Effect Of Phase Dispersion Research Articles

Modified pure-viscoacoustic wave propagation and compensated reverse-time migration in transversely isotropic media

Previous studies demonstrated that seismic attenuation and anisotropy can significantly affect the kinematic and dynamic characteristics of wavefields. If these effects are not incorporated into seismic migration, the resolution of the imaging results will be reduced. Considering the anisotropy of velocity and attenuation, we derive a new pure-viscoacoustic wave equation to simulate P wave propagation in transversely isotropic (TI) attenuating media by combining the complex dispersion relation and modified complex modulus. Compared to the conventional complex modulus, the modified modulus is derived from the optimized relationship between angular frequency and wavenumber, which can improve the modeling accuracy in strongly attenuating media. Wavefield comparisons illustrate that our pure-viscoacoustic wave equation can simulate stable P wavefields in complex geological structures without S-wave artifacts and generate similar P wave information to the pseudo-viscoacoustic wave equation. During the implementation, we introduce two low-rank decompositions to approximate the real and imaginary parts and then use the pseudo-spectral method to solve this new equation. Since the proposed equation can simulate decoupled amplitude attenuation and phase dispersion effects, it is used to perform Q-compensated reverse-time migration (Q-RTM). Numerical examples demonstrate the accuracy and robustness of the proposed method for pure-viscoacoustic wavefield simulations and migration imaging in transversely isotropic attenuating media.

Q-compensated wavefield depth extrapolation-based migration using a viscoacoustic wave equation

In a viscoacoustic medium, intrinsic attenuation causes seismic wavefields to attenuate in amplitude and become dispersed in phase, leading to distorted structural imaging and inaccurate migrated amplitudes. To address this problem, viscoacoustic reverse time migration corrects for dispersion and amplitude attenuation effects in wavefield propagation in forward and backward directions. To provide a parallel alternative approach, the time fractional derivative viscoacoustic wave equation in the depth domain is solved and a Q-compensated wavefield depth extrapolation scheme is established to compensate for viscoacoustic effects during recursive wavefield depth extrapolation. This approach decouples the amplitude attenuation and phase dispersion effects from the viscoacoustic vertical wavenumber. To suppress high wavenumber components and address the problem of exponential amplitude growth during wavefield depth extrapolation, we limit the imaginary part of the vertical wavenumber in the frequency wavenumber domain and use an adaptive stabilization solution. Numerical experiments of impulse responses in a viscoacoustic isotropic medium demonstrate that our proposed scheme can observe calculated wavefields exhibiting amplitude attenuation and phase dispersion effects in comparison to wavefields of the acoustic medium, indicating good agreement with theoretical expectations. We also demonstrate the capability of our proposed scheme to recover imaging amplitudes through a variety of numerical experiments, such as imaging of three-layer, Marmousi, and BP gas reservoir models. The results indicate that our proposed scheme can recover amplitude attenuation and phase dispersion effects more accurately than acoustic migration. We also use marine seismic data featuring natural gas hydrates to indicate that our proposed Q-compensated scheme can generate an enhanced imaging result, especially for the bottom simulating reflectors, compared with the conventional imaging algorithm without Q-compensation.

Discontinuous Galerkin method for solving viscoacoustic wave equations with amplitude dissipation and phase dispersion separation in isotropic and anisotropic media

SUMMARY The standard-linear-solid (SLS) theory works well for viscoelasticity. However, the coupling of amplitude dissipation and phase dispersion makes it impossible to investigate their effects separately by the discontinuous Galerkin method (DGM). In this paper, we have derived newly viscoacoustic wave equations with amplitude dissipation and phase dispersion separation in isotropic and anisotropic media, based on a Fourier method, which is suitable for using a time–space-domain DGM on unstructured meshes. The basic framework of DGM is constructed and the amplitude-dissipation effect and the phase-dispersion effect in viscoacoustic wave equations are investigated. The original equation is first transformed into the frequency–wavenumber domain, where the amplitude dissipation and phase dispersion are separated effectively, and then the decoupled formulation is converted back to the time–space domain. The new equivalent and approximate equations can be obtained. Compared with the original equation, the newly approximated equation enables us to separate the amplitude-loss and phase-delay terms, respectively, and experiences four kinds of effects, namely acoustic effect, only amplitude-dissipation effect, only phase-dispersion effect and both amplitude-dissipation and phase-dispersion effect. Moreover, the stability condition and numerical dispersion for using DGM to solve the new and old equations are presented. Several numerical examples are used to verify the correctness and effectiveness of the modified approximate equations in viscoacoustic isotropic and anisotropic media. The numerical results in a cave and SEG/EAGE salt models demonstrate that the new equations combined with DGM have performances on viscoacoustic media with complex geological structures.

Frequency-Domain Reverse-Time Migration With Attenuation Compensation

Seismic wave suffers from amplitude attenuation and phase distortion when propagating in the attenuating media, thus reverse time migration (RTM) for viscous media should take the attenuation effects into consideration. Compensating for the attenuation effects in RTM may occur numerical instability because of the exponential amplification of the extrapolated wavefields. To obtain stable imaging results, we have developed a stabilized Q-compensated reverse time migration in the frequency domain. This algorithm is implemented by the following steps: first, we use the Kolsky-Futterman model to derive a frequency-domain viscoacoustic wave equation, which can simulate the amplitude loss and phase dispersion effects separately. Then, we calculate the source wavefields in the viscoacoustic media. Next, treating the recorded (viscoacoustic) data as the receiver sources, we can obtain the phase-dispersion-only and viscoacoustic receiver wavefields, which can be used to construct the stabilized Q-compensated receiver wavefields. Finally, we apply the deconvolution imaging condition for obtaining a Q-compensated image. A simple anticline model and gas chimney model are used to verify the effectiveness of the proposed approach. The Q-compensated images for the noise-free data indicate the algorithmic stability and compensation accuracy of the proposed scheme. The noisy data tests for the gas chimney model demonstrate the good anti-noise property of our method. The field data applications further prove its feasibility and practicability.

Electron-acoustic solitons in dense electron–positron–ion plasma: Degenerate relativistic enthalpy function

An enthalpy function Hj∝1+ηj2 is deduced from the density of states for degenerate relativistic charged particulate. Here ηj=pj/mjc2 stands for the relativistic factor, pjmj is the Fermi momentum (mass) for the jth degenerate relativistic species and c designates the vacuum speed of light. The enthalpy function reduces the results Weinberg (1972), Lee and Cheong (2007) obtained for the nondegenerate nonrelativistic (NR) and ultrarelativistic (UR) fluids. The linear and nonlinear propagation characteristics of the electron-acoustic (EA) solitons are examined for dense electron–positron–ion (EPI) plasma that comprises degenerate relativistic electrons/positrons and non-degenerate ions. By using Hj and with the aid of the relativistic hydrodynamic equations, a linear dispersion relation for the EA waves is derived that admits the evolution of the fast and slow electron-acoustic (FEA/SEA) excitations. For the nonlinear EA waves, a Korteweg–de Vries (KdV) equation is obtained; numerical analysis reveals that the FEA mode may ensue in the solitary potential. The propagation of the SEA solitons is forbidden as it suffers from a negative phase dispersion effect. At a relatively low concentration of positrons, the EPI plasma may admit the evolution of electron–holes and therefore, give rise to the compressive FEA solitons. This investigation is important for understanding localized excitations in degenerate relativistic plasmas, as that are relevant in white dwarfs, neutron stars, and high-energy density facilities.

Analysis of Earth’s Ionosphere Effects on Englacial Layering Detectability in Spaceborne Radar Sounders Data

Several studies are in progress for proposing an Earth orbiting radar sounder (EORS) mission. Some of them consider as baseline system architecture a recently proposed distributed radar sounding array in formation flight with enhanced capabilities of clutter suppression. Besides clutter, the detectability of subsurface targets may also be affected by the propagation of the radar signal through Earth’s ionosphere. These effects include frequency-dependent phase dispersion and scintillations. In this letter, we present a subsurface detection performance assessment of an EORS with distributed architecture focusing on the ionospheric effects. The novel contributions of this work are: (i) simulation of the coherent radar response of a representative polar ice target (englacial layering) in the distributed radar sounding case; (ii) inclusion of spatially-dependent ionospheric scintillation effects on the distributed beam pattern; (iii) inclusion of phase dispersion effects for different values of total electron content (TEC), and (iv) analysis of the subsurface detection performance. Detectability analysis is performed after applying a state-of-the-art technique for compensating ionospheric phase-dispersion effects. The results show that the englacial layering is detectable by compensating the dispersion effects in the range between 1 and 21 TECU in the ionosphere. The layering is also detectable at higher values of TEC by improving the accuracy of TEC estimation. Moreover, even without compensation, the worst-case ionospheric phase scintillations of 25° produces a negligible effect on the detectability.

Liouville partial-differential-equation methods for computing 2D complex multivalued eikonals in attenuating media

We have developed a Liouville partial-differential-equation (PDE)-based method for computing complex-valued eikonals in real phase space in the multivalued sense in attenuating media with frequency-independent qualify factors, where the new method computes the real and imaginary parts of the complex-valued eikonal in two steps by solving Liouville equations in real phase space. Because the earth is composed of attenuating materials, seismic waves usually attenuate so that seismic data processing calls for properly treating the resulting energy losses and phase distortions of wave propagation. In the regime of high-frequency asymptotics, the complex-valued eikonal is one essential ingredient for describing wave propagation in attenuating media because this unique quantity summarizes two wave properties into one function: Its real part describes the wave kinematics and its imaginary part captures the effects of phase dispersion and amplitude attenuation. Because some popular ordinary-differential-equation (ODE)-based ray-tracing methods for computing complex-valued eikonals in real space distribute the eikonal function irregularly in real space, we are motivated to develop PDE-based Eulerian methods for computing such complex-valued eikonals in real space on regular meshes. Therefore, we solved novel paraxial Liouville PDEs in real phase space so that we can compute the real and imaginary parts of the complex-valued eikonal in the multivalued sense on regular meshes. We call the resulting method the Liouville PDE method for complex-valued multivalued eikonals in attenuating media; moreover, this new method provides a unified framework for Eulerianizing several popular approximate real-space ray-tracing methods for complex-valued eikonals, such as viscoacoustic ray tracing, real viscoelastic ray tracing, and real elastic ray tracing. In addition, we also provide Liouville PDE formulations for computing multivalued ray amplitudes in a weakly viscoacoustic medium. Numerical examples, including a synthetic gas-cloud model, demonstrate that our methods yield highly accurate complex-valued eikonals in the multivalued sense.

Rotational Molding of Poly(Lactic Acid)/Polyethylene Blends: Effects of the Mixing Strategy on the Physical and Mechanical Properties.

In this study, blends of poly(lactic acid) (PLA)/linear medium density polyethylene (LMDPE) at different weight ratios were prepared by rotational molding. Two mixing strategies were used to evaluate the effect of phase dispersion on the physical and mechanical properties: (i) Dry-blending (DB) using a high shear mixer, and (ii) melt-blending (MB) using a twin-screw extruder. Thermal, morphological, and mechanical analyses were performed on the neat polymers and their blends. The thermal analysis was completed by differential scanning calorimetry (DSC) and thermogravimetric analysis (TGA), and the blends prepared by MB had lower thermal stability than the ones prepared via DB due to some thermo-oxidative degradation through the double thermal process (extrusion and rotomolding). The morphology of the rotomolded parts showed that DB generated larger particle sizes (around 500 µm) compared to MB (around 5 µm) due to the shear and elongational stresses applied during extrusion. The tensile and flexural properties of the rotomolded parts combined the PLA stiffness with the LMDPE toughness independent of the blending technique. Neat PLA presented increments in tensile strength (54%) and flexural strength (111%) for DB compared with MB. A synergistic effect in impact strength was observed in blends with 12 and 25 wt. % of PLA prepared by DB.

An approach for attenuation-compensating multidimensional constant-Q viscoacoustic reverse time migration

Simulation of wave propagation in a constant-[Formula: see text] viscoacoustic medium is an important problem, for instance, within [Formula: see text]-compensated reverse time migration (RTM). Processes of attenuation and dispersion influence all aspects of seismic wave propagation, degrading the resolution of migrated images. To improve the image resolution, we have developed a new approach for the numerical solution of the viscoacoustic wave equation in the time domain and we developed an associated viscoacoustic RTM ([Formula: see text]-RTM) method. The main feature of the [Formula: see text]-RTM approach is compensation of attenuation effects in seismic images during migration by separation of amplitude attenuation and phase dispersion terms. Because of this separation, we are able to compensate the amplitude loss effect in isolation, the phase dispersion effect in isolation, or both effects concurrently. In the [Formula: see text]-RTM implementation, an attenuation-compensated operator is constructed by reversing the sign of the amplitude attenuation and a regularized viscoacoustic wave equation is invoked to eliminate high-frequency instabilities. The scheme is tested on a layered model and a modified acoustic Marmousi velocity model. We validate and examine the response of this approach by using it within an RTM scheme adjusted to compensate for attenuation. The amplitude loss in the wavefield at the source and receivers due to attenuation can be recovered by applying compensation operators on the measured receiver wavefield. Our 2D and 3D numerical tests focus on the amplitude recovery and resolution of the [Formula: see text]-RTM images as well as the interface locations. Improvements in all three of these features beneath highly attenuative layers are evident.

Q-compensated reverse time migration in viscoacoustic media including surface topography

Conventional reverse time migration (RTM) may not produce high-quality images in areas with attenuation and severe topography because severe topographic surfaces have a great impact on seismic wave simulation, resulting in strong scattering and diffraction waves, and anelastic properties of the earth affect the kinematics and dynamics of seismic wave propagation. To overcome these problems, we have developed a [Formula: see text]-compensated topographic RTM method. In this method, a new viscoacoustic quasidifferential equation is introduced to simulate forward- and backward-propagated wavefields. The viscoacoustic equation has a lossy term and a dispersion term without memory variables, and it is solved by a hybrid spatial partial derivative scheme. A new stabilization operator is derived and substituted into the [Formula: see text]-compensated viscoacoustic quasidifferential equation to suppress high-frequency noise during the attenuated wavefield compensation. Numerical tests on a sag attenuating topographic model and an attenuating topographic Marmousi2 model demonstrate that our [Formula: see text]-compensated topographic RTM can produce accurate and high-quality images by correcting the anelastic amplitude loss and phase-dispersion effects. Finally, our method is tested on a field data set.

Effects of RF pulse profile and intra-voxel phase dispersion on MR fingerprinting with balanced SSFP readout

PurposeTo investigate possible errors in T1 and T2 quantification via MR fingerprinting with balanced steady-state free precession readout in the presence of intra-voxel phase dispersion and RF pulse profile imperfections, using computer simulations based on Bloch equations. Materials and methodsA pulse sequence with TR changing in a Perlin noise pattern and a nearly sinusoidal pattern of flip angle following an initial 180-degree inversion pulse was employed. Gaussian distributions of off-resonance frequency were assumed for intra-voxel phase dispersion effects. Slice profiles of sinc-shaped RF pulses were computed to investigate flip angle profile influences. Following identification of the best fit between the acquisition signals and those established in the dictionary based on known parameters, estimation errors were reported. In vivo experiments were performed at 3T to examine the results. ResultsSlight intra-voxel phase dispersion with standard deviations from 1 to 3Hz resulted in prominent T2 under-estimations, particularly at large T2 values. T1 and off-resonance frequencies were relatively unaffected. Slice profile imperfections led to under-estimations of T1, which became greater as regional off-resonance frequencies increased, but could be corrected by including slice profile effects in the dictionary. Results from brain imaging experiments in vivo agreed with the simulation results qualitatively. ConclusionMR fingerprinting using balanced SSFP readout in the presence of intra-voxel phase dispersion and imperfect slice profile leads to inaccuracies in quantitative estimations of the relaxation times.

Viscoelastic reverse time migration with attenuation compensation

We have developed a theory of viscoelastic reverse time migration (RTM). The main feature of viscoelastic RTM is a compensation for P- and S-wave attenuation effects in seismic images during migration. The forward modeling engine is based on a viscoelastic wave equation involving fractional Laplacians. Because of the decoupled attenuation property, wave propagation can be simulated in three scenarios, i.e., only the amplitude loss effect, only the phase dispersion effect, or both effects simultaneously. This separation brings practical flexibility to studying attenuation effects on wave propagation and imaging. The backward modeling operator is constructed by reversing the sign of first-order time derivative amplitude loss operators. Synthetic examples determine the ability of viscoelastic RTM to illuminate degraded areas and shadow zones caused by attenuation. Numerical experiments also reveal that [Formula: see text]-compensated imaging is noticeably more accurate in kinematics and dynamics than elastic imaging in the presence of high attenuation. Results from a synthetic 3D model determine the superiority of viscoelastic RTM over elastic RTM in imaging salt flanks and delineation of salt boundaries, which are dimmed in elastic images.

Q-compensated reverse-time migration

Reduced amplitude and distorted dispersion of seismic waves caused by attenuation, especially strong attenuation, always degrades the resolution of migrated images. To improve image resolution, we evaluated a methodology of compensating for attenuation ([Formula: see text]) effects in reverse-time migration ([Formula: see text]-RTM). The [Formula: see text]-RTM approach worked by mitigating the amplitude attenuation and phase dispersion effects in source and receiver wavefields. Source and receiver wavefields were extrapolated using a previously published time-domain viscoacoustic wave equation that offered separated amplitude attenuation and phase dispersion operators. In our [Formula: see text]-RTM implementation, therefore, attenuation- and dispersion-compensated operators were constructed by reversing the sign of attenuation operator and leaving the sign of dispersion operator unchanged, respectively. Further, we designed a low-pass filter for attenuation and dispersion operators to stabilize the compensating procedure. Finally, we tested the [Formula: see text]-RTM approach on a simple layer model and the more realistic BP gas chimney model. Numerical results demonstrated that the [Formula: see text]-RTM approach produced higher resolution images with improved amplitude and phase compared to the noncompensated RTM, particularly beneath high-attenuation zones.

Dispersion cancellation in high-resolution two-photon interference

The dispersion cancellation observed in Hong-Ou-Mandel (HOM) interference between frequency-entangled photon pairs has been the basis of quantum optical coherence tomography and quantum clock synchronization. Here we explore the effect of phase dispersion on ultranarrow HOM dips. We show that the higher-order dispersion, the line width of the pump laser, and the spectral shape of the parametric fluorescence have a strong effect on the dispersion cancellation in the high-resolution regime with several experimental verifications. Perfect dispersion cancellation with a linewidth of 3\mu m is also demonstrated through 25 mm of water.

On the Transmission and Propagation of Low Attenuation Rate Electromagnetic Pulses in Debye Media

In a dispersive medium, the appearance of the steady-state part of the signal is preceded by oscillations known as precursors. These early oscillations are the product of the interrelated effects of phase dispersion and frequency dependent attenuation. Inside water, the attenuation rate of the Brillouin precursor is sub-exponential, following the inverse square-root of the distance traveled. Based on that, a near-optimal pulse that could achieve this attenuation rate, and, hence, would lend itself to underwater detection and communication applications, was recently proposed. The ldquooptimalityrdquo of this pulse is shown to be related to the temporal support of the pulse and its spectral characteristics, rather than its shape. A family of alternative pulses is found to have the low attenuation feature of the ldquooptimalrdquo pulse, as they eventually evolve into the Brillouin precursor itself shortly after they enter water. In addition, this work considers the practical case when such a pulse would be generated in air, would impinge onto an air-water interface and then propagate inside water. It is shown how the presence of the interface affects the attenuation rate of the pulse inside water and a simple way to recover its low attenuation rate is suggested. The finite-difference time-domain technique is employed in all the simulations.

Asymptotic description of wave propagation in an active Lorentzian medium

In a causally dispersive medium the signal arrival appears in the dynamical field evolution as an increase in the field amplitude from that of the precursor fields to that of the steady-state signal. The interrelated effects of phase dispersion and frequency dependent attenuation and/or amplification alter the pulse in such a fundamental way that results in the appearance of precursor fields. Although superluminal group velocities have been found in various dispersive media, the pulse "front" and associated precursors will never travel faster than c , and hence these are the vehicles through which relativistic causality is preserved. While many rigorous studies of wave propagation and associated abnormal group velocities in passive Lorentzian media have been performed, the corresponding problem in active media has remained theoretically unexplored. This problem is addressed in the present paper, by employing the steepest descent method for the determination of the response of an active Lorentzian medium to a step modulated pulse. The steepest descent method provides a detailed description of the propagation of the pulse inside the dispersive medium in the time domain. Moreover, the evolution of the saddle points illuminates the relation between the medium parameters and the temporal evolution of the propagating pulse within the medium. Hence, useful physical insights are obtained and the interesting differences between the passive and active case are deduced.

Tailoring two-photon interference with phase dispersion

We present a complete analytical formalism of the effect of phase dispersion on the Hong-Ou-Mandel dip with cw pumping, including higher-order terms. We show that the Hong-Ou-Mandel dip is strongly modified by the frequency-dependent phase modulation and even becomes a bump when the phase dispersion is a special step function. We also show that the phase dispersion function can be fully reconstructed via Fourier transformation from the measurement result of the Hong-Ou-Mandel dip when the dispersion function is antisymmetric. An experimental demonstration of the proposed method using a bandpass filter as a test sample is also presented. The two-photon interference phenomenon with a beam splitter was first demonstrated by Hong, Ou, and Mandel HOM1. They used a pair of photons generated via spontaneous parametric downconversion SPDC, in which a single photon of a pumping laser is converted into a pair of photons. These two photons were sent to a 50-50 beam splitter and were detected by single-photon counters after the beam splitter. They measured the number of coincidence events while scanning an optical delay added to the path of one of the photons, and found a sharp dip a HOM dip ,a t which the delay was exactly zero. This two-photon interference phenomenon has become an important tool 2 in a variety of fields. In the field of quantum information, the phenomenon is used in Bell-state analyzers 3 for quantum teleportation 4 and in quantum gates 5,6 and quantum filters 7 for linear optics quantum computation 8. Recently, this phenomenon has come to be used in the field of optical tomography 9. The effect of phase dispersion on two-photon interference, or the HOM dip, has also been attracting attention because the phase dispersion shows interesting characteristics of the phenomenon and of the downconverted photon pairs. Franson 10 analyzed the effect of dispersion up to the second order and found that the width of the HOM dip and

Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics

When an ultrawide-band electromagnetic pulse penetrates into a causally dispersive dielectric, the interrelated effects of phase dispersion and frequency dependent attenuation alter the pulse in a fundamental way that results in the appearance of so-called precursor fields. For a Debye-type dielectric, the dynamical field evolution is dominated by the Brillouin precursor as the propagation depth typically exceeds a single penetration depth at the carrier frequency of the input pulse. This is because the peak amplitude in the Brillouin precursor decays only as the square root of the inverse of the propagation distance. This nonexponential decay of the Brillouin precursor makes it ideally suited for remote sensing. Of equal importance is the frequency structure of the Brillouin precursor. Although the instantaneous oscillation frequency is zero at the peak amplitude point of the Brillouin precursor, the actual oscillation frequency of this field structure is quite different, exhibiting a complicated dependence on both the material dispersion and the input pulse characteristics. Finally, a Brillouin pulse is defined and is shown to possess near optimal (if not optimal) penetration into a given Debye-type dielectric.

Multifrequency viscoacoustic modeling and inversion

Modeling and inversion for seismic wavefields that include the attenuation and phase dispersion effects of Q can be implemented in the space‐frequency domain. The viscoacoustic wave equation is solved by the moment method. Absorbing boundary conditions are implemented by reducing Q and adjusting the complex velocity (to reduce Q‐dependent reflectivity) in a zone around the edges of the model grid. Nonlinear inversion is performed using iterative linearized inversions. The residual wavefield at a single frequency is back projected, using an anticausal Green’s function, along the viscoacoustic wavepath in an estimate of the model, to get updated velocity and Q distributions. The model obtained from data at one frequency becomes input to inversion at the next higher frequency. Velocity and Q are inverted simultaneously as they are interdependent. Both modeling and inversion algorithms are successfully tested with synthetic examples; data at two or three frequencies are sufficient to produce reliable images from noise‐free synthetic data.

Computer-aided design of bioerodible devices with optimal release characteristics: a cellular automata approach

The development of a computational tool to design bioerodible devices with optimal release characteristics is presented. This computational tool uses cellular automata and parallel iterations to model and simulate the release of bioactive agents (drugs) from bioerodible matrices. The simulations can accurately model surface erosion processes in multicomponent systems of arbitrary geometry and with different dissolution rates for each component. Simulation results are analysed to show how the overall release rates are affected by the intrinsic dissolution rates, drug loading, porosity and the dispersion of the drug in the bioerodible matrix. A strongly non-linear dependence of release rates on drug loading and the intrinsic dissolution rates of the solid components is obtained and the effects of phase dispersion on the variability of release rates are elucidated. Finally, guidelines are presented for screening of alternatives to minimize the development effort and experimentation required for designing devices with desired release characteristics.