Path Length Estimation Research Articles

Path length and sediment transport estimation from DEMs of difference: a signal processing approach

Abstract. The difficulties of measuring bedload transport in gravel-bed rivers have given rise to the morphological method wherein sediment transport can be inferred from changes in riverbed elevation and estimates of the distance traveled by sediment: its path length. Because current methods for estimating path length are time- and labor-intensive, we present a method to estimate a characteristic path length from repeat digital elevation models (DEMs of difference, i.e., DoDs). We propose an automated method to extract the spacing between erosional and depositional sites on the DoD by the application of variational mode decomposition (VMD), a signal processing method, to quantify the spacing as a proxy for path length. We developed this method using flume experiments where bed topography and sediment flux were measured and then applied it to published field data with physical path length measured from tracer measurements. Our sediment transport estimates were not significantly different than the measured sediment flux at lower discharges in the lab. However, we observed an underestimation of sediment flux at the higher discharges in the flume study. We interpret this as a limit of the method in confined settings, where sediment transport becomes decoupled from morphological changes. We also explore how the time between survey acquisitions, the morphological active width relative to the channel width, and DoD thresholding techniques affect the proposed method and the potential issues they pose for the morphological method in general.

Working with GNSS collar data. The importance of pre-analysis when setting the sampling interval

The use of tracking collars implemented with Global Positioning System (GPS) has allowed increasingly complex analyses of the movement and space use of tagged animals. Fundamental to these studies is the setting of parameters related to the technology itself. In this study we assessed the effect of the fix interval on certain ecological variables used to describe behavior, animal movement, home range size and habitat use through parameters such as path length, sinuosity, core area or minimum convex polygon. The aim of the study was to select the most reliable time interval in the study area in the northwest province of Jaén (Spain) for a subsequent use of the collars on wild ungulates. For this purpose, we designed and developed static and moving tests for evaluating 6 GNSS collars, one of them improved by using multi-costellation and multifrequency module, programmed with 5 different time intervals. The kinematic results suggest using caution when programming the collars since differences in the sampling interval leads to important differences in the real trajectories. Movement descriptors varied according to time interval, being longest and more winding with the shortest intervals. Similarly, home range approximations were closer to reality as more fixes were obtained. We progressively underestimated the use of the central core zone and the distance reached by the individual as we increased the time interval. However, there is a trade-off between time, autocorrelation and battery expenditure that is almost the only limiting factor. The experimental results show that a fixing interval of 30 min is an optimal and reliable value to studies focused on the estimation of path length or home range estimation. This value also involves a trade-off between high-resolution data and battery life.

Hybrid UWB-Inertial TDoA-Based Target Tracking With Concentrated Anchors

In this paper, hybrid radio/inertial mobile target tracking for accurate and smooth path estimation is considered. The proposed tracking approach builds upon an Ultra WideBand (UWB)-based positioning algorithm, based on the Linear Hyperbolic Positioning System (LinHPS), with Time Difference of Arrival (TDoA) processing and anchors concentrated on a single hotspot at the center of the environment where the target moves. First, we design an Adaptive Radio-based Extended Kalman Filter (AREKF), which does not require a priori statistical knowledge of the noise in the target movement model and estimates the measurement noise covariance, at each sampling time, according to a proper LookUp Table (LUT). In order to improve the performance of AREKF, we incorporate inertial data collected from the target and propose three “hybrid” radio/inertial algorithms, denoted as Hybrid Inertial Measurement Unit (IMU)-aided Radio-based EKF (HIREKF), Hybrid Noisy Control EKF (HNCEKF), and Hybrid Control EKF (HCEKF). Our results on experimentally acquired paths show that the proposed algorithms achieve an average instantaneous position estimation error on the order of a few centimeters. Moreover, the minimum target path length estimation error, obtained with HCEKF, is on the order of 6% and 1% for two paths with lengths equal to approximately 17 m and 46 m, respectively.

Vascular-centric mapping of in vivo blood oxygen saturation in preclinical models.

Assessing intravascular blood oxygen saturation (SO2) is crucial for characterizing in vivo microenvironmental changes in preclinical models of injury and disease. However, most conventional optical imaging techniques for mapping in vivo SO2 assume or compute a single value of the optical path-length in tissue. This is especially detrimental when mapping in vivo SO2 in experimental disease or wound healing models that are characterized by vascular and tissue remodeling. Therefore, to circumvent this limitation we developed an in vivo SO2 mapping technique that utilizes hemoglobin-based intrinsic optical signal (IOS) imaging combined with a vascular-centric estimation of optical path-lengths. In vivo arterial and venous SO2 distributions derived with this approach closely matched those reported in the literature, while those derived using the single path-length (i.e. conventional) approach did not. Moreover, in vivo cerebrovascular SO2 strongly correlated (R2>0.7) with changes in systemic SO2 measured with a pulse oximeter during hypoxia and hyperoxia paradigms. Finally, in a calvarial bone healing model, in vivo SO2 assessed over four weeks was spatiotemporally correlated with angiogenesis and osteogenesis (R2>0.6). During the early stages of bone healing (i.e. day 10), angiogenic vessels surrounding the calvarial defect exhibited mean SO2 that was elevated by10% (p<0.05) relative to that observed at a later stage (i.e., day 26), indicative of their role in osteogenesis. These correlations were not evident with the conventional SO2 mapping approach. The feasibility of our wide field-of-view in vivo SO2 mapping approach illustrates its potential for characterizing the microvascular environment in applications ranging from tissue engineering to cancer.

Deep Machine Learning for Path Length Characterization Using Acoustic Diffraction

Many fields now perform non-destructive testing using acoustic signals for the detection of objects or features of interest. This detection requires the decision of an experienced technician, which varies from technician to technician. This evaluation becomes even more challenging as the object decreases in size. In this paper, we assess the use of both traditional signal-processing machine learning algorithms, Long Short-Term Memory (LSTM), as well as Convolutional Neural Network (CNN) architectures to approximate acoustic anomalies with an eye toward micro-scale applications such as application to biofilms. The probing signal is generated using a continuous sound wave emitted at controlled frequencies of 1 and 5 MHz through metallic specimens of varying heights each containing an anomaly in the form of a hole. Data are collected as the transmitted signal is sampled at several locations as the wave travels through the specimen. We have developed both a CNN and an LSTM architecture for frequency-domain feature detection and approximation. The CNN models, one for phase and one for amplitude data, take short-distance Fourier transforms (SDFTs) representing the change in the signal over multiple observation points as input. The LSTM model takes the change in phase or amplitude points at each lateral location as a comma-separated value (CSV) input. The models analyze the frequency and spatial changes experienced by each specimen and produce an estimation of the acoustic path length of the anomaly in radians. The models are evaluated using mean-square error and the R-square statistic. All models perform with a fairly high R-square score, the amplitude CNN and LSTM models achieving upwards of a 99% fit and the phase CNN achieving a 97% fit on average for the predicted values. With the performance of these models, we demonstrate that utilizing the transfer function phase and amplitude data to analyze acoustic diffraction patterns leads to the ability to extract, with great precision, features in the input signal that describe the nature of the anomaly.

Assessing 3D vs. 2D habitat metrics in a Mediterranean ecosystem for a wiser wildlife management

Although many ecological studies recognize the importance of the three-dimensional complexity of the landscape, only a small number of investigations incorporate it into their methods. Through an empirical study we assess the extent to which habitat metric patterns vary measured by 3D versus 2D methods, and whether it is possible that variations are associated with scale, topographic complexity or specific animal behavior. We analyze areas and potential animals where the use of 3D approaches is most influential for reliable results in ecological studies. This study was carried out in southern Spain, focusing on game estates covering an area of about 2.241 km2. The topographical features of the areas were obtained from a digital terrain model (DTM) and a scalable terrain model (STM) generated from Lidar data. The estates were categorized according to a preliminary 3D vs. 2D surface increment index (SI index). Then an experimental repeated measures design was carried out to compare three nested scales from 1 to 4 km square plots. The influence of plot topography on the SI index was analyzed statistically. Plots were also compared pair-wise through a post hoc comparison analysis. In order to establish how 3D affects the metrics of the trajectories travelled by different animal species, we simulated movements of Oryctolagus cunniculus, Cervus elaphus, Canis lupus and Lynx pardinus, which move and behave differently in the study area. We simulated 15 replicate paths per specie, assuming 24 fixes per path. Variations in animal trajectories were analyzed by fitting general linear models (GLMs), which also provide a basic method for estimating the potential errors produced by a simple 2D analysis. We obtained significant differences between the projected and geometric surface areas, with an increase of up to 11.15% and an increase in animal paths length up to 5% using 3D approaches. Landscape complexity, rather than species type, is related to animal path metrics differences. However, in species such as deer and lynx the 3D spatial component is basic for an accurate estimate of path length, especially in heterogeneous landscapes. Altitude and slope range are determining factors for considering the 3D approaches as advisable. Our results confirm that ecological studies ignoring three-dimensional (3D) components run the risk of misinterpretation due to the fact that the bi-dimensional (2D) approach underestimates distances and landscape area. The use of 3D methods provides more accurate results than with 2D approaches, with far-reaching implications for a wiser wildlife management and conservation in Mediterranean ecosystems.

Accurate arterial path length estimation for pulse wave velocity calculation in growing children and adolescents

Background: Most adult cardiovascular disease begins in childhood. Given the burgeoning obesity pandemic in children worldwide, there is a need for precise and scalable surveillance methods to detect subclinical cardiovascular disease in children and adolescents. Early detection allows early intervention and intensified primary prevention strategies in affected individuals. Carotid-femoral pulse wave velocity (PWV) directly measures arterial wall stiffness, an early feature of atherosclerosis. Calculation of PWV in growing children requires an accurate estimation of the true distance travelled by the aorto-femoral pressure wave, using surface anatomy landmarks. However, a variety of methods are used to estimate this distance, and these have not previously been investigated in growing children and adolescents. We sought to investigate this by comparing true arterial path length measured on computerized tomography (CT) scans, with a variety of estimations based on surface anatomy landmarks. Methods: Arterial path lengths were measured using multi-planar reformation (MPR) imaging software. These measurements were then compared with the surface anatomy measurements obtained using the same MPR imaging software. The fidelity of a variety of arterial path length estimation methods was tested. Results: The surface anatomy distance between the suprasternal notch and the angle of the mandible (PWV recording site in the neck), should be adjusted using the formula y=4.791+(1.0534*x). This value subtracted from the unadjusted distance from the suprasternal notch to the umbilicus, through the mid-inguinal crease to the femoral PWV recording site, provides the simplest reliable approximation of true intraluminal distance travelled. Conclusions: There is high correlation between the surface anatomy distances and the arterial path lengths they represent; however, these are not equal. Most surface anatomy measurements require adjustment using the formulae that we have provided, to accurately estimate the true distance travelled by the pulse wave.

A hyperspectral imaging system for mapping haemoglobin and cytochrome-c-oxidase concentration changes in the exposed cerebral cortex.

We present a novel hyperspectral imaging (HSI) system using visible and near-infrared (NIR) light on the exposed cerebral cortex of animals, to monitor and quantify in vivo changes in the oxygenation of haemoglobin and in cellular metabolism via measurement of the redox states of cytochrome-c-oxidase (CCO). The system, named hNIR, is based on spectral scanning illumination at 11 bands (600, 630, 665, 784, 800, 818, 835, 851, 868, 881 and 894 nm), using a supercontinuum laser coupled with a rotating Pellin-Broca prism. Image reconstruction is performed with the aid of a Monte Carlo framework for photon pathlength estimation and post-processing correction of partial volume effects. The system is validated on liquid optical phantoms mimicking brain tissue haemodynamics and metabolism, and finally applied in vivo on the exposed cortex of mice undergoing alternating oxygenation challenges. The results of the study demonstrate the capacity of hNIR to map and quantify the haemodynamic and metabolic states of the exposed cortex at microvascular levels. This represents (to the best of our knowledge) the first example of simultaneous mapping and quantification of cerebral haemoglobin and CCO in vivo using visible and NIR HSI, which can potentially become a powerful tool for better understanding brain physiology.

Development of a Non-Destructive Method for Detection of the Juiciness of Pear via VIS/NIR Spectroscopy Combined with Chemometric Methods.

Juiciness is a primary index of pear quality and freshness, which is also considered as important as sweetness for the consumers. Development of a non-destructive detection method for pear juiciness is meaningful for producers and sellers. In this study, visible−near-infrared (VIS/NIR) spectroscopy combined with different spectral preprocessing methods, including normalization (NOR), first derivative (FD), detrend (DET), standard normal variate (SNV), multiplicative scatter correction (MSC), probabilistic quotient normalization (PQN), modified optical path length estimation and correction (OPLECm), linear regression correction combined with spectral ratio (LRC-SR) and orthogonal spatial projection combined with spectral ratio (OPS-SR), was used for comparison in detection of pear juiciness. Partial least squares (PLS) regression was used to establish the calibration models between the preprocessing spectra (650–1100 nm) and juiciness measured by the texture analyzer. In addition, competitive adaptive reweighted sampling (CARS) was used to identify the characteristic wavelengths and simplify the PLS models. All obtained models were evaluated via Monte Carlo cross-validation (MCCV) and external validation. The PLS model established by 19 characteristic variables after LRC-SR preprocessing displayed the best prediction performance with external verification determination coefficient (R2v) of 0.93 and root mean square error (RMSEv) of 0.97%. The results demonstrate that VIS/NIR coupled with LRC-SR method can be a suitable strategy for the quick assessment of juiciness for pears.

Accuracy of frame-shaped samples based measurements of magnetoelastic characteristics of soft magnetic materials

Abstract Frame-shaped samples method is commonly used for testing the magnetoelastic characteristics of bulk soft magnetic materials, such as ferrites or steels. However, unusual shape of such samples limits the accuracy of estimation of magnetic path length as well as efficient magnetic cross-section of the samples carried out accordingly to the IEC 60205:2016 standard. Paper presents, how to overcome this drawback with finite elements method. The quality of estimation of magnetic path length and efficient magnetic cross-section of the frame-shaped samples were verified from the point of view of non-linear magnetization characteristics of soft magnetic materials. The results indicate that proper correction of core’s parameters calculated accordingly to the IEC 60205:2016 standard enable significant increase of measurements accuracy of soft magnetic materials’ characteristics.

Comparative assessment on smart pre-processing methods for extracting information in FT-NIR measured data

Fourier transform near infrared (FT-NIR) spectroscopy is a rapid and straightforward technology to analyse soil properties. However, FT-NIR measurement of soil is always accompanied with latent noises coming from masking and overlaps because of systematic bias and manual errors. Several pre-processing methods were investigated as smart pre-modelling modes for extracting data information and getting rid of the noises. Using least squares support vector regression (LSSVR), the FT-NIR calibration models were established for predicting target nutrients (nitrogen, phosphorus and potassium). With the combined investigation on noise-sensing pre-processing methods, the LSSVR models were automatically optimized by tuning its kernel parameters. Results show that the pre-processing methods were optimized in smart ways of polynomial modifications or function derivatives on quasi-continuous data. The effective pre-processing methods for the three target nutrients were originated from optical path length estimation and correction (OPLEC) and Whittaker smoother (WTK), switched to modification of the regulation polynomials and 1st- or 2nd- order derivatives, respectively. Thus we concluded that the modified OPLEC and the derivative WTK had strong potentials to replace the traditional pre-processing methods in FT-NIR analysis of soil nutrients.

Parametric-scaling optimization of pretreatment methods for the determination of trace/quasi-trace elements based on near infrared spectroscopy.

This work proposes a parametric-scaling strategy to optimize the pretreatments of near infrared (NIR) spectroscopic data, so as to cope with the difficulty of NIR technology in detecting trace or quasi-trace elements. This novel strategy helps enhancing the signal to noise ratio and contributes to extracting features from the raw spectrum, so that the information corresponding to the trace elements could be detected much easier. However, due to the complexity of NIR data, it is difficult to comprehensively evaluate and compare the performance of different pretreatment methods, especially when multiple target components are determined simultaneously. For this reason, we create some comprehensive model indicators to define the goodness of pretreatments in simultaneous multiple detection of trace elements. In this paper two near infrared data sets have been investigated, one is used to determinate the key indices in the primary screening of thalassemia and the other one is used to detect the heavy metal pollutants in farmland soil. Results show that the proposed parametric-scaling optimization strategy can improve the effect of pretreatments in the determination of trace/quasi-trace elements, and the model performance with the optimized pretreated data is significantly superior to that with the raw data. The optimized Savitzky-Golay smoother (SGS) keeps its merits in the real data examples. Especially, the newly emerged methods optical path length estimation and correction (OPLEC) and Whittaker smoother (WTK), as well as their parametric-scaling modified methods, show their advantages in the comparison with other pretreatments. According to the results of our experiments, they have shown promising potential in the NIR rapid analysis of trace/quasi-trace elements in the field of biomedical science and agricultural science. This is expected to be tested for other analytes with larger variation.

Random Node Pair Sampling-Based Estimation of Average Path Lengths in Networks

This article describes how the average path length (APL) of a network is an important metric that provides insights on the interconnectivity in a network and how much time and effort would be required for search and navigation on that network. However, the estimation of APL is time-consuming as its computational complexity scales nonlinearly with the network size. In this article, the authors develop a computationally efficient random node pair sampling algorithm that enables the estimation of APL with a specified precision and confidence. The proposed sampling algorithms provide a speed-up factor ranging from 240-750 for networks with more than 100,000 nodes. The authors also find that the computational time required for estimation APL does not necessarily increase with the network size; it shows an inverted U shape instead.

Random Node Pair Sampling-Based Estimation of Average Path Lengths in Networks

The average path length (APL) of a network is an important metric that provides insights on the interconnectivity in a network and how much time and effort would be required for search and navigation on that network. However, the estimation of APL is time-consuming as its computational complexity scales nonlinearly with the network size. In this paper, we develop a computationally efficient random node pair sampling algorithm that enables the estimation of APL with a specified precision and confidence. The proposed sampling algorithms provide a speed-up factor ranging from 240-750 for networks with more than 100,000 nodes. We also find that the computational time required for estimation APL does not necessarily increase with the network size; it shows an inverted U shape instead.

Arterial path length estimation for heart-to-brachium pulse wave velocity.

As an initial step to better define heart-brachial pulse wave velocity (hbPWV), a novel measure of proximal aortic stiffness, we proposed and validated a simple formula to estimate the arterial path length (Lhb) for hbPWV. In 190 patients aged 19-79 years, the reference Lhb was measured by a three-dimensional MRI. Using multiple regression analysis and fivefold cross-validation, estimation formulae were created using age, sex, and height. hbPWV was calculated as Lhb divided by the time interval from the second heart sound to the dicrotic notch on the brachial pressure wave. In a comparison of hbPWV calculated from MRI-based Lhb, strong correlations and accuracy were observed for the hbPWV obtained from the estimated Lhb by sex and height (r = 0.969, -0.05 ± 0.54 m per s of the mean difference), and even greater correlations and accuracy were observed for the estimated value by age, sex, and height (r = 0.976, 0.00 ± 0.43 m per s of the mean difference). hbPWV values obtained from the straight distance method and the height-based equation showed significant biases with mean differences of 2.64 ± 0.94 and 2.53 ± 0.94 m per s. hbPWV could be evaluated accurately by the simple equation to estimate Lhb.

Technical Note: Feasibility of MRI-based estimation of water-equivalent path length to detect changes in proton range during treatment courses.

To evaluate the feasibility of using on-treatment magnetic resonance imaging (MRI) to detect proton range changes during treatment courses. MRI-based virtual computed tomography (vCT) was generated to calculate water-equivalent path length (WEPL) at the distal surface of a clinical target volume. T2-weighted MR images with and without fat suppression were processed by thresholding and fuzzy c-mean clustering to assign a bulk HU (Hounsfield Unit) to each segment of fat, water, and air. Bony tissues in vCT were transferred from planning CT via a region-based registration. We validated this method using images from three patients (aged 9, 12, and 17 yr old) with pelvic sarcomas who underwent proton therapy. MRI-based WEPL was evaluated against those of repeat CT (rCT) and cone beam CT (CBCT). The vCT agreed well with the rCT, wherein the use of bulk HU and imperfect bone registration contributed to discrepancies. The 99th percentile error of the MRI-based WEPL was up to 3.3 mm for posterior fields when we compensated the effect of mobile air. The gamma index (2-mm WEPL/2-mm pixel distance) was greater than 99% for those fields. We observed larger errors for anterior fields, which were due to bowel gas movement, mismatched respiratory motion, and differences in patient posture between the vCT and rCT. Applied to multiple on-treatment MRI of a patient, the estimated WEPL demonstrated an underrange trend, which was consistent with the CBCT results and the increased patient body circumference. The MRI-based vCT produced highly accurate WEPL estimates, demonstrating the potential of using on-treatment MRI for detecting changes in proton ranges.

Improved Model for Estimation of Spatial Averaging Path Length

In mobile communication systems, the transmitted RF signal is subject to mutually independent deterministic path loss and stochastic multipath and shadow fading. As at each spatial location mostly the composite signal samples are measured, their components are distinguished by averaging out the multipath‐caused signal level variations, while preserving just the ones due to shadowing. The prerequisite for this is the appropriateness of the local area averaging path length that enables obtaining the local mean (composed of mean path loss and shadow fading) and the multipath fading as difference between the composite signal sample and the local mean. However, the so far reported analytical approaches to estimation of the averaging path length are based on considering either the multipath or just the shadow fading, with applicability limited to only specific topologies and frequencies. Therefore, in this paper, the most widely used Lee analytical method is generalized and improved by considering multipath and shadowing concurrently, so providing the general closed‐form elementary‐function based estimation of the optimal averaging path length as a function of common multipath and shadow fading parameters characterizing particular propagation environment. The model enables recommendations for the optimal averaging length for all propagation conditions facing the mobile receiver.

A Comparative Investigation of the Combined Effects of Pre-Processing, Wavelength Selection, and Regression Methods on Near-Infrared Calibration Model Performance.

Near-infrared (NIR) spectroscopy is being widely used in various fields ranging from pharmaceutics to the food industry for analyzing chemical and physical properties of the substances concerned. Its advantages over other analytical techniques include available physical interpretation of spectral data, nondestructive nature and high speed of measurements, and little or no need for sample preparation. The successful application of NIR spectroscopy relies on three main aspects: pre-processing of spectral data to eliminate nonlinear variations due to temperature, light scattering effects and many others, selection of those wavelengths that contribute useful information, and identification of suitable calibration models using linear/nonlinear regression . Several methods have been developed for each of these three aspects and many comparative studies of different methods exist for an individual aspect or some combinations. However, there is still a lack of comparative studies for the interactions among these three aspects, which can shed light on what role each aspect plays in the calibration and how to combine various methods of each aspect together to obtain the best calibration model. This paper aims to provide such a comparative study based on four benchmark data sets using three typical pre-processing methods, namely, orthogonal signal correction (OSC), extended multiplicative signal correction (EMSC) and optical path-length estimation and correction (OPLEC); two existing wavelength selection methods, namely, stepwise forward selection (SFS) and genetic algorithm optimization combined with partial least squares regression for spectral data (GAPLSSP); four popular regression methods, namely, partial least squares (PLS), least absolute shrinkage and selection operator (LASSO), least squares support vector machine (LS-SVM), and Gaussian process regression (GPR). The comparative study indicates that, in general, pre-processing of spectral data can play a significant role in the calibration while wavelength selection plays a marginal role and the combination of certain pre-processing, wavelength selection, and nonlinear regression methods can achieve superior performance over traditional linear regression-based calibration.

High‐resolution and low‐complexity dynamic topology estimation for PLC networks assisted by impulsive noise source detection

Power line communication (PLC) network is a promising solution for smart grid and home broadband applications. However, its topology is not always available and is time-varying due to frequent load changes. The existing topology estimation approaches require installation of a large number of PLC modems or suffer limited resolution. Moreover, they are designed for static PLC networks and need to be repeated at a regular basis to combat topology changes. The authors propose a high-resolution and low-complexity dynamic topology estimation scheme for time-varying indoor PLC networks, which consists of three parts: (i) a time-frequency domain reflectometry based path length estimation method, which requires measurement at a single PLC modem and achieves a much higher resolution than the frequency domain reflectometry based method; (ii) a node-by-node greedy algorithm for topology reconstruction, which is much more computationally efficient than the existing peak-by-peak searching algorithm; (iii) an impulsive noise assisted dynamic topology re-estimation method, which results in a significant complexity reduction over fixed-frequency re-estimation.

Estimation of partial optical path length in the brain in subject-specific head models for near-infrared spectroscopy

Three-dimensional head models with the structures constructed from the MR head images of 40 volunteers were constructed to analyze light propagation in the subject-specific head models. The mean optical path length in the head and the partial optical path length in the brain at 13 fiducial points for each volunteer were estimated to evaluate the intersubject and spatial variability in the optical path lengths. Although the intersubject variability in the optical path lengths is very high, the spatial variability in the average of the mean optical path length and partial optical path length is similar to the previously reported data. The mean optical path length in the head increases, whereas the partial optical path length in the brain decreases with an increase in the depth of the brain surface. The partial optical path length is highly correlated with the depth of the brain surface in comparison to the mean optical path length in the head.