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The power to detect linkage disequilibrium with quantitative traits in selected samples.

Results from power studies for linkage detection have led to many ongoing and planned collections of phenotypically extreme nuclear families. Given the great expense of collecting these families and the imminent availability of a dense diallelic marker map, the families are likely to be used in allelic-association as well as linkage studies. However, optimal selection strategies for linkage may not be equally powerful for association. We examine the power to detect linkage disequilibrium for quantitative traits after phenotypic selection. The results encompass six selection strategies that are in widespread use, including single selection (two designs), affected sib pairs, concordant and discordant pairs, and the extreme-concordant and -discordant design. Selection of sibships on the basis of one extreme proband with high or low trait scores provides as much power as discordant sib pairs but requires the screening and phenotyping of substantially fewer initial families from which to select. Analysis of the role of allele frequencies within each selection design indicates that common trait alleles generally offer the most power, but similarities between the marker- and trait-allele frequencies are much more important than the trait-locus frequency alone. Some of the most widespread selection designs, such as single selection, yield power gains only when both the marker and quantitative trait loci (QTL) are relatively rare in the population. In contrast, discordant pairs and the extreme-proband design provide power for the broadest range of QTL-marker-allele frequency differences. Overall, proband selection from either tail provides the best balance of power, robustness, and simplicity of ascertainment for family-based association analysis.

Alleles↗

Proton stopping powers averaged over beam energy spectra.

Stopping powers averaged over the proton energy spectra at various depths in water and a tissue-like material were calculated for proton beams with initial energies between 50 and 250 MeV. The analysis was made for proton beams with a spread-out Bragg peak (SOBP) to assess in particular the variations of stopping power with depth in the SOBP plateau region. Nuclear interactions were accounted for in determination of the proton energy spectra. In modulated beams, stopping power varies considerably with depth along the plateau region of the spread-out peak. Moreover, stopping powers at the same water depth may differ from each other by up to 20% if referring to modulated or non-modulated beams. Calculations of water-air mass stopping power ratios were also performed, with and without the inclusion of nuclear interactions, for modulated and non-modulated beams. The stopping power ratios do not depend significantly on proton energy, and this makes the effect of accounting for nuclear interactions in proton fluence calculation negligible. For the same reason the stopping power ratios for modulated beams do not appreciably differ from those referred to non-modulated beams with the same Emax at the same depth in water. The present results also show that the dependence of stopping power on energy spread and spatial divergence for proton beams is not negligible in some conditions. Some conclusions are drawn on the possibility of using the residual range as a descriptor of the proton beam quality for different experimental beam conditions.

Algorithms↗

High-intensity focused ultrasound for ex vivo kidney tissue ablation: influence of generator power and pulse duration.

BACKGROUND AND PURPOSE: The therapeutic application of noninvasive tissue ablation by high-intensity focused ultrasound (HIFU) requires precise physical definition of the focal size and determination of control parameters. The objective of this study was to measure the extent of ex-vivo porcine kidney tissue ablation at variable generator parameters and to identify parameters to control lesion size. MATERIALS AND METHODS: The ultrasound waves generated by a cylindrical piezoceramic element (1.04 MHz) were focused at a depth of 100 mm using a parabolic reflector (diameter 100 mm). A needle hydrophone was used to measure the field distribution of the sound pressure. The morphology and extent of tissue necrosis were examined at generator powers of up to 400 W (P(el)) and single pulse durations of as long as 8 seconds. RESULTS: The two-dimensional field distribution resulted in an approximately ellipsoidal focus of 32 x 4 mm (-6 dB). A sharp demarcation between coagulation necrosis and intact tissue was observed. Lesion size was controlled by both the variation of generator power and the pulse duration. At a constant pulse duration of 2 seconds, a generator power of 100 W remained below the threshold doses for inducing a reproducible lesion. An increase in power to as high as 400 W induced lesions with average dimensions of as much as 11.2 x 3 mm. At constant total energy (generator power x pulse duration), lesion size increased at higher generator power. CONCLUSIONS: This ultrasound generator can induce defined and reproducible necrosis in ex-vivo kidney tissue. Lesion size can be controlled by adjusting the generator power and pulse duration. Generator power, in particular, turned out to be a suitable control parameter for obtaining a lesion of a defined size.

Animals↗

The fractional rate of change of ventricular power during isovolumic contraction. Derivation of haemodynamic terms and studies in dogs.

Power and the rate of change of power generated by the ventricle during isovolumic contraction can be calculated on the basis of the energy expended upon compression of the blood within the ventricle during this phase of systole. The ratio of the instantaneous rate of change of power, normalized to instantaneous power, appears to be an expression of particular physiological significance and practical value. This ratio, termed the isovolumic fractional rate of change of ventricular power, describes the capability of the ventricle to sustain an acceleration of energy production relative to instantaneous rates of energy production during isovolumic contraction. In dogs, the fractional rate of change of power changed insignificantly with changes of the preload and afterload; yet its magnitude varied prominently with drug-induced alterations of the contractile state. Although the isovolumic fractional rate of change of power takes a form that in some ways is similar to some previously derived indices of performance, it is independent of assumptions of ventricular geometry, fibre orientation, symmetry of contraction, or elasticity of muscle fibres. The isovolumic fractional rate of change of power is both meaningful and theoretically valid. For these reasons, it appears preferable to empirically-derived indices of performance or indices based upon assumptions which are approximations.

Animals↗

Differential increases in average isokinetic power by specific muscle groups of older women due to variations in training and testing.

BACKGROUND: As a person ages, leg speed and power decrease. These changes are associated with increased falls and reduced gait speed. It has been shown that upper leg training in younger persons results in increased strength and power at the specific speed at which resistance training is applied, although there are only limited data concerning speed-specific training effects on lower leg activity. However, because both upper and lower leg speed and power influence gait and balance, it is important to determine the training speeds that selectively improve these variables in older persons. METHODS: No studies have examined selective speed-specific changes in performance for the upper and lower leg muscles in older individuals. Therefore, we compared shifts in the power-velocity relationship after high-speed (HS) and low-speed (LS) isokinetic training of knee extensors (KE) and flexors (KF), dorsiflexors (DF), and plantar flexors (PF) in community-dwelling women (ages 61 to 75). Subjects were randomly assigned to a HS training, LS training, or control (C) group. Training occurred three times a week for 12 weeks. HS training occurred at 4.73 rad.s(-1) (knee) and 3.14 rad.s(-1) (ankle); LS training for both joint actions was at 1.05 rad.s(-1). RESULTS: HS training improved KE power at intermediate (3.14 rad.s(-1); p =.0007) and high (5.24 rad.s(-1); p =.0004) testing speeds. Neither the HS nor LS group showed any change in KF as a result of the training. Both LS and HS training improved DF power at all testing speeds; however, PF power improved only with LS training and only at 1.05 rad.s(-1) (p =.0132) and 3.14 rad.s(-1) (p =.0310). CONCLUSIONS: Our results suggest that, in older women, lower leg training should occur at lower training speeds than upper leg training. Additionally, attention to differential speed-specific training of knee and ankle actions could improve power production, mobility, balance, and other functional measures in older persons.

Age Factors↗

Maximum and sustained power of extensor digitorum longus muscles from young, adult, and old mice.

We tested the hypothesis that extensor digitorum longus (EDL) muscles in old mice would generate lower maximum powers and sustain lower powers than EDL muscles in young or adult mice. Power was measured in situ at 35 degrees C during single and repeated isovelocity shortening contractions through 10% of fiber length at optimum velocity for power. Sustained power was measured during repeated contractions at a stimulation frequency of 150 Hz. The train rate, and consequently the duty cycle, of repeated contractions was incremented every 5 to 10 min until force could not be maintained throughout the shortening contraction. During single contractions, the maximum absolute powers of muscles in young and old mice were 30% lower than that of adult mice, and power normalized by muscle mass (watts/kg) was 20% lower for muscles in old than for those in adult mice (p less than .05). During repeated contractions, EDL muscles in old and adult mice tolerated lower train rates and duty cycles and sustained lower absolute and normalized powers than EDL muscles in young mice.

Aging↗

Hartley transforms and narrow bessel bandpass filters produce similar power spectra of multiple frequency oscillators and all-night EEG.

Frequency specific power obtained from time and frequency domain analyses are explored in simulated signals and all-night electroencephalogram (EEG). Signals were subjected to a fast Hartley transformation (FHT) and to digital sixth-order Bessel bandpass filters (BDF) of the infinite impulse response type. Numeric values of FHT, BDF and, if suited, authentic frequency specific power were subjected to a Pearson correlation. Frequency bins at 1.6-2.4 Hz (delta), 4.75-5.9 Hz (theta), 9.3-11.5 Hz (alpha), 12.5-14.9 Hz (sigma) and 16.6-19.5 Hz (beta) were investigated. When compared with true power of single frequency oscillators (256-sample windows), frequency specific power of the FHT correlated functionally (1.0) and BDF correlated highly (0.85, delta; 0.99, other bins). For analyses of "white noise", a multiple frequency oscillator and all-night EEG, four rectangular window sizes were applied (256, 512, 1,024 or 2,048 samples). The FHT power correlated better with authentic frequency specific power of "white noise" (256-sample windows) (0.61-0.98) than BDF power (0.67-0.89). With 512-sample windows of "white noise", the estimate of both the FHT (0.69-0.99) and BDF (0.71-0.93) improved. Direct comparison between FHT and BDF frequency specific power obtained from "white noise" or all-night EEG revealed a high degree of compliance between methods for all frequency bins (up to 0.99). For delta, the accord was relatively low for the 256-sample window (EEG, 0.68; "white noise", 0.72), but increased with lengthening window size (2,048-sample: 0.97; 0.99). Averaging of multiple EEG 256-sample windows also increased the agreement between methods.(ABSTRACT TRUNCATED AT 250 WORDS)

Brain↗

Stability of the heart rate power spectrum over time in the conscious dog.

The purpose of this experiment was to test the stability of the heart rate (HR) power spectrum over time in conscious dogs. HR was recorded for 1 h for each of six animals on 2 days. A Fast Fourier transform was used to derive the HR power spectrum for the 12 contiguous 5-min epochs comprising the 1-h recordings. Changes in frequency and amplitude of the various spectral peaks were quantitatively examined. We confirm the presence of two major concentrations of power centered around 0.02 (low frequency peak) and 0.32 Hz (high frequency peak). However, we observed variations in these spectral peaks, especially their amplitudes, both within each hour and from day 1 to day 2. The amplitudes of these two spectral peaks tended to vary reciprocally. HR power spectra based on 5 min of recorded data were also derived from an additional eight animals in both the lying and standing positions; the power spectra from these short recordings were sufficiently sensitive to detect redistributions in power due to changes in posture in all eight dogs. We conclude that: 1) data should be recorded for relatively long periods (e.g., 1 h) to characterize the HR power spectrum; 2) some variability in frequency and amplitude will persist across spectra even when based on longer data bases; 3) care should be taken to ensure that the subject's behavioral state is stable within the recording period; 4) shorter (e.g., 5 min) data bases are not suitable except for detecting relatively robust changes in the HR power spectrum.

Animals↗

Power capability of skeletal muscle to pump blood.

This article describes the various ways in which an electrically stimulated, muscle-wrapped pouch can be used to pump blood. The fundamental equations that allow calculation of the output and input power are developed. The output power in watts is 23.5 X 10(-4) (Pm - PDP)F, where Pm is mean arterial pressure and PDP is pouch diastolic pressure (mmHg); F is the pouch output flow (liters/min). The input power in watts is the oxygen consumed by the muscle in milliliters per second multiplied by 20.1. The oxygen consumed is equal to the product of muscle capillary flow and the A-VO2 difference across the muscle. The ratio of output power to input power is the efficiency; typical values range from 7 to 15%. At rest the typical 350 gm heart produces an output power of about 3.5 milliwatts/gram. Skeletal muscle can produce 10-15 milliwatts per gram; however, it is not known how long conditioned or unconditioned muscle can sustain this power level. In addition, it is not known what power level an electrically stimulated skeletal muscle can sustain for a prolonged period; this represents a gap in our knowledge of cardiac assistance using skeletal muscle.

Assisted Circulation↗

A straightforward algorithm for the quantification of power Doppler signals.

RATIONALE AND OBJECTIVES: To document a simple algorithm that quantifies power Doppler signals by computer-based statistical image analysis. METHODS: Personal computer-based software was developed. Color pixels representing power Doppler signals were separated from the background gray pixels in power Doppler images converted in BMP format. To avoid misregistration during this segmentation, a threshold value is required to adequately distinguish between gray and color pixels. Equating the intensity of each color pixel to a Power Value (PV), a histogram of Power Values in power Doppler images was obtained. RESULTS: The software was designed to allow users to adjust two thresholds (Difference Threshold and Intensity Threshold) by observing the segmentation results in real time. The software calculates the following indexes within any region of interest as defined by the operator: mean PV, peak PV, standard deviation of PV, percentage power Doppler area, and integrated PV per unit area. CONCLUSIONS: Power Doppler signals can be objectively quantified using this straightforward algorithm. This simple and practical method can be applied to comparative or longitudinal studies upon vascularity or blood flow.

Algorithms↗

A model for optimizing cycling performance by varying power on hills and in wind.

The effect of varying power, while holding mean power constant, would have on cycling performance in hilly or windy conditions was analyzed. Performance for a 70-kg cyclist on a 10-km time trial with alternating 1-km segments of uphill and downhill was modeled, with mean VO2 (3, 4, 5 L.min-1), variation in VO2 (5, 10, 15%), and grade (0, 5, 10, 15%) used as independent variables. For the conditions of 4 L.min-1, 10% variation, and 10% grade, results were as follows: finishing time of 22:47.2 with varied power, versus 24:20.3 at constant power, for a time savings of 1 min 33.1 s. Separately, a 40-km time trial with alternating 5-km segments of headwind and tailwind (0, 8, 16, 24 km.h-1) was modeled, with the following results for the conditions of 4 L.min-1, 10% variation, and wind speed of 16 km.h-1: finishing time of 60:21.2 with power variation vs 60:50.2 at constant power, for a time savings of 29 s. Time saved was directly proportional to variation in VO2, grade, and wind speed and was indirectly proportional to mean VO2. In conclusion, the model predicts that significantly time savings could be realized on hilly and windy courses by slightly increasing power on uphill or headwind segments while compensating with reduced power on downhill or tailwind segments.

Bicycling↗

Cross-validation of three jump power equations.

UNLABELLED: The vertical jump-and-reach score is used as a component in the estimation of peak mechanical power in two equations put forth by Lewis and Harman et al. PURPOSE: The purpose of the present study was to: 1) cross-validate the two equations using the vertical jump-and-reach test, 2) develop a more accurate equation from a large heterogeneous population, 3) analyze gender differences and jump protocols, and 4) assess Predicted Residual Sum of Squares (PRESS) as a cross-validation procedure. METHODS: One hundred eight college-age male and female athletes and nonathletes were tested on a force platform. They performed three maximal effort vertical jumps each of the squat jump (SJ) and countermovement jump (CMJ) while simultaneously performing the vertical jump-and-reach test. Regression analysis was used to predict peak power from body mass and vertical jump height. RESULTS: SJ data yielded a better power prediction equation than did CMJ data because of the greater variability in CMJ technique. The following equation was derived from SJ data: Peak Power (W) = 60.7x (jump height cm]) +45.3x(body mass [kg])-2055. This equation revealed greater accuracy than either the Lewis or previous Harman et al. equations and underestimated peak power by less than 1%, with a SEE of 355.0 W using SJ protocol. The use of one equation for both males and females resulted in only a slight (5% of power output) difference between genders. Using CMJ data in the SJ-derived equation resulted in only a 2.7% overestimation of peak power. Cross-validation of regression equations using PRESS reveals accurate and reliable R2 and SEE values. CONCLUSIONS: The SJ equation is a slightly more accurate equation than that derived from CMJ data. This equation should be used in the determination of peak power in place of the formulas developed by both Harman et al. and Lewis. Separate equations for males and females are unnecessary.

Adult↗

Signs of surface torsion and torsional dioptric power.

Although there is agreement in the literature over the magnitude of torsion and torsional dioptric power, there is ambiguity over the signs of those quantities. The purpose of this paper is to define terms in such a way that the ambiguity is removed. Explicit equations are presented for torsion and torsional power along a meridian of a surface. In keeping with common practice in other disciplines, right-handed torsion is chosen to be positive. The components of the dioptric power matrix of thin systems and of the reduced vergence matrix are reinterpreted in terms of curvital and torsional power. In this reinterpretation the off-diagonal components of the matrices remain the torsional power and the reduced torsion along the meridian orthogonal to the reference meridian. However, they become the negatives of those quantities along the reference meridian. In particular, the top-right component can be interpreted as the reduced torsion or the torsional power along the meridian orthogonal to the reference meridian and the bottom-left as the negative of those quantities along the reference meridian. Torsion and torsional power along a meridian, as well as curvature and curvital power, are invariant under change of reference meridian and under spherocylindrical transposition.

Eyeglasses↗

On- and off-eye spherical aberration of soft contact lenses and consequent changes of effective lens power.

BACKGROUND: Soft contact lenses produce a significant level of spherical aberration affecting their power on-eye. A simple model assuming that a thin soft contact lens aligns to the cornea predicts that these effects are similar on-eye and off-eye. METHODS: The wavefront aberration for 17 eyes and 33 soft contact lenses on-eye was measured with a Shack-Hartmann wavefront sensor. The Zernike coefficients describing the on-eye spherical aberration of the soft contact lens were compared with off-eye ray-tracing results. Paraxial and effective lens power changes were determined. RESULTS: The model predicts the on-eye spherical aberration of soft contact lenses closely. The resulting power change for a +/- 7.00 D spherical soft contact lens is +/- 0.5 D for a 6-mm pupil diameter and +/- 0.1 D for a 3-mm pupil diameter. Power change is negligible for soft contact lenses corrected for off-eye spherical aberration. CONCLUSIONS: For thin soft contact lenses, the level of spherical aberration and the consequent power change is similar on-eye and off-eye. Soft contact lenses corrected for spherical aberration in air will be expected to be aberration-free on-eye and produce only negligibly small power changes. For soft contact lenses without aberration correction, for higher levels of ametropia and large pupils, the soft contact lens power should be determined with trial lenses with their power and p value similar to the prescribed lens. The benefit of soft contact lenses corrected for spherical aberration depends on the level of ocular spherical aberration.

Adult↗

Improved diagnosing of small hepatocellular carcinomas by echo-enhanced power Doppler sonography in patients with cirrhosis.

BACKGROUND AND AIMS: Echo-enhanced power Doppler sonography is an increasingly used procedure for the differentiation of liver tumours. The aim of this prospective study was to investigate the accuracy of echo-enhanced power Doppler sonography in comparison with conventional ultrasound and fundamental power Doppler sonography in diagnosing hepatocellular carcinomas and regenerative nodules in patients with cirrhosis. SUBJECTS AND METHODS: Eighty-seven patients with cirrhosis and 90 liver tumours at conventional ultrasound were included in the study, selected from 103 consecutive patients with a mean age of 60 years (range 23-87 years) who presented to our department from January 1998 through January 2002. Sonography was performed by an experienced examiner, who was unaware of the clinical diagnosis. The exact diagnosis was based upon histological evidence from biopsy examination, laboratory results, and/or a follow-up of at least 18 months. RESULTS: There were 65 hepatocellular carcinomas, 21 regenerative nodules, and four metastases in the study group. Only 43% of the carcinomas (mainly nodules > 3 cm in diameter) could be classified correctly by conventional ultrasound or fundamental power Doppler sonography. However, 46% of the malign lesions (mainly nodules < 3 cm in diameter) were not differentiable. All non-differentiable tumours were classified correctly by echo-enhanced power Doppler sonography. The overall sensitivity of echo-enhanced power Doppler sonography with respect to diagnosing hepatocellular carcinoma was 89%; its specificity was 60%. The corresponding values for regenerative nodules were 71% and 100%, respectively. CONCLUSIONS: Only hepatocellular carcinomas greater than 3 cm in size can be diagnosed with a high accuracy by conventional ultrasound or fundamental power Doppler sonography. However, small malignant lesions in cirrhotic livers are often not differentiable with these techniques. With echo-enhanced power Doppler sonography, the differentiation of small hepatocellular carcinomas can be improved. However, histology is the standard of reference.

Adult↗

Non-invasive detection of coronary artery stenosis: a comparison among power-Doppler contrast echo, 99Tc-Sestamibi SPECT and echo wall-motion analysis.

BACKGROUND: Power-Doppler imaging is a recently developed method for myocardial contrast echocardiography (MCE). It can selectively evaluate the signal coming from an ultrasound contrast agent, allowing myocardial perfusion studies. OBJECTIVE: To compare the ability of power-Doppler MCE with stress-echo wall-motion and nuclear scan imaging (SPECT) to assess myocardial ischaemia during pharmacological stress, using coronary angiography as reference. METHODS: In 25 patients the three non-invasive imaging modalities were acquired during a single dipyridamole stress test (so as to avoid stress variations). Power-Doppler MCE was acquired using continuous intravenous infusion of Levovist. Echo wall-motion was acquired too. At peak stress 99Tc-Sestamibi was injected; stress SPECT images were acquired 30 min after injection. RESULTS: Power-Doppler MCE and SPECT showed 84% concordance (21 of 25 patients; kappa=0.67) for detection of ischaemia. Concordance based on coronary artery territories for normal perfusion versus fixed defects versus reversible defects was 92% (69 of 75; kappa=0.81), with 100% for left anterior descending, 92% for right coronary artery and 84% for circumflex. Power-Doppler MCE had lower sensitivity than SPECT (89 versus 100%) but higher specificity (100 versus 88%) for identification of stenotic (> or = 70%) coronary arteries as assessed by angiography. Echo wall-motion analysis showed the lowest sensitivity (68%) with 100% specificity. Accuracy was 94% for both power-Doppler MCE and SPECT, and 83% for wall-motion analysis. CONCLUSION: Power-Doppler MCE is a sensitive and specific method for identification of myocardial perfusion during pharmacological stress. Accuracy of power-Doppler MCE for stenotic coronary arteries appears to be slightly higher than stress-echo wall-motion and similar to SPECT.

Aged↗

How safe is high-power ultrasonic dissection?

OBJECTIVE: To evaluate the safety of ultrasonic dissection. SUMMARY BACKGROUND DATA: High-power ultrasonic dissection is in widespread use for both open and laparoscopic operations and is generally perceived to carry a low risk of collateral damage, but there is no published evidence for this. METHODS: Under controlled experimental conditions, ultrasonic dissections were performed in pigs using Ultracision (Ethicon) or Autosonix (Tyco/USSC) at the three power settings (3, 4, and 5) in random fashion to mobilize the cardia and fundus, bile duct, hepatic artery, portal vein, aorta from the inferior vena cava, renal vessels, colon, and ureters. The dissections (open and laparoscopic) were carried out on pigs at each power setting with each device. Thermal mapping of the tissues during dissection was performed with an infrared thermal camera and associated software. The animals were killed at the end of each experiment and specimens were harvested for quantitative histology. RESULTS: Extreme and equivalent temperature gradients were generated by ultrasonic dissection with both systems. Heat production was directly proportional to the power setting and the activation time. The core body temperature of the animals after completion of the laparoscopic dissections rose by an average of 2.3 degrees C. The zone around the jaws that exceeded 60 degrees C with continuous ultrasonic dissection for 10 to 15 seconds at level 5 measured 25.3 and 25.7 mm for Ultracision and Autosonix, respectively. At this power setting and an activation time of 15 seconds, the temperature 1.0 cm away from the tips of the instrument exceeded 140 degrees C. Although there was no discernible macroscopic damage, these thermal changes were accompanied by significant histologic injury that extended to the media of large vessels and caused partial- to full-thickness mural damage of the cardia, ureter, and bile duct. Collateral damage was absent or insignificant after dissections at power level 3 with both systems and an activation time not exceeding 5 seconds. CONCLUSIONS: High-power ultrasonic dissections at level 5 and to a lesser extent level 4 result in considerable heat production that causes proximity collateral damage to adjacent tissues when the continuous activation time exceeds 10 seconds. Ultrasonic dissections near important structures should be conducted at level 3. At power levels of 4 and 5, the ultrasonic energy bursts to the tissue should not exceed 5 seconds at any one time.

Animals↗

Power-law tail probabilities of drainage areas in river basins.

We examine the appearance of power-law behavior in rooted tree graphs in the context of river networks. It has long been observed that the tails of statistical distributions of upstream areas in river networks, measured above every link, obey a power-law relationship over a range of scales. We examine this behavior by considering a subset of all links, defined as those links which drain complete Strahler basins, where the Strahler order defines a discrete measure of scale, for self-similar networks with both deterministic and random topologies. We find an excellent power-law structure in the tail probabilities for complete Strahler basin areas, over many ranges of scale. We show analytically that the tail probabilities converge to a power law under the assumptions of (1) simple scaling of the distributions of complete Strahler basin areas and (2) application of Horton's law of stream numbers. The convergence to a power law does not occur for all underlying distributions, but for a large class of statistical distributions which have specific limiting properties. For example, underlying distributions which are exponential and gamma distributed, while not power-law scaling, produce power laws in the tail probabilities when rescaled and sampled according to Horton's law of stream numbers. The power-law exponent is given by the expression phi=ln(R(b))/ln(R(A)), where R(b) is the bifurcation ratio and R(A) is the Horton area ratio. It is commonly observed that R(b) approximately equal R(A) in many river basins, implying that the tail probability exponent for complete Strahler basins is close to 1.0.

Journal Article↗