Gaussian Response Function Research Articles

Hazard Model: Epidemic-Type Aftershock Sequence (ETAS) Model for Hungary

In this article we present a space–time epidemic-type aftershock sequence (ETAS) model for the area of Hungary, motivated by the goal of its application in insurance risk models. High-quality recent instrumental data from the period 1996–2021 are used for model parameterization, including data from the recent nearby Zagreb and Petrinja event sequences. In the earthquake-triggering equations of our ETAS model, we replace the commonly used modified Omori law with the more recently proposed stretched exponential time response form, and a Gaussian space response function is applied with a variance add-on for epicenter error. After this model was tested against the observations, an appropriate overall fit for magnitudes M≥3.0 was found, which is sufficient for insurance applications, although the tests also show deviations at the M=2.5 threshold. Since the data used for parameterization are dominated by Croatian earthquake sequences, we also downscale the model to regional zones via parameter adjustments. In the downscaling older historical data are incorporated for a better representation of the key events within Hungary itself. Comparison of long-term large event numbers in simulated catalogues versus historical data shows that the model fit by zone is improved by the downscaling.

Linear multi-scale modeling of diffusion MRI data: A framework for characterization of oriented structures across length scales.

Diffusion-weighted magnetic resonance imaging (DW-MRI) has evolved to provide increasingly sophisticated investigations of the human brain's structural connectome in vivo. Restriction spectrum imaging (RSI) is a method that reconstructs the orientation distribution of diffusion within tissues over a range of length scales. In its original formulation, RSI represented the signal as consisting of a spectrum of Gaussian diffusion response functions. Recent technological advances have enabled the use of ultra-high b-values on human MRI scanners, providing higher sensitivity to intracellular water diffusion in the living human brain. To capture the complex diffusion time dependence of the signal within restricted water compartments, we expand upon the RSI approach to represent restricted water compartments with non-Gaussian response functions, in an extended analysis framework called linear multi-scale modeling (LMM). The LMM approach is designed to resolve length scale and orientation-specific information with greater specificity to tissue microstructure in the restricted and hindered compartments, while retaining the advantages of the RSI approach in its implementation as a linear inverse problem. Using multi-shell, multi-diffusion time DW-MRI data acquired with a state-of-the-art 3T MRI scanner equipped with 300 mT/m gradients, we demonstrate the ability of the LMM approach to distinguish different anatomical structures in the human brain and the potential to advance mapping of the human connectome through joint estimation of the fiber orientation distributions and compartment size characteristics.

On Low-Energy Tail Distortions in the Detector Response Function of X-Ray Microcalorimeter Spectrometers

We use narrow spectral lines from the X-ray spectra of various highly charged ions to measure low-energy tail-like deviations from a Gaussian response function in a microcalorimeter X-ray spectrometer with Au absorbers at energies from 650 to 3320 eV. We review the literature on low-energy tails in other microcalorimeter X-ray spectrometers and present a model that explains all the reviewed tail fraction measurements. In this model, a low-energy tail arises from the combination of electron escape and energy trapping associated with Bi X-ray absorbers.

Ion temperature measurements of indirect-drive implosions with the neutron time-of-flight detector on SG-III laser facility

The accuracy of the determination of the burn-averaged ion temperature of inertial confinement fusion implosions depends on the unfold process, including deconvolution and convolution methods, and the function, i.e., the detector response, used to fit the signals measured by neutron time-of-flight (nToF) detectors. The function given by Murphy et al. [Rev. Sci. Instrum. 68(1), 610-613 (1997)] has been widely used in Nova, Omega, and NIF. There are two components, i.e., fast and slow, and the contribution of scattered neutrons has not been dedicatedly considered. In this work, a new function, based on Murphy's function has been employed to unfold nToF signals. The contribution of scattered neutrons is easily included by the convolution of a Gaussian response function and an exponential decay. The ion temperature is measured by nToF with the new function. Good agreement with the ion temperature determined by the deconvolution method has been achieved.

Evaluation of spectral scale effects in estimation of vegetation leaf area index using spectral indices methods

Spectral index methodology has been widely used in Leaf Area Index (LAI) retrieval at different spatial scales. There are differences in the spectral response of different remote sensors and thus spectral scale effect generated during the use of spectral indices to retrieve LAI. In this study, PROSPECT, leaf optical properties model and Scattering by Arbitrarily Inclined Layers (SAIL) model, were used to simulate canopy spectral reflectance with a bandwidth of 5 nm and a Gaussian spectral response function was employed to simulate the spectral data at six bandwidths ranging from 10 to 35 nm. Additionally, for bandwidths from 5 to 35 nm, the correlation between the spectral index and LAI, and the sensitivities of the spectral index to changes in LAI and bandwidth were analyzed. Finally, the reflectance data at six bandwidths ranging from 40 to 65 nm were used to verify the spectral scale effect generated during the use of the spectral index to retrieve LAI. Results indicate that Vegetation Index of the Universal Pattern Decomposition (VIUPD) had the highest accuracy during LAI retrieval. Followed by Normalized Difference Vegetation Index (NDVI), Modified Simple Ratio Indices (MSRI) and Triangle Vegetation Index (TVI), although the coefficient of determination R2 was higher than 0.96, the retrieved LAI values were less than the actual value and thus lacked validity. Other spectral indices were significantly affected by the spectral scale effect with poor retrieval results. In this study, VIUPD, which exhibited a relatively good correlation and sensitivity to LAI, was less affected by the spectral scale effect and had a relatively good retrieval capability. This conclusion supports a purported feature independent of the sensor of this model and also confirms the great potential of VIUPD for retrieval of physicochemical parameters of vegetation using multi-source remote sensing data.

A NON-PARAMETRIC APPROACH TO CONSTRAIN THE TRANSFER FUNCTION IN REVERBERATION MAPPING

ABSTRACT Broad emission lines of active galactic nuclei stem from a spatially extended region (broad-line region, BLR) that is composed of discrete clouds and photoionized by the central ionizing continuum. The temporal behaviors of these emission lines are blurred echoes of continuum variations (i.e., reverberation mapping, RM) and directly reflect the structures and kinematic information of BLRs through the so-called transfer function (also known as the velocity-delay map). Based on the previous works of Rybicki and Press and Zu et al., we develop an extended, non-parametric approach to determine the transfer function for RM data, in which the transfer function is expressed as a sum of a family of relatively displaced Gaussian response functions. Therefore, arbitrary shapes of transfer functions associated with complicated BLR geometry can be seamlessly included, enabling us to relax the presumption of a specified transfer function frequently adopted in previous studies and to let it be determined by observation data. We formulate our approach in a previously well-established framework that incorporates the statistical modeling of continuum variations as a damped random walk process and takes into account long-term secular variations which are irrelevant to RM signals. The application to RM data shows the fidelity of our approach.

SU‐F‐T‐548: An Extraction Method for Correction of Detector Volume Effect in Small Field Profile Measurement

Purpose:To investigate the appropriateness of Gaussian detector response function in SRS profile measurement.Methods:In‐air profile data for a 6MV beam were collected by using ionization chambers and radiochromic film. Two ionization chambers (FC65‐G(IBA) and PinPoint(PTWTW31009)) were used to scan the beam profile. Film was used to measure the beam penumbra. A half‐blocked beam was acquired with stepping size of 0.2mm and gradient of the measured profile represents the detector response function. The field widths of 2cm and 3cm were measured at SAD. Two detector response functions were evaluated, i.e. a traditional single Gaussian and a three‐Gaussian symmetric function. Two corresponding models were used to fit measured data. One was error functions with one shaping parameter. The other model was a symmetrical superposition of three sets of error functions with fixed separation and two different shaping parameters. In both fitting, resultant Gaussian kernels and corresponding σs could be calculated analytically. SNR of the fitted curves were obtained and compared.Results:Our measured data indicated that the ionization chambers had a three‐peak response function. From the first modeling, SNR for FC65‐G were 36.0dB and 38.3dB for 2cm and 3cm field respectively. The corresponding SNR for the PinPoint were 43.6dB and 42.5dB. From the second modeling, SNR for FC65‐G were 42.5dB and 47.4dB for 2cm and 3cm field respectively. The corresponding SNR for the PinPoint were 44.0dB and 44.5 dB. Three‐gaussian model had an observable improvement for FC65‐G chamber. Film measured penumbra was 3–5 times smaller than that of ionization‐chambers.Conclusion:Superposition of three Gaussian kernels resembles the detector response due to central electrode and chamber wall and results in a better fitting to the measurement data than a single Gaussian kernel. Our study indicated that the Three‐Gaussian model was more suitable for the detector with sigma comparable to penumbra width.

A Fourier approach to pulse pile-up in photon-counting x-ray detectors.

An analytic Fourier approach to predict the expected number of counts registered in a photon-counting detector subject to pulse pile-up for arbitrary photon flux, detector response function, and pulse-shape is presented. The analysis provides a complete forward model for energy-sensitive, photon-counting x-ray detectors for spectral computed tomography. The formalism of the stochastic theory of the expected frequency of level crossings of shot noise processes is applied to the pulse pile-up effect and build on a recently published analytic Fourier representation of the level crossing frequency of shot noise processes with piece-wise continuous kernels with jumps. The general analytic result is validated by a Monte Carlo simulation for pulses of the form g(t) = e(-t/τ) (t > 0) and a Gaussian detector response function. The Monte Carlo simulations are in excellent agreement with the analytic predictions of photon counts within the numerical accuracy of the calculations. The phenomenon of pulse pile-up is identified with the level-crossing problem of shot noise processes and an exact, analytic formula for the expected number of counts in energy-sensitive, photon-counting x-ray detectors for arbitrary photon flux, response function, and pulse-shapes is derived. The framework serves as a theoretical foundation for future works on pulse pile-up.

Effect of competing cubic-quintic nonlinearities on the modulational instability in nonlocal Kerr-type media

We investigate analytically and numerically the modulational instability (MI) of plane waves under competing nonlocal cubic-local quintic nonlinearities. The generic properties of the MI gain spectra are then demonstrated for the Gaussian response function, exponential response function, and rectangular response function. Special attention is paid to competing nonlocal cubic-local quintic nonlinearities on the MI. We observe that the focusing local quintic nonlinearity increases the growth rate and bandwidth of instability contrary to the small values of defocusing local quintic nonlinearity which decrease the growth rate and bandwidth of instability. Numerical simulations of the full model equation describing the dynamics of the waves are been carried out and leads to the development of pulse trains, depending upon the sign the quintic nonlinearity.

Detailed modeling of the statistical uncertainty of Thomson scattering measurements

The uncertainty of electron density and temperature fluctuation measurements is determined by statistical uncertainty introduced by multiple noise sources. In order to quantify these uncertainties precisely, a simple but comprehensive model was made of the noise sources in the MST Thomson scattering system and of the resulting variance in the integrated scattered signals. The model agrees well with experimental and simulated results. The signal uncertainties are then used by our existing Bayesian analysis routine to find the most likely electron temperature and density, with confidence intervals.In the model, photonic noise from scattered light and plasma background light is multiplied by the noise enhancement factor (F) of the avalanche photodiode (APD). Electronic noise from the amplifier and digitizer is added. The amplifier response function shapes the signal and induces correlation in the noise. The data analysis routine fits a characteristic pulse to the digitized signals from the amplifier, giving the integrated scattered signals. A finite digitization rate loses information and can cause numerical integration error.We find a formula for the variance of the scattered signals in terms of the background and pulse amplitudes, and three calibration constants. The constants are measured easily under operating conditions, resulting in accurate estimation of the scattered signals' uncertainty. We measure F ≈ 3 for our APDs, in agreement with other measurements for similar APDs. This value is wavelength-independent, simplifying analysis. The correlated noise we observe is reproduced well using a Gaussian response function. Numerical integration error can be made negligible by using an interpolated characteristic pulse, allowing digitization rates as low as the detector bandwidth. The effect of background noise is also determined.

Performance parameters of a liquid filled ionization chamber array

In this work, the properties of the two-dimensional liquid filled ionization chamber array Octavius 1000SRS (PTW-Freiburg, Germany) for use in clinical photon-beam dosimetry are investigated. Measurements were carried out at an Elekta Synergy and Siemens Primus accelerator. For measurements of stability, linearity, and saturation effects of the 1000SRS array a Semiflex 31013 ionization chamber (PTW-Freiburg, Germany) was used as a reference. The effective point of measurement was determined by TPR measurements of the array in comparison with a Roos chamber (type 31004, PTW-Freiburg, Germany). The response of the array with varying field size and depth of measurement was evaluated using a Semiflex 31010 ionization chamber as a reference. Output factor measurements were carried out with a Semiflex 31010 ionization chamber, a diode (type 60012, PTW-Freiburg, Germany), and the detector array under investigation. The dose response function for a single detector of the array was determined by measuring 1 cm wide slit-beam dose profiles and comparing them against diode-measured profiles. Theoretical aspects of the low pass properties and of the sampling frequency of the detector array were evaluated. Dose profiles measured with the array and the diode detector were compared, and an intensity modulated radiation therapy (IMRT) field was verified using the Gamma-Index method and the visualization of line dose profiles. The array showed a short and long term stability better than 0.1% and 0.2%, respectively. Fluctuations in linearity were found to be within ±0.2% for the vendor specified dose range. Saturation effects were found to be similar to those reported in other studies for liquid-filled ionization chambers. The detector's relative response varied with field size and depth of measurement, showing a small energy dependence accounting for maximum signal deviations of ±2.6% from the reference condition for the setup used. The σ-values of the Gaussian dose response function for a single detector of the array were found to be (0.72±0.25) mm at 6 MV and (0.74±0.25) mm at 15 MV and the corresponding low pass cutoff frequencies are 0.22 and 0.21 mm(-1), respectively. For the inner 5×5 cm2 region and the outer 11×11 cm2 region of the array the Nyquist theorem is fulfilled for maximum sampling frequencies of 0.2 and 0.1 mm(-1), respectively. An IMRT field verification with a Gamma-Index analysis yielded a passing rate of 95.2% for a 3 mm∕3% criterion with a TPS calculation as reference. This study shows the applicability of the Octavius 1000SRS in modern dosimetry. Output factor and dose profile measurements illustrated the applicability of the array in small field and stereotactic dosimetry. The high spatial resolution ensures adequate measurements of dose profiles in regular and intensity modulated photon-beam fields.

The dose response functions of ionization chambers in photon dosimetry – Gaussian or non-Gaussian?

This study is concerned with the spatial resolution of air-filled ionization chambers in photon-beam dosimetry, i.e. with their dose response functions. These act as convolution kernels K(x,y), transforming true dose profiles D(x,y) into the measured signal profiles M(x,y). One-dimensional dose response functions have been experimentally determined for nine types of cylindrical ionization chambers both in their lateral and longitudinal directions, as well as across two plane-parallel chambers and for the single chambers of two 2D arrays. All these 1D dose response functions are closely described by Gaussian functions. The associated energy-dependent values of the standard deviations σ have been measured for 6 and 15 MV photons with an uncertainty of 0.02mm. At depths beyond secondary electron fluence build-up, there was no detectable depth dependence of the σ values. The general occurrence of Gaussian dose response functions, their extension beyond the geometrical boundaries of the chambers, and the energy dependence of their standard deviations can be understood by considering the underlying system of convolutions, which is the origin of the influences of secondary electron transport. Monte-Carlo simulations of the convolution kernels for a cylindrical, a square, and a flat ionization chamber and their Fourier analysis have been employed to show that the Gaussian convolution kernels are approximations to the true dose response functions, valid in the clinically relevant domain of the spatial frequency. This paper is conceived as the starting point for the deconvolution methods to be described in a further publication.

SU-E-T-177: Deconvolution of Line Dose Profiles with Ionization Chambers - Experiences with the First Software Implementation in Mephisto 3.0.

To evaluate the first implementation of a deconvolution algorithm in a commercial water phantom scanning software. Line dose profile measurements in a water phantom are an essential part of a quality assurance system and in base data measurements in radiotherapy. Usually these measurements are performed with waterproof ionization chambers of various sizes. These dose profile measurements are broadened by the Gaussian response functions of the detectors. In recent studies we showed that the undisturbed line dose profiles can be reconstruced by iterative deconvolution of the measured signal profiles with the Gaussian detector response functions. Recently, the proposed method was implemented in Mephisto 3.0. In this work we analyze the applicability and the limits of the deconvolution algorithm for several chambers by comparing the result with diode measurements. As long as the dose gradient becomes not too steep the deconvolution algorithm is able to reconstruct the undisturbed dose profiles with sufficient accuracy. Deviations occur for smallest field sizes in which the width of the detector's lateral response functions reaches the dimensions of the field. A simple chart for those limits is derived. The implemented deconvolution algorithm allows a fast and simple correction of measured dose profiles broadened by the volume effect of the ionization chambers. It offers therefore for the first time a clinical deconvolution of the profiles on a regular base and by this the implementation of undisturbed base data in the treatment planning systems as well as in the quality assurance process in modern radiotherapy.

SU-E-T-114: Characterization of the Spatial Response Functions of Ionization Chambers for Photon Beam Dosimetry.

The finite extension of an ionization chamber gives rise to a spatial averaging effect, known as the "volume effect". In order to provide the appropriate corrections, the response functions along its lateral and longitudinal directions are characterized using Gaussian distributions, whose standard deviations slat and slong have been determined for a large set of clinical dosimeters. Nine cylindrical ionization chambers, two parallel-plate chambers and two 2D ionization chamber arrays have been examined by scanning rectangular photon fields along their short axes. The true profiles D(x) were known from scans with a small Si diode. The ionization chambers were aligned with their symmetry axes either perpendicular or parallel to the scan direction in order to obtain slat and slong separately. In a search process, D(x) was numerically convolved with normalized one-dimensional Gaussian kernels K(x) of varying s. The best fit between the convolution product D(x) * K(x) and the measured profile M(x) of the ionization chamber was used to determine parameters slat and slong of the Gaussian kernels. For both the lateral and longitudinal directions, very good agreement was found between M(x) and the convolution products of D(x) with Gaussian kernels K(x). For all chambers, their 2s values are similar to the cavity dimensions, which means that the "tails" of the Gaussian response functions reach into the exterior of the chambers, - an effect of the ranges of the secondary electrons. At higher photon energies response functions K(x) are slightly wider, but no detectable depth dependence has been observed. We have shown that the response functions of ionization chambers can be described by Gaussian distributions, confirming earlier observations, and we determined their standard deviations in both the lateral and longitudinal directions. Using these response functions, appropriate correction methods determined to eliminate the volume effect can be applied.

Three-dimensional spatiotemporal solitary waves in strongly nonlocal media

We construct a class of three-dimensional strongly nonlocal spatiotemporal solitary waves of the nonlocal nonlinear Schrödinger equation, by using superpositions of single accessible solitons as initial conditions. Evolution of such solitary waves, termed the accessible light bullets, is studied numerically by choosing specific values of topological charges and other solitonic parameters. Our numerical results reveal that in strongly nonlocal nonlinear media with a Gaussian response function, different classes of accessible spatiotemporal solitons can be generated and controlled by tailoring different soliton parameters.

Accessible solitary wave families of the generalized nonlocal nonlinear Schrödinger equation

Two-dimensional accessible solitary wave families of the generalized nonlocal nonlinear Schrodinger equation are obtained by utilizing superpositions of various single accessible solitary solutions. Specific values of soliton parameters are selected as initial conditions and the superposition of known single solitary solutions in the highly nonlocal regime are launched into the nonlocal nonlinear medium with a Gaussian response function, to obtain novel numerical solitary solutions of improved stability. Our results reveal that in nonlocal media with the Gaussian response the higher-order spatial accessible solitary families can exist in various forms, such as asymmetric necklace, asymmetric fractional, and symmetric multipolar necklace solitons.

SPATIAL SOLITARY WAVES IN GENERALIZED NON-LOCAL NONLINEAR MEDIA

We introduce a very general self-trapped beam solution to the generalized non-local nonlinear Schrödinger equation in cylindrical coordinates, by combining superpositions of the known single accessible soliton solutions. Specific values of soliton parameters are selected as initial conditions and superpositions of the single soliton solutions in the highly non-local regime are launched into the non-local nonlinear medium with Gaussian response function, to obtain novel numerical solitary wave solutions. Novel solitary waves have been constructed that exhibit unique features whose intensity pattern is formed by various figures.

Receptive Field Dynamics Underlying MST Neuronal Optic Flow Selectivity

Optic flow informs moving observers about their heading direction. Neurons in monkey medial superior temporal (MST) cortex show heading selective responses to optic flow and planar direction selective responses to patches of local motion. We recorded MST neuronal responses to a 90 x 90 degrees optic flow display and to a 3 x 3 array of local motion patches covering the same area. Our goal was to test the hypothesis that the optic flow responses reflect the sum of the local motion responses. The local motion responses of each neuron were modeled as mixtures of Gaussians, combining the effects of two Gaussian response functions derived using a genetic algorithm, and then used to predict that neuron's optic flow responses. Some neurons showed good correspondence between local motion models and optic flow responses, others showed substantial differences. We used the genetic algorithm to modulate the relative strength of each local motion segment's responses to accommodate interactions between segments that might modulate their relative efficacy during co-activation by global patterns of optic flow. These gain modulated models showed uniformly better fits to the optic flow responses, suggesting that coactivation of receptive field segments alters neuronal response properties. We tested this hypothesis by simultaneously presenting local motion stimuli at two different sites. These two-segment stimuli revealed that interactions between response segments have direction and location specific effects that can account for aspects of optic flow selectivity. We conclude that MST's optic flow selectivity reflects dynamic interactions between spatially distributed local planar motion response mechanisms.

Assessment of the setup dependence of detector response functions for mega‐voltage linear accelerators

Accurate modeling of beam profiles is important for precise treatment planning dosimetry. Calculated beam profiles need to precisely replicate profiles measured during machine commissioning. Finite detector size introduces perturbations into the measured profiles, which, in turn, impact the resulting modeled profiles. The authors investigate a method for extracting the unperturbed beam profiles from those measured during linear accelerator commissioning. In-plane and cross-plane data were collected for an Elekta Synergy linac at 6 MV using ionization chambers of volume 0.01, 0.04, 0.13, and 0.65 cm3 and a diode of surface area 0.64 mm2. The detectors were orientated with the stem perpendicular to the beam and pointing away from the gantry. Profiles were measured for a 10 x 10 cm2 field at depths ranging from 0.8 to 25.0 cm and SSDs from 90 to 110 cm. Shaping parameters of a Gaussian response function were obtained relative to the Edge detector. The Gaussian function was deconvolved from the measured ionization chamber data. The Edge detector profile was taken as an approximation to the true profile, to which deconvolved data were compared. Data were also collected with CC13 and Edge detectors for additional fields and energies on an Elekta Synergy, Varian Trilogy, and Siemens Oncor linear accelerator and response functions obtained. Response functions were compared as a function of depth, SSD, and detector scan direction. Variations in the shaping parameter were introduced and the effect on the resulting deconvolution profiles assessed. Up to 10% setup dependence in the Gaussian shaping parameter occurred, for each detector for a particular plane. This translated to less than a +/- 0.7 mm variation in the 80%-20% penumbral width. For large volume ionization chambers such as the FC65 Farmer type, where the cavity length to diameter ratio is far from 1, the scan direction produced up to a 40% difference in the shaping parameter between in-plane and cross-plane measurements. This is primarily due to the directional difference in penumbral width measured by the FC65 chamber, which can more than double in profiles obtained with the detector stem parallel compared to perpendicular to the scan direction. For the more symmetric CC13 chamber the variation was only 3% between in-plane and cross-plane measurements. The authors have shown that the detector response varies with detector type, depth, SSD, and detector scan direction. In-plane vs. cross-plane scanning can require calculation of a direction dependent response function. The effect of a 10% overall variation in the response function, for an ionization chamber, translates to a small deviation in the penumbra from that of the Edge detector measured profile when deconvolved. Due to the uncertainties introduced by deconvolution the Edge detector would be preferable in obtaining an approximation of the true profile, particularly for field sizes where the energy dependence of the diode can be neglected. However, an averaged response function could be utilized to provide a good approximation of the true profile for large ionization chambers and for larger fields for which diode detectors are not recommended.

Semi‐empirical procedures for correcting detector size effect on clinical MV x‐ray beam profilesa)

The measured radiation beam profiles need to be corrected for the detector size effect to derive the real profiles. This paper describes two new semi-empirical procedures to determine the real profiles of high-energy x-ray beams by removing the detector size effect from the measured profiles. Measured profiles are corrected by shifting the position of each measurement point by a specific amount determined from available theoretical and experimental knowledge in the literature. The authors developed two procedures to determine the amount of shift. In the first procedure, which employs the published analytical deconvolution procedure of other investigators, the shift is determined from the comparison of the analytical fit of the measured profile and the corresponding analytical real profile derived from the deconvolution of the fitted measured profile and the Gaussian detector response function. In the second procedure, the amount of shift at any measurement point is considered to be proportional to the value of an analytical function related to the second derivative of the real profile at that point. The constant of proportionality and a parameter in the function are obtained from the values of the shifts at the 90%, 80%, 20%, and 10% dose levels, which are experimentally known from the published results of other investigators to be approximately equal to half of the radius of the detector. These procedures were tested by correcting the profiles of 6 and 18 MV x-ray beams measured by three different ionization chambers and a stereotactic field diode detector with 2.75, 2, 1, and 0.3 mm radii of their respective active cylindrical volumes. The corrected profiles measured by different detectors are found to be in close agreement. The detector size corrected penumbra widths also agree with the expected values based on the results of an earlier investigation. Thus, the authors concluded that the proposed procedures are accurate and can be used to derive the real profiles of clinical high-energy x-ray beams.