Our results show that we can separate prograde and retrograde signals to independently process fundamental and higher mode signals, in turn allowing us to identify lower frequency dispersion when compared with single component data. These examples demonstrate that the polarity mute approach can improve estimates of S-wave velocities with depth.

Performance comparison of Rayleigh and STW modes on quartz crystal for strain sensor application

In this study, we compare two kinds of strain sensors based on Rayleigh wave and surface transverse wave (STW) modes, respectively. First, we perform a strain-and-stress analysis using the finite element method, and we consider the contribution to a surface acoustic wave (SAW) velocity shift. Prior to fabrication, we use a coupling-of-modes model to simulate and optimize two-port SAW resonators for both modes. We use a network analyzer to measure and characterize the two devices. Further, we perform an experiment using a strain-testing system with a tapered cross-section cantilever beam. The experimental results show that the ratio of the frequency shift to the strain for the Rayleigh wave mode is −1.124 ppm/με in the parallel direction and 0.109 ppm/με in the perpendicular direction, while the corresponding values for the STW mode are 0.680 ppm/με and 0.189 ppm/με, respectively.

Near‐Surface Velocity Structure of Pacaya Volcano, Guatemala, Derived from Small‐Aperture Array Analysis of Seismic Tremor

Knowledge of the typically complicated near‐surface structure on volcanoes is critical for determining accurate seismic‐event locations and seismic source mechanisms associated with precursory and eruptive activity. To generate a near‐surface velocity model for the active cone of Pacaya volcano, Guatemala, we recorded seismic tremor and ambient noise with a small‐aperture array of 11 short‐period seismometers in January 2011. The diffuse wavefield was investigated using the spatial autocorrelation (SPAC) method, and we computed Rayleigh and Love dispersion curves at the array location. The phase velocities for Rayleigh waves range from 1000 m/s at 2 Hz to 230 m/s at 10 Hz; those for Love waves range from 600 to 250 m/s over the same frequency band. Such low velocities are consistent with the presence of unconsolidated tephra on the surface of the volcano interspersed with lava flows. Assuming they represent the fundamental modes of Rayleigh and Love waves, we inverted the dispersion curves to produce a shear‐wave velocity model of the upper 500 m beneath the array. Online Material: S ‐wave velocity models obtained from inversion, and Love‐ and Rayleigh‐wave dispersion curves.

Reason and Condition for Mode Kissing in MASW Method

Identifying correct modes of surface waves and picking accurate phase velocities are critical for obtaining an accurate S-wave velocity in MASW method. In most cases, inversion is easily conducted by picking the dispersion curves corresponding to different surface-wave modes individually. Neighboring surface-wave modes, however, will nearly meet (kiss) at some frequencies for some models. Around the frequencies, they have very close roots and energy peak shifts from one mode to another. At current dispersion image resolution, it is difficult to distinguish different modes when mode-kissing occurs, which is commonly seen in near-surface earth models. It will cause mode misidentification, and as a result, lead to a larger overestimation of S-wave velocity and error on depth. We newly defined two mode types based on the characteristics of the vertical eigendisplacements calculated by generalized reflection and transmission coefficient method. Rayleigh-wave mode near the kissing points (osculation points) change its type, that is to say, one Rayleigh-wave mode will contain different mode types. This mode type conversion will cause the mode-kissing phenomenon in dispersion images. Numerical tests indicate that the mode-kissing phenomenon is model dependent and that the existence of strong S-wave velocity contrasts increases the possibility of mode-kissing. The real-world data shows mode misidentification caused by mode-kissing phenomenon will result in higher S-wave velocity of bedrock. It reminds us to pay attention to this phenomenon when some of the underground information is known.

On the shaping factors of the secondary microseismic wavefield

AbstractSeismic noise in the period band 3–10 s is known as secondary microseism, and it is generated at the ocean surface by the interaction of ocean gravity waves coming from nearly opposite directions. In this paper, we investigate the seismic content of the wavefield generated by a source at the ocean surface and three of the major wavefield shaping factors using the 2‐D spectral element method: the ocean‐continent boundary, the source site effect, and the thickness of seafloor sediments. The seismic wavefield recorded on the vertical component seismograms below the seafloor is mainly composed of the fundamental mode and the first overtone of Rayleigh waves. A mode conversion from the first overtone to the fundamental mode of Rayleigh waves occurs at the ocean‐continent boundary. The presence of a continental shelf at the ocean‐continent boundary produces a negligible effect on land‐recorded seismograms, whereas the source site effect, i.e., the source location with respect to the local ocean depth and sediment thickness, plays the major role. A source in shallow water mostly enhances the fundamental mode of Rayleigh waves, whereas a source in deep water mainly enhances the first overtone of Rayleigh waves. Land‐recorded long‐period signals (T > 6 s) are mostly due to deep water sources, whereas land‐recorded short‐period signals (T < 6 s) are due to sources in relatively shallow water, located close to the shelf break. Seafloor sediments around the source region trap seismic waves reducing the amplitude of land‐recorded signals, especially at long periods (T > 6 s).

Self-adaptive method for high frequency seismic surface wave method

When the high frequency multi-channel analysis of surface waves (MASW) method is conducted to explore soil properties in the vadose zone, existing rules for selecting the near offset and spread lengths cannot satisfy the requirements of planar dominant Rayleigh waves for all frequencies of interest and will inevitably introduce near and far field effects as well as spatial aliases at certain frequencies. To solve these problems, a self-adaptive method is developed to determine high frequency dispersion trends. In this method, an initial dispersion curve is obtained by a fixed-offset MASW method and used to estimate wavelengths at all frequencies of interest. At each frequency, the near offset and spread lengths are then set to be 0.2–0.6 and 2–4 wavelengths respectively. In other words, the near offset and spread lengths are self-adaptive to the corresponding wavelength. Furthermore, in order to avoid spatial aliases and minimize the number of sensors or sensor locations, a variable sensor spacing configuration is proposed, in which the spacing for the first 120 steps is set to be 0.5cm and the next 40 steps start with 1cm spacing followed by 1cm incremental spacing for each subsequent step. In this study, an accelerometer was used as a sensor to detect surface vibrations generated by an electrodynamic shaker operating in chirp mode. Two field tests were conducted and the results from the fixed-offset, self-adaptive, and variable spacing MASW methods were compared. The study demonstrates the ability of the self-adaptive MASW method to preferentially identify the dispersion trends of either the fundamental mode or higher modes of Rayleigh waves. It is also found that the dispersion trends can be determined beyond the source frequency range due to nonlinear effects that generate high harmonics of surface waves. This nonlinear phenomenon deserves future investigation.

Multi-component joint analysis of surface waves

Propagation of surface waves can occur with complex energy distribution amongst the various modes. It is shown that even simple VS (shear-wave velocity) profiles can generate velocity spectra that, because of a complex mode excitation, can be quite difficult to interpret in terms of modal dispersion curves.In some cases, Rayleigh waves show relevant differences depending on the considered component (radial or vertical) and the kind of source (vertical impact or explosive). Contrary to several simplistic assumptions often proposed, it is shown, both via synthetic and field datasets, that the fundamental mode of Rayleigh waves can be almost completely absent. This sort of evidence demonstrates the importance of a multi-component analysis capable of providing the necessary elements to properly interpret the data and adequately constrain the subsurface model. It is purposely shown, also through the sole use of horizontal geophones, how it can be possible to efficiently and quickly acquire both Love and Rayleigh (radial-component) waves.The presented field dataset reports a case where Rayleigh waves (both their vertical and radial components) appear largely dominated by higher modes with little or no evidence of the fundamental mode.The joint inversion of the radial and vertical components of Rayleigh waves jointly with Love waves is performed by adopting a multi-objective inversion scheme based on the computation of synthetic seismograms for the three considered components and the minimization of the whole velocity spectra misfits (Full Velocity Spectra – FVS – inversion).Such a FVS multi-component joint inversion can better handle complex velocity spectra thus providing a more robust subsurface model not affected by erroneous velocity spectra interpretations and non-uniqueness of the solution.

Shear-Wave Velocity Tomography of the Lithosphere-Asthenosphere System beneath the Iranian Plateau

We conducted a tomographic inversion of Rayleigh-wave dispersion to obtain 2D phase and group velocity tomographic images in the 10-100 s period range and shear-wave velocity structures for the Iranian plateau. For this purpose, the fun- damental mode of Rayleigh waves, recorded along 1586 paths by 29 broadband sta- tions, was identified by applying the frequency time analysis (FTAN) to each epicenter-station path which simultaneously satisfies the two-station method condi- tions. The fundamental modes identified by FTAN have been used to determine the path-average interstation phase and group velocities at selected periods. With this pro- cedure, 243 group and phase velocity dispersion curves were processed to obtain to- mographic maps by applying the Yanovskaya-Ditmar formulation for periods in the 10-100 s range. Averaged dispersion curves of phase and group velocities, which represent six rather homogeneous regions, are computed. Finally, we used a fully non- linear inversion procedure to derive tomographic images of the elastic structure of the lithosphere and asthenosphere of the six main structural and seismotectonic features of the Iranian plateau. The dense path coverage in the Iranian plateau permits us to pro- duce images that have substantially higher lateral resolution compared to images cur- rently available from global and regional group velocity studies. Tomographic maps at high frequencies are well correlated with the upper crust structure, especially with sediment layers thickness. Estimated shear-wave velocity structures are well corre- lated with the shield-like lithosphere structure in Zagros. A low-velocity zone (LVZ) is not detected in Alborz or the south Caspian basin, which can imply thrusting of the oceanic crust of the southern Caspian Sea under the Alborz to the south. LVZs are derived for the region east of Iran, central Iran, and Kopeh Dagh.

Structural model of the lithosphere–asthenosphere system beneath the Qinghai–Tibet Plateau and its adjacent areas

The deep structure of the lithosphere–asthenosphere system, as imaged from geophysical data, of the Qinghai–Tibet Plateau, the highest on the Earth, provides important clues in understanding its orogenic processes. Here we reconstruct the main features of the structure of the crust and upper mantle from surface wave tomography in the Qinghai–Tibet Plateau and its adjacent areas, in order to understand the modality of the convergence and collision process between the Indian and Eurasian plates. Based on Rayleigh waves dispersion theory, we collected long period and broad-band seismic data from the global and regional seismic networks surrounding the study area (20°N–50°N, 70°E–110°E). After applying instrument response calibration and filtering, group velocities of the fundamental mode of Rayleigh waves are measured using the frequency–time analysis (FTAN). Combining the published dispersion data, a 2-D surface-wave tomography method is applied to calculate the lateral variations of group velocity distribution at different periods, in the range from 8s to 150s. The Hedgehog non-linear inversion method is performed to obtain shear wave velocity (Vs) versus depth models of the crust and upper mantle for 181 cells, with size 2°×2°. In order to identify the cellular representative models, we applied the local smoothness optimization method (LSO). Fairly detailed structural models of the lithosphere–asthenosphere system have been defined. The Vs models demonstrate the lateral variation of the thickness of the metasomatic lid between the south and north of the Bangong–Nujiang Suture (BNS) and the west and east of Tibet. The variation in thickness of the metasomatic lid may suggest that the leading edge of the subducting Indian slab reaches up to BNS.

On dispersive propagation of surface waves in patchy saturated porous media

Frequency-dependent velocity and attenuation for Rayleigh-wave propagation along a vacuum/patchy saturated porous medium interface are investigated in the low frequency band (0.1–1000 Hz). Conventional patchy saturation models for compressional waves are extended to account for Rayleigh wave propagation along a free surface. The mesoscopic interaction of fluid and solid phases, as a dominant loss mechanism in patchy saturated media, significantly affects Rayleigh-wave propagation and attenuation. Researches on the dispersion characteristics at low frequencies with different gas fractions in patchy saturated media also demonstrate a strong correlation between the Rayleigh-wave mode and the fast compressional wave. Especially, the strongest attenuation with the maximum value of 1/Q for Rayleigh waves are obtained in the frequency range of 1–200 Hz. Numerical results show that the significant dependence of velocity and attenuation on frequencies and gas fractions presents a distinctive dynamical response of Rayleigh waves in the time domain.

Discrete microfluidics based on aluminum nitride surface acoustic wave devices

To date, most surface acoustic wave (SAW) devices have been made from bulk piezoelectric materials, such as quartz, lithium niobate or lithium tantalite. These bulk materials are brittle, less easily integrated with electronics for control and signal processing, and difficult to realize multiple wave modes or apply complex electrode designs. Using thin film SAWs makes it convenient to integrate microelectronics and multiple sensing or microfluidics techniques into a lab-on-a-chip with low cost and multi-functions on various substrates (silicon, glass or polymer). In the work, aluminum nitride (AlN)-based SAW devices were fabricated and characterized for discrete microfluidic (or droplet based) applications. AlN films with a highly c-axis texture were deposited on silicon substrates using a magnetron sputtering system. The fabricated AlN/Si SAW devices had a Rayleigh wave mode at a frequency of 80.3 MHz (with an electromechanical coupling coefficient k 2 of 0.24 % and phase velocity v p of 5,139 m/s) and a higher-frequency-guided wave mode at 157.3 MHz (with a k 2 value of 0.22 % and v p of 10,067 m/s). Both modes present a large out of band rejection of ~15 dB and were successfully applied for microfluidic manipulation of liquid droplets, including internal streaming, pumping and jetting/nebulization, and their performance differences for microfluidic functions were discussed. A detailed investigation of the influences of droplet size (ranging from 3 to 15 μL) and RF input power (0.25–68 W) on microdroplet behavior has been conducted. Results showed that pumping and jetting velocities were increased with an increase of RF power or a decrease in droplet size.

Effects of source and cavity depths on wave fields in layered media

Abstract The multiple modes of Rayleigh waves (R-waves) or leaky Lamb waves (L-waves) may be present in the activated and/or the back-scattered wave fields. The presence of these waves in the wave fields is related to the source and the cavity depths. Two types of layered media are used to investigate the influences of the source and the cavity depths. One is the layered half space where the half space is stiffer than all the overlying layers; the other is the pavement system where there is significant velocity contrast between layers and the shear velocity of layers decreases with the layer depth. The discrete displacement expressions developed for spherical point sources are used to study the effects of the source depth on the excitability and the propagation behavior of R-waves or leaky L-waves. In layered half spaces, the multiple modes of R-waves can be activated by the sources within a certain depth. The effective phase velocity of these activated modes is derived. In pavement systems, the leaky L-waves can be activated by the sources at any depth. The discrete displacement expressions of the scattered waves are derived and used to evaluate the effects of the cavity depth on the presence of R-waves or the leaky L-waves in the back-scattered wave field.

Lithospheric VS models in the Campanian Plain (Italy) by integrating Rayleigh wave dispersion data from noise cross-correlation functions and earthquake recordings

Cross-correlation functions of long noise recordings with two broadband stations and earthquake recordings in the Campanian Plain have been processed with frequency time analysis to extract the fundamental mode of Rayleigh waves. Group velocities have been combined with regional group and phase velocity data in the non-linear inversion, with Hedgehog method, in order to get average shear wave velocity models for lithospheric structures extending to 73km of depth.The structural model below the central part of the Campanian Plain is characterized by a covering of pyroclastics and alluvial sediments, about 2km thick, on the carbonate platform with VS ranging from 2.30–2.40 to 2.85–3.15km/s. However, the presence of lava bodies within the carbonates cannot be excluded in the light of the same density and seismic velocities. At greater depths, a main feature is represented by a sharp increment of velocity around 8–9km of depth (VS of 3.85km/s), which can be attributed to the presence of metamorphic rocks, overlying a low VS layer (5% velocity reduction), at about 14–15km of depth. Such structural model resembles those found below the quiescent Roccamonfina and Colli Albani volcanoes, and can be interpreted as the signature of a cooling magma chamber.Moreover, a low VS layer is detected at 8–9km of depth towards the Apennines and at 6km of depth in the southernmost part of the Campanian Plain, nearby Mt. Vesuvius. Such low velocity layer seems to be a regional feature since it has been found below Roccamonfina in the North, Campi Flegrei, bay of Napoli and Mt. Vesuvius in the South, and can be explained by the widespread presence of partially melted material below the whole Campanian area.

Automatic ultrasonic inspection of large-diameter pipes

A device and a principle of operation of some ultrasound scanners-flaw detectors for diagnosing the basic metal mass of main oil and gas pipelines in reinsulation is considered. Different methods, materials, and devices for efficient excitation and reception of ultrasound waves in pipelines with wall thicknesses of 6–20 mm and the selection of Lamb and Rayleigh wave modes for testing as a function of the pipe-wall thickness are discussed. The principle of operation of a unidirectional primary EMA transducer is described.

Sensitivity study of surface waves for CO2 storage monitoring

Carbon dioxide capture and geological storage (CCS) is a potential technology to reduce greenhouse gas emission. Suitable techniques are essential for site characterization as well as CO2 injection and storage monitoring. A surface wave seismic method was explored in this study to investigate its feasibility for this purpose. Elastic wave responses of CO2 flooded rock were first investigated numerically in two types of rocks, carbonate rock and sandstone. It is indicated that elastic wave velocities change more significantly as there is greater difference for bulk modulus between the injected CO2 and the existing media in pore spaces. With the wave velocity and density variation ranges estimated from the rock sample study, the sensitivity of surface wave velocity was examined by perturbing parameters of the CO2 storage layer in two layered reservoir models. It is found that the surface waves are more sensitive to the changes of shear wave velocity and thickness of CO2 storage layer; but they are less sensitive to density and compressional wave velocity variations. The fundamental mode of Rayleigh waves is most sensitive to the physical parameter perturbation of the CO2 storage layer for the carbonate case. However, high frequency modes were observed to be more active for shear wave velocity and thickness variation scenarios in the sandstone reservoir simulations. The simulations demonstrate that the monitoring feasibility increases as the CO2 reservoir layer becomes thicker and the bury depth goes shallower. However, with the geological setting parameters found in existing CCS projects, it is concluded to be a challenge to detect abnormalities in a CO2 storage reservoir by comparing the shift of shear wave velocity profiles that are derived from analysis of surface wave response data. It is, therefore, proposed to consider other microtremor attributes during the development of CO2 monitoring techniques based on passive measurement of microseismicity, which is explored by some researchers.

Theoretical investigation of acoustic wave devices based on different piezoelectric films deposited on silicon carbide

Performances of acoustic wave (AW) devices based on silicon carbide (SiC) substrates are theoretically studied, in which two types of piezoelectric films of ZnO and AlN deposited on 4H-SiC and 3C-SiC substrates are adopted. The phase velocities (PV), electromechanical coupling coefficients (ECC), and temperature coefficients of frequency (TCF) for three AW modes (Rayleigh wave, A0 and S0 modes of Lamb wave) often used in AW devices are calculated based on four types of configurations of interdigital transducers (IDTs). It is found that that the ZnO piezoelectric film is proper for the AW device operating in the low-frequency range because a high ECC can be realized using a thin ZnO film. The AlN piezoelectric film is proper for the device operating in the high-frequency range in virtue of the high PV of AlN, which can increase the finger width of the IDT. Generally, in the low-frequency Lamb wave devices using ZnO piezoelectric films with small normalized thicknesses of films to wavelengths hf/λ, thin SiC substrates can increase ECCs but induce high TCFs simultaneously. In the high-frequency device with a large hf/λ, the S0 mode of Lamb wave based on the AlN piezoelectric film deposited on a thick SiC substrate exhibits high performances by simultaneously considering the PV, ECC, and TCF.

New Features in the Subsurface Structure Model of El Hierro Island (Canaries) from Low-Frequency Microseismic Sounding: An Insight into the 2011 Seismo-Volcanic Crisis

To study the deep structure of El Hierro Island, Canarian Archipelago, we have used a microseismic sounding method (MSM) based on the fact that heterogeneities of the Earth’s crust disturb the spectrum of the low-frequency microseismic field in their vicinity. So, at the Earth’s surface, the spectral amplitudes of definite frequency f above the high-velocity heterogeneities are decreasing, and above the low-velocity ones they are increasing. Moreover, the frequency f is connected with the depth of a heterogeneity H and the velocity of the fundamental mode of Rayleigh waves V R(f) through the relation H ≈ 0.4V R(f)/f. From these relations, the MSM lets us model the subsurface structure in a 3D context by inverting the amplitude-frequency spatial distribution of the microseismic field of low frequency. The validity of the method is shown through of numerical simulations and previous applications with known or verified solutions. This MSM is now used to invert the microseismic data registered in El Hierro Island. The obtained subsurface model reveals two large intrusive bodies beneath the island. Joint interpretation of microseismic and gravimetric data and their comparison with the available geological studies relate the central-eastern intrusive body to the early stage of the island formation. With respect to the western intrusive body, at the depths of 15–25 km, an area with lowest seismic velocities is identified, where we suggest that a modern magmatic reservoir is located. This reservoir could be associated with the recent submarine eruption in October 2011 and the accompanying seismic swarm, which started in July 2011. Several correlations between the shallowest structures identified by the gravity and MSM approaches are also found. Besides the numerical simulation and previous studies of this method, the correlation between gravity results, the MSM model, the geological information and the possible explanation of the features of the seismic swarm through the model obtained offer us a valid proof about the plausibility of the subsurface structures identified from MSM.

FEM simulation of Rayleigh waves for CMOS compatible SAW devices based on AlN/SiO2/Si(1 0 0)

A simulation study of Rayleigh wave devices based on a stacked AlN/SiO2/Si(100) device was carried out. Dispersion curves with respect to acoustic phase velocity, reflectivity and electromechanical coupling efficiency for tungsten W and aluminium Al electrodes and different layer thicknesses were quantified by 2D FEM COMSOL simulations. Simulated acoustic mode shapes are presented. The impact of these parameters on the observed Rayleigh wave modes was discussed. High coupling factors of 2% and high velocities up to 5000m/s were obtained by optimizing the AlN/SiO2 thickness ratio.

Modelling secondary microseismic noise by normal mode summation

Secondary microseisms recorded by seismic stations are generated in the ocean by the interaction of ocean gravity waves. We present here the theory for modelling secondary microseismic noise by normal mode summation. We show that the noise sources can be modelled by vertical forces and how to derive them from a realistic ocean wave model. We then show how to compute bathymetry excitation effect in a realistic earth model by using normal modes and a comparison with Longuet-Higgins approach. The strongest excitation areas in the oceans depends on the bathymetry and period and are different for each seismic mode. Seismic noise is then modelled by normal mode summation considering varying bathymetry. We derive an attenuation model that enables to fit well the vertical component spectra whatever the station location. We show that the fundamental mode of Rayleigh waves is the dominant signal in seismic noise. There is a discrepancy between real and synthetic spectra on the horizontal components that enables to estimate the amount of Love waves for which a different source mechanism is needed. Finally, we investigate noise generated in all the oceans around Africa and show that most of noise recorded in Algeria (TAM station) is generated in the Northern Atlantic and that there is a seasonal variability of the contribution of each ocean and sea.

Investigation into Mass Loading Sensitivity of Sezawa Wave Mode-Based Surface Acoustic Wave Sensors

In this work mass loading sensitivity of a Sezawa wave mode based surface acoustic wave (SAW) device is investigated through finite element method (FEM) simulation and the prospects of these devices to function as highly sensitive SAW sensors is reported. A ZnO/Si layered SAW resonator is considered for the simulation study. Initially the occurrence of Sezawa wave mode and displacement amplitude of the Rayleigh and Sezawa wave mode is studied for lower ZnO film thickness. Further, a thin film made of an arbitrary material is coated over the ZnO surface and the resonance frequency shift caused by mass loading of the film is estimated. It was observed that Sezawa wave mode shows significant sensitivity to change in mass loading and has higher sensitivity (eight times higher) than Rayleigh wave mode for the same device configuration. Further, the mass loading sensitivity was observed to be greater for a low ZnO film thickness to wavelength ratio. Accordingly, highly sensitive SAW sensors can be developed by coating a sensing medium over a layered SAW device and operating at Sezawa mode resonance frequency. The sensitivity can be increased by tuning the ZnO film thickness to wavelength ratio.