1. N.J. McCormick and I. Kuscer, "On the inverse problem in radiative transfer", J. Math. Phys. 15, 926-927 (1974).

2. N.J. McCormick and J.A.R. Veeder, "On the inverse problem of transport theory with azimuthal dependence, J. Math. Phys. 19, 994-998 (1978) and 20, 216 (1979).

Papers 1 and 2 describe inverse methods that would require measurements everywhere throughout space away from a localized plane source, and hence would be impractical to implement.

3. N.J. McCormick, "Transport scattering coefficients from reflection and transmission measurements," J. Math. Phys. 20, 1504-1507 (1979).

This is the first development of two independent sets of linear inverse equations that potentially could be used to estimate any number of expansion coefficients of the phase function using only radiance measurements on the surfaces of a slab target.

4. N.J. McCormick, "Determination of the single-scattering albedo of a dense Rayleigh-scattering atmosphere with true absorption," Astrophys. Space Sci. 71, 235-238 (1980).

This gives a set of two coupled equations with which to estimate two parameters of an atmosphere from polarized radiance measurements.

5. N.J. McCormick and R. Sanchez, "Inverse problem transport calculations for anisotropic-scattering coefficients," J. Math. Phys. 22, 199-208 (1981).

Numerical tests of the methods of paper 3 were performed for the highly idealized case of simulated noise-free radiance measurements made continuously as a function of direction.

6. R. Sanchez and N.J. McCormick, "General solutions to inverse transport problems," J. Math. Phys. 22, 847-855 (1981).

This gives a general deductive method for developing more than two independent sets of inverse equations, but the new sets of equations are highly nonlinear, rather than linear, in the unknowns; the linear equations of paper 3 are the first two sets of equations derived from the general procedure.

7. N.J. McCormick, "A critique of inverse solutions to slab geometry transport problems," Prog. Nucl. Energy 8, 235-246 (1981).

This provides a state-of-the-art summary of the preceding papers, and the work of others, through early 1981.

8. N.J. McCormick, "Remote characterization of a thick slab target with a pulsed laser," J. Opt. Soc. Am. 72, 756-759 (1982).

This provides a brand new set of inverse equations that relies on the use of backscattered radiance measurements long after a slab target has been pulsed by a light source.

9. R. Sanchez and N.J. McCormick, "Numerical evaluation of optical single- scattering properties using multiple-scattering transport methods," J. Quant. Spectrosc. Rad. Transfer 28, 169-184 (1982).

This extends the numerical work in paper 5 and examines some of the effects on the estimated parameters arising from simulated radiance measurements at only a finite number of angular directions.

10. R. Sanchez and N.J. McCormick, "Inverse problem calculations for multigroup diffusion theory," Nucl. Sci. Eng. 83, 63-71 (1983).

This shows how an inverse method can be used with angle-integrated radiance measurements; the matrix approach used is more appropriate for Raman scattering or neutron transport applications.

11. N.J. McCormick, "Inverse methods for remote determination of properties of optically thick atmospheres," Appl. Optics 22, 2556-2558 (1983).

This summarizes and compares the work in papers 3 and 8 and suggests a new way to evaluate the integrals over angle when radiance measurements are available at only a finite number of directions, as in paper 9.

12. R. Sanchez and N.J. McCormick, "Solutions to inverse problems for the Boltzmann-Fokker-Planck Equation," Transport Theory and Statist. Phys. 12, 129-155 (1983).

This develops special inverse equations that might be more appropriate for extremely anisotropic scattering and relies on a decomposition of the phase function into one portion for the nearly- forward scattering plus another for the moderately-anisotropic scattering in the remaining directions.

13. N.J. McCormick and R. Sanchez, "Solutions to an inverse problem in radiative transfer with polarization II," J. Quant Spectrosc. Rad. Transfer 30, 527-535 (1983).

This generalizes the methods developed in papers 3 and 4 to the case of general anisotropic scattering where four components of the measured radiance are to be used (i.e., the four Stokes parameters I, Q, U, and V); part I of the paper was published independently by C.E. Siewert.

14. N.J. McCormick, "Recent developments in inverse scattering transport methods," Transport Theory and Statist. Phys. 13, 15-28 (1984).

This updates paper 7 and reviews the literature from 1981 to 1983, and it also presents the general sets of inverse equations that could be used to analyze coupled-frequency radiative transfer (i.e., Raman scattering) or multigroup neutron transport.

15. N.J. McCormick, "Methods for estimating the similarity parameter of clouds from internal measurements of the scattered radiation field," J. Quant. Spectrosc. Rad. Transfer 33, 63-70 (1985).

This develops two-altitude inverse equations for estimating only one parameter of an atmosphere from radiance measurements made deep within a cloud where the radiation transport is in the diffusion regime.

16. K.K. Hunt and N.J. McCormick, "Numerical test of an inverse method for estimating single-scattering parameters from multiple-scattering experiments," J. Opt. Soc. Am. A 2, 1965-1971 (1985).

This numerically tests the equations proposed in paper 8 and concludes that the albedo of single scattering, the asymmetry factor, and perhaps another coefficient might be obtainable from the long-time dieaway of the backscattered radiance at only three azimuthal angles for a fixed polar angle when random radiance measurement inaccuracies do not exceed 1%.

17. J.C. Oelund and N.J. McCormick, "Sensitivity of multiple-scattering inverse transport methods to measurement errors," J. Opt. Soc. Am. A 2, 1972-1978 (1985).

This extends the numerical work in paper 9 by analytically examining the reason for poor estimates of the scattering parameters in some instances when simulated radiance measurements are available at only a finite number of angular directions, and it includes results of simulated random measurement errors performed with a Monte Carlo sampling scheme.

18. T. Duracz and N.J. McCormick, "Equations for estimating the similarity parameter from radiation measurements within weakly absorbing optically thick clouds," J. Atmos. Sci. 43, 486-492 (1986).

This extends the analytical work in paper 15 and provides series expansions for estimating the similarity parameter from zenith/nadir radiance or upward/downward irradiance measurements made deep within a weakly absorbing cloud.

19. N.J. McCormick, "Methods for solving inverse problems for radiation transport -- an update," Transport Theory and Statist. Phys. 15, 759-772 (1986).

This updates paper 14 and reviews the literature from 1983 to 1985, and it also presents some new approximate solution techniques for the time-independent method that may help reduce the poor conditioning observed in paper 17 of the numerical solution of the formally-exact equations.

20. T. Duracz and N.J. McCormick, "Analytical error estimates for the time-dependent radiative-transfer inverse problem," J. Opt. Soc. Am. A 3, 1871-1875 (1986).

This extends paper 8 and gives analytical estimates for errors induced by assuming that a finite slab is infinitely thick, the detector is perfectly timed with respect to the incident pulse, the pulse width is infinitesimally short, and the detector is perfectly oriented.

21. N.J. McCormick, "Inverse radiative transfer with a delta-Eddington phase function," Astrophys. Space Sci. 129, 331-334 (1987).

This uses equations from papers 3 and 6 and a change of variables to obtain inverse equations valid for the delta-Eddington scattering model.

22. T. Duracz and N.J. McCormick, "Numerical study of the time-dependent radiative transfer inverse problem," J. Opt. Soc. Am. A 4, 1849-1854 (1987).

This builds upon papers 8 and 20 and shows that a curve fitting procedure should be used to estimate the coefficients; numerical results are presented for the effects of simulated broadening of the incident pulse in time and of systematic experimental errors.

23. R.A. Elliott, T. Duracz, N.J. McCormick, and D.R. Emmons, "Experimental test of a time-dependent inverse radiative transfer algorithm for estimating scattering parameters," J. Opt. Soc. Am. A 5, 366-373 (1988).

This tests the time-dependent inversion algorithm (papers 8, 11, 16, and 20) with experimental data and shows that estimates of the single scattering albedo are within 1% for the experiments on nonabsorbing spheres.

24. T. Duracz and N.J. McCormick, "Radiative transfer calculations for characterizing obscured surfaces using time-dependent backscattered pulses," Appl. Opt. 28, 544-552 (1989).

This uses direct problem transport calculations to develop a set of graphical inversion maps to estimate the location and surface albedo of an object having a diffusely reflecting surface that is obscured by a multiple-scattering medium of known properties.

25. R.A. Elliott, T. Duracz, N.J. McCormick, and D.J. Bossert, "Experimental test of a time-dependent inverse radiative transfer algorithm for estimating scattering parameters: addendum," J. Opt. Soc. Am. A 6, 603-606 (1989).

This extends the inversion algorithm (papers 8, 11, 16, and 20) with experimental data improved over that in paper 23 and shows that the single scattering albedo and scattering asymmetry factor can be estimated to within 1% for the experiments on nonabsorbing spheres.

26. N.J. McCormick and G.E. Rinaldi, "Seawater optical property estimation from in situ irradiance measurements," Appl. Opt. 26, 2605-2613 (1989).

This extends the time-dependent inversion diffusion algorithm (paper 15) to seawater and uses it with numerically simulated data to test for the accuracy of the algorithm for shallow depths for which it is not strictly applicable, and tests it with numerically simulated experimental errors.

27. N.J. McCormick, "Estimation of two similarity parameters from polarized radiation measurements within a medium," J. Quant. Spectrosc. Rad. Transfer 42, 303-309 (1989).

This generalizes paper 15 to the case of polarized radiation with four Stokes parameters I, Q, U, and V, and shows that only two parameters can be estimated with detectors deep within an optically-thick medium.

28. N.J. McCormick, "Particle size distribution retrieval from backscattered polarized radiation measurements: a proposed method," J. Opt. Soc. Am. A 7, 1811-1816 (1990).

This generalizes paper 8 to the case of polarized radiation (with four Stokes parameters I, Q, U, and V) and shows that measurements with an azimuthally-symmetric detector, long after a thick medium has been pulsed by a light source, possibly can be used to estimate two parameters that may enable the particle size distribution to be estimated.

29. T. Duracz and N.J. McCormick, "Multiple scattering corrections for lidar detection of obscured objects," Appl. Opt. 29, 4170-4175 (1990).

This numerically solves for the time-dependent backscattered return from a spherical object obscured by an optically-thick atmosphere using the Monte Carlo method.

30. R. Sanchez, N.J. McCormick, and H.C. Yi, "Iterative inverse radiative transfer method to estimate optical thickness and surface albedo," Transport Theory and Statist. Phys. 19, 357-385 (1990).

This develops and numerically tests an algorithm for simultaneously estimating the optical thickness of a homogeneous, plane-geometry medium and the albedo of an obscured underlying surface by using remote measurements of only the ingoing and outgoing radiance outside the medium.

31. H.C. Yi, N.J. McCormick, and R. Sanchez, "Cloud optical thickness estimation from irradiance measurements," J. Atm. Sci. 47, 2567-2579 (1990).

This derives two approximate algorithms (asymptotic and transport-corrected diffusion) to estimate the optical thickness of clouds and gives numerical tests of the algorithms against FN calculations, as well as a sensitivity analysis.

32. H.C. Yi, R. Sanchez, and N.J. McCormick, "Bioluminescence estimation from ocean in situ irradiances," Appl. Opt. 31, 822-830 (1992).

This numerically tests the use of the photon conservation equation for estimating the spatial profile of a source using irradiance measurements at two positions to characterize the source strength of the layer between.

33. N.J. McCormick, "Inverse radiative transfer problems: a review," Nucl. Sci. Eng. 112, 185-198 (1992).

This review follows up those of references 7, 14, and 19 and covers developments in 1986-1992.

34. Z. Tao, N.J. McCormick, and R. Sanchez, "Ocean source and optical property estimation using explicit and implicit algorithms," Appl. Opt. 33, 3265-3275 (1994).

This develops an asymptotic two-stream approximation and extends the method developed in reference 32 so that full use is made of measurements of the downward and upward lowest two angular moments of the radiance.

35. L.J. Holl and N.J. McCormick, "Ocean optical-property estimation with the Zaneveld-Wells algorithm," App. Opt. 34, 5433-5441 (1994).

This algorithm is for estimating optical properties from a set of specially-designed detectors located within a medium, and requires that derivatives of the measurements be estimated.

36. N.J. McCormick, "Analytical transport theory applications in optical oceanography," Ann. Nucl. Energy 23, 381-395 (1996).

This shows how the absorption and scattering coefficients can be estimated from a variety of measurements made deep within a medium.

37. E. Steinfelds, M.A. Samuel, N.J. McCormick, and J.R. Reid, "Radiative Transfer Single-Scattering Albedo Estimation with a Super-Pade Approximation of Chandrasekhar's H-Function," Int. J. Theor. Phys. 36, 997-1007 (1997).

Three algorithms for evaluating the single scattering albedo for a semi-infinite medium that scatters isotropically are developed, with an emphasis on minimizing the number of measurements needed.

38. N.J. McCormick and R. Sanchez, "Two-Region Inverse Transport Analysis with Solutions of the Two-Region Milne Problem," Transport Theory and Statist. Phys. 26, 607-618 (1997).

This algorithm is for estimating from exterior measurements the single scattering albedo values of two thick layers that scatter isotropically.

39. N.J. McCormick, R. Sanchez, and H.C. Yi, "Cloud Optical Thickness Estimation from Ground-Level Measurements," J. Atmos. Oceanic Phys., 10, 702-708 (1997).

Two algorithms for estimating the unknown optical thickness of a homogeneous cloudy atmosphere using a ground-level downward radiance or irradiance measurements are presented; this work is closely related to some of the work in references 30 and 31.

40. R.A. Leathers and N.J. McCormick, "Ocean inherent optical property estimation from irradiances," Appl. Opt. 36, 8685-8698 (1997).

This numerically tests an algorithm from reference 36 that uses the ratio of the upward to downward irradiances and also the downward diffuse attentuation coefficient.

41. L.K. Sundman, R. Sanchez, and N.J. McCormick, "Ocean optical source estimation with widely spaced irradiance measurements," Appl. Opt. 37, 3793-3803 (1998).

This tests an algorithm that uses irradiance measurements located an arbitrary distance apart in sea water and solves for two parameters that fit a globally exponential or linear source shape.

42. N.J. McCormick, "Unified approach to analytical solutions of three inverse transport problems", Prog. Nucl. Energy 34, 425-430 (1999).

The problems considered are the determination of the single scattering albedo plus scattering properties of the medium, the determination of the incident illumination, and the determination of a spatially - dependent interior source within the medium.

43. R.A. Leathers and N.J. McCormick, "Algorithms for ocean-bottom albedo determination from in-water natural-light measurements," Appl. Opt. 38, 3199-3205 (1999).

This presents a new algorithm for estimating the bottom albedo and numerically tests it against another algorithm.

44. R.A. Leathers, C.S. Roesler, and N.J. McCormick, "Ocean inherent optical property determination from in-water light field measurements," Appl. Opt. 38, 5096-5103 (1999).

This algorithm extends the work in reference 40 to use the nadir-viewing radiance and downward irradiance. Estimates of the absorption coefficient agree well with those from an in situ reflecting tube instrument.

45. N.J. McCormick, "Ocean optics phase function inverse equations," Appl. Opt. 41, 4958-4961 (2002).

Equations are obtained for determining the Legendre polynomial expansion coefficient of the phase function from angular or angular-spatial integrals of the radiance measured either in water or at the surface.

46. A.H. Hakim and N.J. McCormick, "Ocean optics estimation for absorption, backscattering, and phase function parameters," Appl. Opt. 42, 931-938 (2003).

For the Henyey-Greenstein scattering phase function model, radiance measurements at one depth can be used to estimate the single scattering albedo and the scattering asymmetry factor. With measurements at two depths those parameters plus the absorption, scattering, and backscattering coefficients can be determined. Bottom parameters also can be estimated from radiance measurements near the bottom.

47.  T. Viik and N. J. McCormick, “Numerical test of an inverse polarized radiative transfer algorithm,” J. Quant. Spectrosc. Rad. Transfer 78, 235-241 (2003).

The procedure in paper 4 is tested with which to determine the single-scattering albedo from polarization measurements of the angle-dependent intensity at two locations within, or on the boundaries of, a homogeneous finite or infinite atmosphere that scatters radiation according to the Rayleigh law with true absorption.

48.   N. J. McCormick, “Analytic inverse radiative transfer equations for atmospheric and hydrologic optics,” J. Opt. Soc. Am. A 21, 1009-1017 (2004).

Two new sets of analytical equations are derived with which the albedo of single scattering and the coefficients of a Legendre polynomial expansion of the scattering phase function can be determined for a source-free, homogeneous plane-parallel medium uniformly illuminated over the surfaces.

49.   T. Viik and N. J. McCormick, “Quadratic integrals in inverse problems with multiple scattering," in Photopolarimetry in Remote Sensing (G. Videen et al., eds.), Kluwer Academic Publishers (Netherlands), 125-136 (2004).

Procedures for determining the single-scattering albedo of an isotropically scattering semi-infinite medium for unpolarized radiation and for polarized radiation in an atmosphere that scatters according to the Rayleigh-Cabannes law are numerically tested.

50.   N. J. McCormick, “Gas-surface accommodation coefficients from viscous slip and temperature jump coefficients,” Phys. Fluids 17, 107104-1 to 107104-8 (2005). [DOI: 10.1063/1.2111133]

Analytical equations for the viscous slip and temperature jump coefficients for a gas flowing over a surface are numerically tested against highly accurate numerical results.  The gas-surface interaction is assumed to be governed by either the elastic-diffuse (Maxwell) one-parameter model or the Cercignani-Lampis two-parameter model.  Approximate analytical equations obtained for solving the inverse problem to estimate the accommodation coefficient(s) are obtained and numerically tested.

51.   A. Kaskas, M. C. Gulecyuz, C. Tezcan, and N. J. McCormick, “Analytic algorithms for determining radiative transfer optical properties of ocean waters,” Appl. Opt. 45, 7698-7705 (2006).

A synthetic model for the scattering phase function is used to develop simple algebraic equations, valid for any water type, for evaluating the ratio of the backscattering to absorption coefficients of spatially uniform, very deep waters.  The algorithms are compared with other analytic correlations that were previously derived from extensive numerical simulations, and they are also numerically tested with forward problem results computed with a modified F_N method.

52.   R. Sanchez and N. J. McCormick, “On the uniqueness of the inverse source problem for linear particle transport theory,” Transport Theory and Statist. Phys. 37, 236-263 (2008) [DOI:10.1080/00411450802526301].

Inverse source problems for time-independent linear transport with data from invasive and noninvasive detectors are analyzed.  The former inverse problem is proven to have a unique solution, while for the latter we construct counterexamples that prove that the problem is underdetermined for the general case of anisotropic sources and prove that it has a unique solution for isotropic sources and scattering. 

53.   E. Rehm and N. J. McCormick, “Inherent Optical Property Estimation in Deep Water,''  Optics Express 19, 24,986-25,005 (2011).

A synthetic model for the scattering phase function with one fitted parameter, which generalizes the approach used in reference 51, is used to develop one algorithm for estimating the ratio of the backscattering to absorption coefficients and a second algorithm for estimating the absorption coefficient from measurements of the downward planar irradiance and the vertically upward radiance.  To implement the first algorithm it is assumed that the light measurements are deep enough that the asymptotic eigenmode of linear transport theory can be used to estimate the fitted parameter, and an analogous assumption is used in implementing the second algorithm.  Hydrolight calculations are used to generate and test the sensitivity of the fitted parameter to different waters and then used with NAV08 experimental data, along with independent measurements of the absorption coefficient and estimated backscattering coefficients, to test the efficacy of the algorithms for other water types.

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