User's Guide

LIDORT - User’s Guide to Version 2.3 1 

User’s Guide 

LIDORT, Version 2.3 

A Linearized Discrete Ordinate Radiative Transfer 

Model for Atmospheric Retrieval 

Author: Robert Spurr 

Harvard-Smithsonian Center for Astrophysics 

60 Garden Street, Cambridge, MA 02138, USA 

Tel. +1 617 496 7819 

email: rspurr@cfa.harvard.edu 

January 2001


Table of Contents 

1. Introduction to LIDORT Versions V2.3S and V2.3E 

1.1 Background and General Introduction 

1.2 Scope of Version 2.3 

1.3 Summary of User’s Guide to Version 2.3 

2. LIDORT Structures; Input/Output and Initialization File 

2.1 Standard Structures 

2.2 Extended Structures 

2.3 Initialization File Example 

3. LIDORT Package; Environment and Master Module 

3.1 LIDORT Package Organization 

3.2 Use of LIDORT Master Module in an External Environment 

3.3 Computational Sequence in LIDORT_V23E_MASTER 

4. Geophysical Input Preparation 

4.1 Preparation for LIDORT_GEOPHYS.VARS (Standard) 

4.2 Preparation for LIDORT_L_GEOPHYS.VARS (Extended) 

5. Installation, Makefile and Test Example 

5.1 Installation Procedure 

5.2 Makefile Example in Directory V23E_RELEASE 

5.3 Test-case Example in Directory V23E_RELEASE 

6. Error Handling, Utilities, Validation, References 

6.1 Error Handling 

6.2 Utilities 

6.3 A Note on Validation 

6.4 References


1. Introduction to LIDORT Versions 2.3S and 2.3E 

1.1 Background and History 

LIDORT [1] [7] is a generic discrete ordinate radiative transfer package designed to produce both backscatter 

intensities and weighting functions for a general atmospheric scenario. In addition to the usual 

backscatter intensity simulation, LIDORT can generate simultaneously whole fields of analytic weighting 

functions, without the need for repeated RT model calls based on finite-difference estimations. It is thus 

ideally suited to retrieval applications in the earth’s atmosphere. The model is a pure scattering formalism 

using optical depth as the vertical coordinate; there are no databases, and the user must define and set up 

the required inputs before calling the package as a subroutine within a test environment. 

Version 1.0 of LIDORT was developed in Summer 1999 for a plane-parallel multi-layer atmosphere with 

full multiple scattering, beam and thermal emission source terms, and with a general reflecting boundary 

condition. Version 1.0 may be termed the “satellite application”, that is, output is restricted to upwelling 

intensities and weighting functions at the top-of-the-atmosphere (TOA) reference level. As far as intensities 

are concerned, the scope of Version 1.0 is almost the same as that for the DISORT plane-parallel package 

[2], except for the restriction to TOA output. A number of modifications and additional features were 

developed for Version 1.1, which was given public release in November 1999 (there is now a website for 

LIDORT - see Chapter 5). For details, see [1]. This version was validated against DIOSRT Version 1. 

Version 2.1 of LIDORT was developed over the period October 1999 to July 2000. The basic package of 

Version 1.1 was completely reorganized in order to incorporate a number of extensions which are 

described in the next section. Version 2.1 was released in October 2000. As with Version 1, the LIDORT 

package was written in FORTRAN 77; double precision arithmetic was used throughout. For a description 

of the theory and some applications of Version 2.1, see [7]. Version 1.1 was based on a complete perturbation 

analysis of the plane-parallel discrete ordinate solution in a multi-layered atmosphere; the emphasis in 

Version 2.1 is on explicit differentiation of the discrete ordinate solution in order to find Jacobian derivatives 

of the intensity field; the techniques are equivalent since the discrete ordinate differential equations 

are linear. We use the term linearization to denote the process of differentiation. 

Version 2.3 contains an extension to generate weighting functions with respect to phase function expansion 

coefficients. This is of great potential importance for the retrieval of aerosol properties, and this extends the 

scope of the model to a new area of retrieval problems. The inputs have been simplified for this version; 

principally, only the total single scattering albedo and phase function moments (and their parameter derivatives 

for the linearization) are taken as input - references to individual scatterers have been dropped. The 

inputs are now exactly the same as those for DISORT. Version 2.3 also comes with a single scatter correction 

package based on the Nakajima-Tanaka (NT) algorithm [8]. Partial validation against DISORT Version 

2.0 (which has the NT single scatter correction) has been carried out for a number of situations [9]. 

Fast and accurate 4 and 6 stream versions of LIDORT Version 2.3 have been developed for use in the ozone 

profile retrieval algorithms for GOME and related instruments [9]. When the single scatter correction is 

performed, it is only necessary to do the main LIDORT discrete ordinate calculation for the multiple-scatter 

contributions. Some saving in time is gained in this manner, as the Fourier azimuth convergence is 

faster for the multiple-scatter parts.


1.2 Scope of Version 2.3 

In addition to the classical treatment of the beam particular integral discrete ordinate solution developed by 

Chandrasekhar [3] and used in the DISORT work [2], Versions 2.1 and 2.3 feature an alternative treatment 

of this source term using the powerful Green’s function technique developed by Siewert and co-workers 

[4]. Both Versions 2.1 and 2.3 have the ability to generate weighting functions and intensity fields anywhere 

in the atmosphere, for upwelling and downwelling directions, for arbitrary optical depths and stream 

angles. This extends the scope of the model to cover a much wider range of geophysical scenarios, including 

all ground-based and airborne applications. There is also a facility for output of layer-integrated multiple 

scatter source terms and their weighting functions; this is useful for limb scattering retrieval 

applications, and for the sphericity correction required for wide-angle off-nadir viewing in the GOME-2 

instrument [9]. 

In addition to the plane parallel formulation, Versions 2.1 and 2.3 have the ‘pseudo-spherical’ approximation 

of the radiative transfer equation, in which the direct beam attenuation is treated in a curved atmosphere. 

In Version 2.1, all slant path optical thickness values were input to the model (no ray tracing). In 

Version 2.3, these are (optionally) calculated inside LIDORT for a non-refractive atmosphere; for a refracting 

atmosphere, they must be input. Both weighting functions and intensities can be generated using the 

‘average secant’ approximation to the solar beam attenuation - this is valid for most applications with solar 

zenith angles < 90 degrees. The Delta-M scaling option [5] is available for the treatment of sharply peaked 

forward scattering scenarios. The NT single scatter correction package for Version 2.3 is dependent on the 

use of results from LIDORT, and is performed after the main discrete ordinate calculation. 

The Green’s function technique allows the spherical-atmosphere direct beam attenuation to be treated in a 

more sophisticated manner using a Fourier sine series to represent the attenuation. This additional feature 

is the subject of a separate research development for intensity-only calculations [6]; it will not be included 

in the present package. 

Following discussions with a number of users of the model, it has been decided in Versions 2.1 and 2.3 to 

make a division between an intensity-only model (Versions 2.1S and 2.3S) and an extended model with the 

additional weighting function capability (Versions 2.1E and 2.3E). Apart from the three topmost high-level 

modules, all code in the “S” package is a subset of that used in the “E” packages; the extended versions 

have been constructed around the core of the “S” packages. 

This user’s guide is a description of the Extended Version 2.3E. In view of the above remarks, the guide 

will cover all the necessary requirement for the standard “S” version. Additional remarks on installation of 

the standard version will be given where appropriate.Version 2.3E has the following features: 

• Quasi-spherical (average secant) and plane-parallel treatments for an external beam source; 

• Particular integral solution using the “classical” technique or the Green’s function method; 

• Full multiple scattering formalism for several scatterers; 

• Bi-directional reflectance at the lower boundary, including surface thermal emission (isotropic); 

• Intensity output at arbitrary optical depth, for any set of azimuth and elevation angles, for upwelling 

and/or downwelling directions; 

• Additional options for mean value output (flux, mean intensity) at arbitrary optical depth; 

• Additional options for layer-integrated multiple scatter source terms and their weighting functions; 

• Weighting function output at arbitrary optical depth, for any set of azimuth and elevation angles, for 

upwelling and/or downwelling directions;


• Weighting functions with respect to layer atmospheric variables (including phase function quantities) 

defined by the user, weighting functions for albedo and surface emission. 

• The Nakajima-Tanaka single scatter module, for both intensity and weighting function correction. 

1.3 Summary of User’s Guide to Version 2.3 

The layout of the present document is similar to that used in the earlier User’s Guide for Version 2.1. Chapter 

2 deals with the structures (collections of variables) used in the model, with particular emphasis on 

input and output structures; this is followed by an example of the initialization file which controls the handling 

of input variables. Chapter 3 starts with a summary of the contents of the LIDORT package, and discusses 

the environment for a call to LIDORT. A “pseudo-code” description of the top-level calculation 

module is given. 

The preparation of geophysical atmospheric inputs (single scattering albedos, vertical optical thickness 

values, phase function moments, albedos etc., plus variational derivatives required for the linearization) is 

application-specific, and must be done by the user. This is the most important part of the input procedure, 

and we deal with this aspect in detail in Chapter 4, using a number of simple examples including one taken 

from a terrestrial atmosphere application (the forward model component of the ozone profile retrieval from 

nadir backscatter measurements in the UV). 

Chapters 5 deals with installation aspects, from the use of a “makefile” in the compilation and linking, to 

the verification procedure for a given test-case example. The last chapter describes briefly the LIDORT 

error handling procedure and auxiliary modules required for numerical calculations.


2. LIDORT Structures; Input/Output and Initialization File 

Structures are just files with variable declarations and a storage facility (in FORTRAN 77 we use COM- 

MON block statements). The set of LIDORT structures is divided in two groups - one for the intensity-only 

calculations in V2.3S (the Standard or core structures, dealt with in Section 2.1), and a second group 

which is required for the weighting function applications in V2.3E (the Extended or supplementary structures, 

Section 2.2). The Initialization file is described with an example in Section 2.3. Here are summary 

lists. 

First Group - Standard Structures 

Structures in this group are required for both the standard and the extended versions of LIDORT. All structures 

in this Section are found in a directory labeled include_s. The following is a complete list: 

Constants, symbolic dimensions and fixed indices 

• LIDORT.PARS 

Input structures (control and model/geophysical variables) 

• LIDORT_CONTROL.VARS 

• LIDORT_MODEL.VARS 

• LIDORT_GEOPHYS.VARS 

Output structure (intensities) 

• LIDORT_RESULTS.VARS 

Lidort internal structures 

• LIDORT_LEGENDRE.VARS 

• LIDORT_MULTIPLIERS.VARS 

• LIDORT_SETUPS.VARS 

• LIDORT_SOLUTION.VARS 

• LIDORT_SSCORRECTION.VARS 

Second Group - Extended Structures 

Structures in this group are required only for the extended version V2.3E of LIDORT. All structures in this 

Section are found in a directory labeled include_e. The following is a complete list: 

Additional symbolic dimensions 

• LIDORT_L.PARS 

Additional Input structures (control and geophysical variables) 

• LIDORT_L_CONTROL.VARS 

• LIDORT_L_GEOPHYS.VARS 

Additional Output structure (weighting functions) 

• LIDORT_L_RESULTS.VARS 

Lidort internal structures


• LIDORT_L_MULTIPLIERS.VARS 

• LIDORT_L_SETUPS.VARS 

• LIDORT_L_SOLUTION.VARS 

• LIDORT_L_SSCORRECTION.VARS 

2.1 Standard Structures 

The first 5 files in the above list of standard structures cover the input/output requirements for all applications, 

and they must be included in any interface and environment modules. The fixed structure 

LIDORT.PARS must be declared before all other structures to ensure consistency with the dimensioning. 

These 5 files are described in detail below; the remaining four internal structures are not required in the 

input/output description, but are noted briefly here. 

2.1.1 Fixed Structure LIDORT.PARS 

We leave out fixed numbers such as π, 0.0, 1.0, etc., and some fixed character strings used for output formatting. 

File unit numbers are also omitted. The following parameter indices are used to indicate the error 

status for the package or any part of it: 

LIDORT_SUCCESS = 0 (status index for a successful execution). 

LIDORT_SERIOUS = 1 (status index for an aborted execution). 

We now describe the basic set of dimensioning parameters. These numbers should be altered to suit memory 

requirements and/or a particular application. For example, if a calculation with clouds is required, 

allowance should be made for a large number of phase function moments and sufficient quadrature 

streams, so MAXSTRM and MAXMOMENT should be increased in value. All other dimensioning parameters 

are combinations of this basic set, and are not described here. 

MAXSTRM 

Maximum possible number of half-space quadrature streams. 

MAXLAYER 

Maximum allowed number of layers in the atmosphere. 

MAXMOMENT 

Maximum number of Legendre expansion coefficients. This should be at least twice MAXSTRM. 

MAXFOURIER 

Maximum number of allowed terms in the Fourier cosine azimuth series. 

This should be at least twice MAXSTRM. 

MAX_DIRECTIONS 

Maximum number of directions (= 2, “upwelling” or “downwelling”). 

MAX_USER_STREAMS 

Maximum allowed number of user-defined off-quadrature stream angles desired for output. 

MAX_USER_RELAZMS


Maximum allowed number of user-defined relative azimuth angles desired for output. 

MAX_OUT_USERTAUS 

Maximum allowed number of user-defined optical depths at which output is required. 

MAX_OFFGRID_USERTAUS 

Maximum allowed number of off-grid optical depths at which output is required (Off-grid means optical 

depths between layer boundaries). This number should always be less than MAX_OUT_USERTAUS. 

There are two dimensioning parameters here for the optical depth output, as this gives maximum amount of 

choice in output specification. 

The following two parameters are used to ‘toggle’ special cases. 

HOPITAL_TOLERANCE 

If the difference between a user-stream cosine and (say) the solar zenith angle cosine is less than 

HOPITAL_TOLERANCE, L’Hopital’s Rule will be invoked to provide a limit value. Currently set at 10 -5 . 

OMEGA_SMALLNUM 

If any total layer single scattering albedo is within OMEGA_SMALLNUM of unity, then its value will be 

reset to 1.0 - OMEGA_SMALLNUM. Currently set at 10 -6 . 

The following parameters are used to deal with underflow in the transmittance factors. 

MAX_TAU_SPATH 

If the solar path optical thickness exceeds this limit, then corresponding transmittances (negative exponentials) 

will be set to zero. Current value 32. 

MAX_TAU_UPATH, MAX_TAU_QPATH 

If (respectively) the path optical thickness values for the user-defined and quadrature directions exceed 

these limits, then corresponding transmittances (negative exponentials) will be set to zero. Current values 

32. 

2.1.2 Input Structure LIDORT_CONTROL.VARS 

This file contains major control variables for the execution of the package. All variables may be specified 

using explicitly-coded statements written by the user, or they may be assigned by reading entries in the Initialization 

file (this task is done using the LIDORT_V23S_INPUT or LIDORT_V23E_INPUT master 

modules). In this structure, all entries are Boolean values (with the exception of the “FILENAME” variables). 

Where appropriate, all variables are checked for consistency inside the LIDORT package, before 

execution of the main radiative transfer modules. 

DO_DIRECT_BEAM 

Control for the inclusion of the direct beam contribution in the output. Should be set .TRUE. This variable 

will be replaced in later version. 

DO_CLASSICAL_SOLUTION 

Flag for using the Chandrasekhar (classical) solution of the discrete ordinate particular integral for the 

beam source. If not set, the Green’s function solution will be used.


DO_QUASPHER_BEAM 

Flag for use of the quasi-spherical approximation for the direct beam attenuation. If not set, the atmosphere 

will be plane-parallel. 

DO_QSREFRAC_BEAM 

Flag for using refractive geometry input in the quasi-spherical approximation. DO_QUASPHER_BEAM 

must be set first (this is checked). Ray tracing in a refractive atmosphere is not done inside LIDORT, and if 

this flag is set, the slant path optical thickness values TAUTHICK_INPUT (see below) must be calculated 

by the user. If set, a small internal adjustment will be made to the direct beam zenith angles at each layer. 

DO_CHAPMAN_FUNCTION 

Flag for making an internal calculation of TAUTHICK_INPUT for either plane-parallel or non-refractive 

pseudo-spherical geometry. This will not work with DO_QSREFRAC_BEAM (a check is made for this). If 

called, you must specify a height grid and earth radius in order that the goniometry can be done inside 

LIDORT. (New to Version 2.3). 

DO_RAYLEIGH_ONLY 

Flag for performing simulations in a Rayleigh atmosphere with only one scatterer (in this case, molecules). 

If set, a maximum of 3 Fourier terms will be used in the azimuth expansion (Rayleigh phase function has 

cos 2 Θ behavior). (Checked internally). 

DO_ISOTROPIC_ONLY 

Flag for performing simulations in an isotropically scattering atmosphere. If set, then only the azimuthindependent 

contribution is required (first Fourier component). DO_NO_AZIMUTH must be set “true”. 

(Checked internally). 

DO_LAMBERTIAN_ALBEDO 

Flag for using the Lambertian surface reflection condition at the lower boundary. If “false”, boundary is 

assumed fully bidirectional. 

DO_SURFACE_EMISSION 

Flag for inclusion of the black-body surface emission term. 

DO_NO_AZIMUTH 

Flag for controlling inclusion of azimuth dependence in the output. If set, then only the fundamental harmonic 

(azimuth-independent) contribution will be calculated. Must be set for the DO_ISOTROPIC_ONLY 

option to be taken. 

DO_FULL_QUADRATURE 

Flag controlling the use of a single quadrature scheme over complete elevation space (both hemispheres). 

If not set, quadrature will default to the standard double scheme of DISORT and other models. Not recommended 

for practical use. This will be removed in the next version. 

DO_ALL_FOURIER 

Flag controlling calculation of Fourier azimuth terms. If set, LIDORT will calculate all possible Fourier 

components, regardless of any accuracy convergence criterion. If not set, usual convergence criterion will 

apply. Should only be required for test purposes.


DO_DELTAM_SCALING 

Flag for controlling use of the Delta-M scaling option. In most circumstances, this flag will be set. Delta-M 

scaling is now automatic in DISORT Version 2.0, but we prefer to leave the option open. 

DO_ADDITIONAL_MVOUT 

Flag to produce mean-value output in addition to intensities and weighting functions. This is now independent 

of DO_QUAD_OUTPUT and DO_USER_STREAMS flag settings. The model will produce mean 

value results on-the-fly. 

DO_MVOUT_ONLY 

Flag to generate mean-value output only. Since these quantities are hemisphere-integrated, there is no need 

for any angular output, and in addition, only the fundamental harmonic (azimuth-independent) contributes. 

Thus the DO_USER_STREAMS and DO_NO_AZIMUTH flags will be checked for consistency. 

SAVE_LAYER_MSST 

Flag to produce layer-integrated multiple scatter source term output in addition to total intensities and 

weighting functions. This output is restricted to whole layers and can only be generated if flag 

DO_LBOUND_TAUS (see below) is set at the same time. (Output is also restricted to off-quadrature 

angles). 

This is a specialist option and should be used only in certain circumstances, principally the following: (a) 

for a limb-scattering application, in which the single scatter source terms are generated by ray tracing 

along the limb paths, then added to a suitable combination of layer-integrated multiple scatter source 

terms to generate the backscatter field; and (b) for nadir applications with wide viewing angles, where the 

upwelling single scatter contributions may be replaced by NT-corrected accurate values. 

DO_FULLRAD_MODE 

If set, LIDORT will give a full radiance calculation of intensities and weighting functions. The only time 

this flag should not be set is when SAVE_LAYER_MSST output is required. 

DO_SSCORRECTION 

If set, LIDORT will perform the single scatter correction. DO_FULLRAD_MODE must be set first. This 

only works for user-defined stream output, so DO_QUAD_OUTPUT should not be set (this is checked). 

When DO_SSCORRECTION and DO_FULLRAD_MODE are both set, LIDORT will do an internal discrete 

ordinate calculation only of the multiple-scatter field, leaving the single scatter field to be evaluated in 

the SSCORRECTION post-processing module. This is the normal mode of operation; it saves considerably 

on time, since Fourier azimuth convergence is tested only on the multiple-scatter field. 

DOUBLE_CONV_TEST 

(New to Version 2.3). If set, the Fourier azimuth series will be examined twice for convergence; in previous 

versions the double test was fixed. If not set, a single test is made; in this case one saves on an additional 

Fourier computation. However in normal mode, LIDORT examines convergence of the multiple scatter 

field; one test may well be enough for this. Testing is currently being done to verify this is the case. 

DO_QUAD_OUTPUT 

If set, there will be output at the quadrature zenith angles used in the discrete ordinate solution. Not normally 

required, but may be of use. Cannot be used with the single scatter correction 

DO_SSCORRECTION.


DO_USER_STREAMS 

If set, there will be output at a number of off-quadrature zenith angles to be specified by the user. This 

would normally be the output flag to use for stream angle output. If this and the previous flag are not set, 

then there will be no output - this is checked. 

DO_USER_TAUS 

Flag for arbitrary optical depth output. This would normally be used in conjunction with a number of specified 

levels of output. This is the usual flag to set for optical depth. 

DO_LBOUND_TAUS 

Flag for output at layer boundaries. This is complementary to the previous DO_USER_TAUS flag; the two 

cannot be set together. If you want just layer boundary output, this is the one to use. 

DO_UPWELLING, DO_DNWELLING 

Flag for obtaining upwelling and/or downwelling output. One or both must be set! 

DO_DEBUG_INPUT 

Flag for writing-to-file for debugging. Do not use. 

DO_WRITE_INPUT 

Flag for writing-to-file of all input variables (including those in the present structure). Optional. 

DO_WRITE_SCENARIO 

Flag for writing a summary of the atmospheric geophysical input to file. Optional. 

DO_WRITE_RESULTS 

Flag for writing output to file. This is not mandatory, as LIDORT may be embedded in a wavelength loop 

(for example) and the results of each call may be stored and written at a later stage in the test environment. 

DO_WRITE_RESULTS_IDL 

Flag for writing results to IDL file. Not recommended for use in the present version. 

DO_WRITE_FOURIER 

Flag for output of Fourier components. Not normally required. 

INPUT_WRITE_FILENAME, IDLRESULTS_WRITE_FILENAME, SCENARIO_WRITE_FILENAME 

FOURIER_WRITE_FILENAME, RESULTS_WRITE_FILENAME 

These 5 character strings are for the specification of 5 output files corresponding to the above 5 flags controlling 

output. 

LIDORT_ERROR_FILENAME 

Character string indicating name of file to contain error messages (if any). This file will be opened at the 

first serious error encounter and closed before exiting the package. This string will be assigned at the start 

of the initialization file-read, to the string passed as an argument into the file-read master module (see Section 

3.2 for an example). If you do not specify a name, the error file will be “fort.25”. 

LIDORT_ERROR_INIT


This Boolean flag will be set “false” at the start of the input file-read. LIDORT will set this flag internally if 

any errors have occurred. It is not read from the Initialization File. 

2.1.3 Input Structure LIDORT_MODEL.VARS 

This file contains major variables controlling the execution. In addition to such key numbers as 

NSTREAMS and NLAYERS (see below), this file contains all the geometrical input quantities - the solar 

zenith angle cosine, the quadrature stream angles and weights, the fields of user-defined azimuth angles 

and off-quadrature zenith angles, as well as optical depth input. The file is divided into those variables 

which must be specified beforehand by the user (either explicitly or using the file-read capability), and 

those which are derived internally (for example, the Gaussian quadrature weights and cosines which are 

computed afresh with each call to LIDORT). 

User-defined (file-read) variables 

NSTREAMS 

Number of quadrature streams in the zenith cosine half space [0,1]. Must be less than or equal to the symbolic 

dimension MAXSTRM set in the fixed-numbers structure LIDORT.PARS. 

NLAYERS 

Number of layers in the atmosphere (NLAYERS=1isallowed). Must be less than or equal to the symbolic 

dimension MAXLAYER. 

NMOMENTS_INPUT 

Number of moment coefficients characterizing the phase function. When using the delta-M scaling 

NMOMENTS_INPUT should be at least equal to 2*NSTREAMS to ensure that the delta-M truncation 

factor is meaningfully calculated. NMOMENTS_INPUT need not be the same as the internal variable 

NMOMENTS which governs the Legendre phase function expansions in the full discrete ordinate computations 

(see below). NMOMENTS_INPUT will be used in the single scatter correction to generate accurate 

phase functions, so it should be set accordingly. Must be less than MAXMOMENTS. 

FLUX_FACTOR 

Beam source irradiance value. Normally set equal to 1 for “sun-normalized” output. 

LIDORT_ACCURACY 

This is the accuracy criterion for convergence of the Fourier cosine series in relative azimuth. If for each 

output stream, addition of the m th Fourier harmonic changes the total (Fourier-summed) intensity by less 

than LIDORT_ACCURACY of the total, then we have passed the convergence test for the first time (the 

double test is controlled by flag DOUBLE_CONV_TEST). Convergence is tested for all output intensities 

at all output stream angles, optical depths and azimuth angles. 

SUN0 

Beam source zenith angle (degrees). (Checked internally to be in the range [0,90)). 

EARTH_RADIUS 

Earth’s radius in [km]. Only required when DO_CHAPMAN_FUNCTION has been set. This number is 

needed to do the geometry internally in this case. Checked internally to be in the range [6320,6420].


SUNLOCAL_INPUT(MAXLAYER) 

Beam source zenith angles, defined one for each layer. Only required for refractive geometry attenuation of 

the solar beam (DO_QSREFRAC_BEAM), in which case the user must define it. For non-refractive geometry 

SUN0 will be used for every layer. 

ZENITH_TOLERANCE 

Zenith tolerance (nearness of output zenith cosine to 1.0) 

N_USER_STREAMS 

Number of user-defined zenith streams (off-quadrature). Must be not greater than symbolic dimension 

MAX_USER_STREAMS set in LIDORT.PARS. 

USER_ANGLES_INPUT(MAX_USER_STREAMS) 

Array of user-defined angles at which off-quadrature output is to be generated (values in degrees). The 

ordering is not important (LIDORT orders and checks this input internally). 

N_USER_RELAZMS 

Number of relative azimuth angles at which output is to be generated. Must be not greater than symbolic 

dimension MAX_USER_RELAZMS set in LIDORT.PARS. 

USER_RELAZMS(MAX_USER_RELAZMS) 

Values of relative azimuth angles in degrees. 

N_OUT_USERTAUS 

Number of optical depth output values. Required if DO_USER_TAUS is set. Should be less than or equal 

to the symbolic dimension MAX_OUT_USERTAUS. 

USER_TAUS_INPUT(MAX_OUT_USERTAUS) 

Output optical depth values. These can be in any order (LIDORT sorts them in ascending order internally). 

They are checked to see if any value exceeds the optical depth grid defined below as part of the geophysical 

input structure. Repetition of input values is also checked. 

LBOUND_TAUS_INPUT(MAX_OUT_USERTAUS) 

Integer values indicating which layer boundaries are required for optical depth output. Can be specified in 

any order; the value 0 indicates top of the atmosphere, the value 1 indicates the bottom boundary of the first 

layer, etc. Should not be greater than NLAYERS (this is checked). Should only be used if the associated 

flag is set (DO_LBOUND_TAUS). Repetition of input values is also checked. 

Derived variables computed internally in LIDORT 

DO_MSMODE_LIDORT 

Internal flag controlling execution of main LIDORT discrete ordinate calculation. This will be set when 

DO_FULLRAD_MODE and DO_SSCORRECTION are both set. Basically it tells LIDORT to compute 

multiple scatter field only. 

NSTR2 = 2 * NSTREAMS 

NTOTAL = NSTR2 * NLAYERS 

N_SUBDIAG, N_SUPDIAG


NSTR2 is the total number of upward and downward streams; NTOTAL is the actual dimension of boundary-value 

matrix problem. The other two integers are the umber of super- and sub- diagonals used in the 

band-matrix storage scheme required for the boundary value problem. (Both = 3*NSTREAMS - 1). 

X(MAXSTRM), A(MAXSTRM) 

Gaussian half-space quadrature abscissae and weights. AX(MAXSTRM) is the product of X and A. 

XANG(MAXSTRM) are the quadrature angles in degrees computed from the cosine abscissae X. 

NMOMENTS 

This variable is set internally. Normally NMOMENTS = 2*NSTREAMS - 1. The following settings are 

also used, depending on their respective input flags: NMOMENTS = 0 for isotropic case, NMOMENTS = 

2 for Rayleigh only. 

USER_STREAMS(MAX_USER_STREAMS) 

USER_SECANTS(MAX_USER_STREAMS) 

Cosines and inverse cosines of the angles in USER_ANGLES array. 

X0 

Cosine of solar zenith angle SUN0 

SUN_SZA_COSINES(MAXLAYER) 

Cosines of solar zenith angles in SUNLOCAL_INPUT (refractive geometry only). 

N_DIRECTIONS, WHICH_DIRECTIONS(MAX_DIRECTIONS) 

Number of directions (1 or 2) and directional array. 

N_OUT_STREAMS 

Total number of streams for output. This equals N_USER_STREAMS if there is no quadrature output, 

NSTREAMS if there is only quadrature output, and the sum of the two otherwise. 

OUT_ANGLES(MAX_OUT_STREAMS) 

All output angles (quadrature and/or user-defined) arranged in ascending order. 

QUADOUTPUT_INDEX(MAXSTRM) 

Integer mask indicating which of the output angles (if any) are quadrature values. 

USEROUTPUT_INDEX(MAX_USER_STREAMS) 

Integer mask indicating which of the output angles (if any) are user-defined values. 

N_CONVTESTS 

Number of convergence tests which must be satisfied before exiting the Fourier loop. Depends on the number 

of output streams, relative azimuth angles and optical depth output points. 

N_CONV_STREAMS 

Number of zenith streams used in the convergence. Elevation streams within ZENITH_TOLERANCE of 

the nadir position will be omitted from the convergence testing. 

LOCAL_UM_START


Integer indicating start of loop over user-defined angles. This is normally 1, but if one or more of the output 

angles is zero (nadir) or tolerably close to zero (as defined by ZENITH_TOLERANCE), then there is no 

need to repeat calculations for azimuthally-dependent Fourier terms - in this case LOCAL_UM_START > 

1. 

N_OFFGRID_USERTAUS 

OFFGRID_UTAU_LAYERIDX(MAX_OFFGRID_USERTAUS) 

OFFGRID_UTAU_VALUES(MAX_OFFGRID_USERTAUS) 

OFFGRID_UTAU_OUTFLAG(MAX_OUT_USERTAUS) 

OFFGRID_UTAU_OUTINDEX(MAX_OUT_USERTAUS) 

UTAU_LEVEL_MASK_UP(MAX_OUT_USERTAUS) 

UTAU_LEVEL_MASK_DN(MAX_OUT_USERTAUS) 

Optical depth output values are sorted internally into those that coincide with one of the boundary values, 

and those that lie inside a given layer (the OFFGRID_UTAU values). These arrays are masks and indicators 

which control this division. 

N_LAYERSOURCE_UP, N_LAYERSOURCE_DN 

N_ALLLAYERS_UP, N_ALLLAYERS_DN 

STERM_LAYERMASK_UP(MAXLAYER) 

STERM_LAYERMASK_DN(MAXLAYER) 

In the ‘post-processing’ of the solution (integration of the source function), these integers indicate the number 

of whole layers (N_LAYERSOURCE_UP/DN) which must be so integrated. For example if we require 

only upwelling intensity at a point somewhere in the middle of the atmosphere, the source function integration 

will start at the bottom boundary and work upwards until the output level is reached - we do not need 

to do any further integration upwards of this point. The Boolean arrays STERM_LAYERMASK_UP/DN 

control the execution of layer source function integration. 

2.1.4 Input Structure LIDORT_GEOPHYS.VARS 

This file has all geophysical variables required for input to the LIDORT package. The first group contains 

all atmospheric variables required for an intensity calculation (without reference to linearization and 

weighting function output - see Section 2.2.4 for this). This group includes total phase function moment 

coefficients in each layer, total single scattering albedos in each layer, and layer optical depth coordinates 

(expansion coefficients for the layer thermal blackbody function were also specified in Version 1.1). The 

second group comprises all surface information (albedo, bidirectional reflection components, surface emission 

Planck function value, emissivity values). In Chapter 4 we discuss the creation of these geophysical 

inputs for a number of simple examples. 

The input has been simplified to exclude any reference to scatterers - the single scattering albedo is a total 

value and the phase function moments are weighted means of the individual scatterer moments. These total 

geophysical inputs are exactly the same as those required for DISORT, and this has facilitated validation 

comparisons with the models (“apples” and “apples”). 

Group 1. Main atmospheric input. 

OMEGA_TOTAL_INPUT(MAXLAYER)


Single scattering albedo for each layer. Note that none of these values should be equal to one. The values 

are checked internally, and LIDORT will abort if any single scattering albedo is within 

OMEGA_SMALLNUM of unity (this small number is set in the file LIDORT.PARS). 

TAUGRID_INPUT(0:MAXLAYER) 

Vertical optical depth grid τ v (n). The first entry is zero (TOA = top of the atmosphere); the other values are 

the optical depths at the lower boundaries of the layers. 

HEIGHT_GRID(0:MAXLAYER) 

Heights in [km] at the layer boundaries, measured from TOA, thus HEIGHT_GRID(0) is TOA, and 

HEIGHT_GRID(NLAYERS) is the surface level. Only required when the Chapman function calculation of 

TAUTHICK_INPUT is done internally. Must be monotonically decreasing from TOA (this is checked). 

(New to Version 2.3). 

TAUTHICK_INPUT(MAXLAYER,MAXLAYER) 

This quantity τ(n,k) is the slant optical thickness in layer k for a solar beam atmospheric path ending at the 

lower boundary of layer n; thus 

n 

∑ 

k = 1 

τ( nk , ) 

= 

where τ slant (n) is the total slant optical depth for the sun’s ray through all layers up to and including n. For 

a plane-parallel atmosphere, τ slant (n) =τ v (n) /µ 0 , τ(n,k) =(τ v (k) −τ v (k−1)) / µ 0 . For the quasi-spherical 

option in a refracting curved atmosphere, it is necessary to use an external ray tracing module in the preparation 

interface. For a non-refractive atmosphere (straightforward shell geometry), τ(n,k) can also be calculated 

externally, but by turning on flag DO_CHAPMAN_FUNCTION and setting EARTH_RADIUS and 

the height grid HEIGHT_GRID, the calculation will be done internally within LIDORT (this latter option 

is new to Version 2.3). 

PHASMOMS_TOTAL_INPUT(0:MAXMOMENT,MAXLAYER) 

For each layer, we define the quantities β l which are the Legendre moments of the phase function expansion 

multiplied by (2l+1). The first coefficient (indexed by 0) should always be 1 (an internal check is done 

for this). For the isotropic case, there is only this value. Don’t forget to supply enough coefficients if you 

are using the Delta-M scaling option and if you want the single scatter correction using an accurate (nontruncated) 

form of the phase function. These are total phase function moments. 

Group 2. Surface input. 

τ slant 

( n) 

ALBEDO 

This should always be specified, for Lambertian and bidirectional cases. The bidirectional coefficients 

should always be albedo-normalized (see Section 4.1 and [1]). 

BIREFLEC(0:MAXMOMENT,MAXSTRM,MAXSTRM) 

Albedo-normalized bidirectional reflection Fourier expansion coefficients ρ m (−µ i ,µ j ) for scattering from 

incident quadrature direction −µ i to reflecting quadrature direction µ j . This is required for the reflection of 

the downward radiation. 

BIREFLEC_0(0:MAXMOMENT,MAXSTRM)


Albedo-normalized bidirectional reflection Fourier expansion coefficients, for scattering from incident 

direction −µ 0 to reflecting quadrature direction µ j . This is required for the reflection of the direct beam at 

the surface. 

EMISSIVITY(MAXSTRM) 

Surface emission is assumed isotropic so is only present for the fundamental Fourier harmonic. The directional 

emissivity is given by Kirchhoff’s law, so this quantity is to be computed by integrating the reflection 

function (see [1]). For an assumed Lambertian albedo R, the emissivity is 1 − R for all directions. Only 

required if surface emission flag is set. 

SURFBB 

Black-body Planck function at the surface. Only required if surface emission flag is set. 

USER_BIREFLEC(0:MAXMOMENT,MAX_USER_STREAMS,MAXSTRM) 

These are the albedo-normalized bidirectional reflection Fourier expansion coefficients, for scattering from 

incident quadrature direction −µ i to reflecting off-quadrature direction µ α (one of the user-defined stream 

angles) This is required for the reflection of the field at the surface. 

USER_BIREFLEC_0(0:MAXMOMENT,MAX_USER_STREAMS) 

Albedo-normalized bidirectional reflection Fourier expansion coefficients, for scattering from incident 

direction −µ 0 to reflecting off-quadrature direction µ α . This is required for the reflection of the direct beam 

at the surface. 

USER_EMISSIVITY(MAX_USER_STREAMS) 

Directional emissivity for off-quadrature streams. Again follows from Kirchhoff’s law. Only required if 

surface emission flag is set. 

2.1.5 Output Structure LIDORT_RESULTS.VARS 

This file contains all output variables divided into four groups - (i) the complete set of Fourier-summed 

intensities and the single Fourier-term values of intensity; (ii) the mean-value output; (iii) layer-integrated 

multiple scatter source term output (Fourier component and summed values); and (iv) direct and diffuse 

source terms at the lower boundary (Fourier and summed). The latter two groups should only be used in 

conjunction with the flag SAVE_LAYER_MSST. The component terms are local to the Fourier loop inside 

LIDORT_MASTER - they are added to the summed solution values inside the module which performs the 

convergence test on the intensity. Fourier component results may be printed to file (optional), and will then 

be overwritten after the next Fourier component is calculated. 

Group 1. Intensity output 

INTENSITY(MAX_OUT_USERTAUS, MAX_OUT_STREAMS, 

MAX_DIRECTIONS, MAX_USER_RELAZMS) 

Backscatter intensity I(t,u,d,a) at any optical depth t, output stream angle u, direction d (upwelling and/or 

downwelling), and for azimuth angle a. 

INTENSITY_F(MAX_OUT_USERTAUS,MAX_OUT_STREAMS,MAX_DIRECTIONS) 

Fourier component of the backscatter intensity I m (t,u,d) at any optical depth t, output stream angle u, direction 

d (upwelling and/or downwelling).


Group 2. Mean-value output. 

MEAN_INTENSITY(MAX_OUT_USERTAUS,MAX_DIRECTIONS) 

FLUX(MAX_OUT_USERTAUS,MAX_DIRECTIONS) 

Values of flux and mean intensity at any optical depth (upwelling and/or downwelling). 

Group 3. Layer-integrated multiple scatter source term output. 

MSCATSTERM(MAXLAYER,MAX_USER_STREAMS, 


Layer-integrated multiple scatter source term L(n,u,d,a) for layer n, off-quadrature user angle u, direction d 

and relative azimuth angle a. 

MSCATSTERM_F(MAXLAYER,MAX_USER_STREAMS,MAX_DIRECTIONS) 

Fourier component L m (n,u,d) of layer-integrated multiple scatter source term for layer n, off-quadrature 

user angle u, and direction d. 

Group 4. Diffuse and direct-beam bottom-of-atmosphere source term output. 

MSCATBOA_SOURCETERM(MAX_USER_STREAMS, MAX_USER_RELAZMS) 

Bottom-of-the-atmosphere (BOA) multiple scatter source term B(u,a) for off-quadrature user angle u and 

relative azimuth angle a (upwelling direction only). 

MSCATBOA_SOURCETERM_F(MAX_USER_STREAMS) 

Fourier component B m (u) of BOA multiple scatter source term for off-quadrature user angle u. 

DIRECTBOA_SOURCETERM(MAX_USER_STREAMS, MAX_USER_RELAZMS) 

Bottom-of-the-atmosphere (BOA) direct-beam source term B*(u,a) for off-quadrature user angle u and relative 

azimuth angle a (upwelling direction only). 

DIRECTBOA_SOURCETERM_F(MAX_USER_STREAMS) 

Fourier component B* m (u) of BOA direct-beam source term for off-quadrature user angle u. 

2.1.6 Internal Standard Structures 

LIDORT_LEGENDRE.VARS 

Contains all Legendre polynomials required for the discrete ordinate solution (at quadrature angles), at 

arbitrary stream angles (post-processed solution), and for the direct beam zenith angles. 

LIDORT_MULTIPLIERS.VARS 

A number of storage arrays required for the post-processing of the RT solution at arbitrary stream angles 

and optical depths. 

LIDORT_SETUPS.VARS 

Storage for local geophysical variables (the single scattering albedos and other geophysical inputs will be 

changed if the Delta-M scaling option is set). All transmittance factors (optical depth exponential transmittances) 

with the exception of the eigenstream transmittances are pre-calculated before the RT equation is


solved, and the results stored in this structure. Other set-up quantities required for the RT solution are also 

stored here; these are computed in the associated module LIDORT_MISCSETUPS, called only for the first 

(m=0) Fourier component. 

LIDORT_SOLUTION.VARS 

All variables associated with the discrete ordinate solution, including the homogeneous solution separation 

constants and eigenvectors, the particular integral solution vectors, and the matrices and column vectors 

required for the boundary value problem. Many of these quantities are required repeatedly in the weighting 

function computations which follow the solution for the original intensity field. 

LIDORT_SSCORRECTION.VARS 

Storage for variables computed in the single scatter post-processing for the intensity. Many of these quantities 

are required repeatedly in the single scatter weighting function computations which follow. 

2.2 Extended Structures 

The first 4 files in this list cover the I/O requirements for all extended (weighting function) applications, 

and they must be included in any interface and environment modules. The fixed structure 

LIDORT_L.PARS must be declared before the following 3 I/O structures. These 4 files are described in 

detail below; the remaining 3 internal structures are not part of the I/O description, and are given brief 

mention here. 

2.2.1 Fixed Structure LIDORT_L.PARS 

MAX_PARAMETERS 

This is the only additional symbolic dimension required for the extended version. It denotes the maximum 

number of atmospheric parameters for which weighting functions will be available. 

2.2.2 Input Structure LIDORT_L_CONTROL.VARS 

As before, all variables may be specified using explicitly-coded statements written by the user, or they may 

be assigned by reading entries in the Initialization file (this task is done using the LIDORT_V23E_INPUT 

master module). There are 4 Flags which must be set by the user - the final flag is set internally. Version 2.3 

addition: A list of weighting function names has been included for ease of identification. 

DO_SIMULATION_ONLY 

If set, then no weighting functions will be computed. The model will then compute the intensities alone 

using the standard code (which can of course be run in stand-alone mode in a Version 2.1S application). 

DO_LAYER_LINEARIZATION 

Indicates that atmospheric weighting functions should be output. 

DO_ALBEDO_LINEARIZATION 

Control for albedo weighting functions. Will not execute for a black surface (albedo = 0). 

DO_SURFBB_LINEARIZATION 

Control for weighting functions with respect to the surface Planck function. Will only execute if the corresponding 

flag is set in the core input control.


DO_LINEARIZATION 

Set internally when one or more of the above 3 linearization flags has been set. 

TOTAL_LAYERWF 

Total number of desired weighting functions. Not every weighting function will be output for every layer 

(though this is usually the case) and this number represents a maximum number. Must be less than dimensioning 

parameter MAX_PARAMETERS. 

LAYERWF_NAMES(MAX_PARAMETERS) 

A character-string array (31 characters presently specified) for descriptions of the weighting functions. 

This is for labelling purposes only. 

2.2.3 Input Structure LIDORT_L_GEOPHYS.VARS 

This file contains all additional variables that specify the layer weighting function output. They are only 

required if the general flag DO_LAYER_LINEARIZATION has been set. They include inputs 

LAYER_VARY_FLAG and LAYER_VARY_NUMBER which determine the extent of weighting function 

calculations for atmospheric parameters. 

LAYER_VARY_FLAG(MAXLAYER) 

If DO_LAYER_LINEARIZATION has been set, then this flag indicates which layers will be varied in the 

linearization, i.e. weighting functions will be output only for those layers which have been thus flagged. 

LAYER_VARY_NUMBER(MAXLAYER) 

Controls the number of weighting functions to be computed for a given layer (number of parameters varied 

in given layer). Each entry must be less than the dimensioning parameter MAX_PARAMETERS. 

OMEGA_VARS_TOTAL_INPUT(MAX_PARAMETERS,MAXLAYER) 

OMEGA_VARS_TOTAL_INPUT(q,n) is the relative variation in the total single scattering albedo in layer 

n, with respect to varying parameter q in that layer. (Version 2.3: no reference to scatterer). 

EXT_VARS_INPUT(MAX_PARAMETERS, MAXLAYER). 

Relative variation in extinction coefficient for layer n with respect to varying parameter q in that layer. 

PHASMOM_VARS_TOTAL_INPUT(MAX_PARAMETERS, MAXLAYER). 

Relative variation in phase function moment coefficients. For Legendre moment l and for layer n with 

respect to varying parameter q in that layer. (New feature in Version 2.3). 

2.2.4 Output Structure LIDORT_L_RESULTS.VARS 

As with the standard results structure (Section 2.1.5), the variables are classified in four groups. Again, 

groups (iii) and (iv) are only required for the specialist option flagged by SAVE_LAYER_MSST. 

Group 1. General and Fourier-component Weighting function output. 

ATMOSWF (MAX_PARAMETERS, MAXLAYER, MAX_OUT_USERTAUS, 

MAX_OUT_STREAMS, MAX_DIRECTIONS, MAX_USER_RELAZMS)


Weighting functions K(q,n,t,u,d,a) with respect to atmospheric variable q in layer n, at optical depth t, output 

stream angle u, upwelling and/or downwelling direction d, and for relative azimuth angle a. 

ALBEDOWF (MAX_OUT_USERTAUS, MAX_OUT_STREAMS, 


Weighting functions K R (t,u,d,a) with respect to surface albedo, at optical depth t, output stream angle u, 

upwelling and/or downwelling direction d, and for relative azimuth angle a. 

SURFBBWF (MAX_OUT_USERTAUS, MAX_OUT_STREAMS, 


Weighting functions K S (t,u,d,a) with respect to surface black-body emission, at optical depth t, output 


ATMOSWF_F (MAX_PARAMETERS, MAXLAYER, MAX_OUT_USERTAUS, 

MAX_OUT_STREAMS, MAX_DIRECTIONS) 

Fourier component weighting functions K m (q,n,t,u,d) with respect to atmospheric variable q in layer n, at 

optical depth t, output stream angle u, upwelling and/or downwelling direction d. 

ALBEDOWF_F (MAX_OUT_USERTAUS, MAX_OUT_STREAMS, 


Fourier component weighting functions K Rm (t,u,d) with respect to surface albedo, at optical depth t, output 


Group 2. Mean-value Weighting function output. 

MINT_ATMOSWF (MAX_PARAMETERS, MAXLAYER, 

MAX_OUT_USERTAUS, MAX_DIRECTIONS) 

Weighting functions of Mean intensity K I (q,n,t,d) with respect to atmospheric variable q in layer n, at optical 

depth t, and upwelling and/or downwelling direction d. 

MINT_ALBEDOWF (MAX_OUT_USERTAUS, MAX_DIRECTIONS) 

Weighting functions of Mean intensity K IR (t,d) with respect to surface albedo, at optical depth t, direction 

d. 

MINT_SURFBBWF (MAX_OUT_USERTAUS, MAX_DIRECTIONS) 

Weighting functions of Mean intensity K IS (t,d) with respect to surface black-body emission, at optical 

depth t, direction d. 

FLUX_ATMOSWF (MAX_PARAMETERS, MAXLAYER, 

MAX_OUT_USERTAUS, MAX_DIRECTIONS) 

Weighting functions of Flux K F (q,n,t,d) with respect to atmospheric variable q in layer n, at optical depth t, 

and upwelling and/or downwelling direction d. 

FLUX_ALBEDOWF (MAX_OUT_USERTAUS, MAX_DIRECTIONS) 

Weighting functions of Flux K FR (t,d) with respect to surface albedo, at optical depth t, direction d. 

FLUX_SURFBBWF (MAX_OUT_USERTAUS, MAX_DIRECTIONS)


Weighting functions of Flux K FS (t,d) with respect to surface black-body emission, at optical depth t, direction 

d. 

Group 3. Multiple scatter layer source term weighting function output. 

ATMOSWF_MSCATSTERM (MAX_PARAMETERS, MAXLAYER, MAXLAYER, 

MAX_OUT_STREAMS, MAX_DIRECTIONS, MAX_USER_RELAZMS) 

Layer-integrated multiple scatter source term weighting functions L(q,n,k,u,d,a) for layer k, off-quadrature 

stream angle u, upwelling and/or downwelling direction d and relative azimuth angle a, with respect to 

atmospheric variable q which is varying in layer n. 

ALBEDOWF_MSCATSTERM (MAXLAYER, MAX_USER_STREAMS, 


Layer-integrated multiple scatter source term weighting functions L R (k,u,d,a) with respect to surface 

albedo for layer k, off-quadrature stream angle u, upwelling and/or downwelling direction d and relative 

azimuth angle a. 

SURFBBWF_MSCATSTERM (MAXLAYER, MAX_USER_STREAMS, 


Layer-integrated multiple scatter source term weighting functions L S (k,u,d,a) with respect to surface blackbody 

emission for layer k, off-quadrature stream angle u, upwelling and/or downwelling direction d and 

relative azimuth angle a. 

ATMOSWF_MSCATSTERM_F (MAX_PARAMETERS, MAXLAYER, MAXLAYER, 

MAX_USER_STREAMS, MAX_DIRECTIONS) 

Layer-integrated multiple scatter source term Fourier component weighting functions L m (q,n,k,u,d) for 

layer k, off-quadrature stream angle u and direction d, with respect to atmospheric variable q which is varying 

in layer n. 

ALBEDOWF_MSCATSTERMF_F (MAXLAYER, MAX_USER_STREAMS, 


Layer-integrated multiple scatter source term Fourier component weighting functions L Rm (k,u,d) with 

respect to surface albedo for layer k, off-quadrature stream angle u and upwelling and/or downwelling 

direction d. 

Group 4. Bottom-of-atmosphere source term weighting function output. 

ATMOSWF_MSCATBOA_STERM (MAX_PARAMETERS, MAXLAYER, 

MAX_OUT_STREAMS, MAX_USER_RELAZMS) 

BOA multiple scatter source term weighting functions B(q,n,u,a) for off-quadrature stream angle u and relative 

azimuth angle a, with respect to atmospheric variable q varying in layer n. 

ATMOSWF_DIRECTBOA_STERM (MAX_PARAMETERS, MAXLAYER, 

MAX_OUT_STREAMS, MAX_USER_RELAZMS) 

BOA direct-beam source term weighting functions B(q,n,u,a) for off-quadrature stream angle u and relative 

azimuth angle a, with respect to atmospheric variable q varying in layer n.


ALBEDOWF_MSCATBOA_STERM (MAX_USER_STREAMS, MAX_USER_RELAZMS) 

BOA multiple scatter source term weighting functions B R (u,a) with respect to surface albedo for offquadrature 

stream angle u and relative azimuth angle a. 

SURFBBWF_BOA_STERM (MAX_USER_STREAMS, MAX_USER_RELAZMS) 

BOA source term weighting functions B S (u,a) with respect to surface black-body emission for off-quadrature 

stream angle u and relative azimuth angle a. 

ATMOSWF_MSCATBOA_STERM_F 

(MAX_PARAMETERS, MAXLAYER,MAX_USER_STREAMS) 

BOA multiple scatter source term Fourier component weighting functions B m (q,n,u) for off-quadrature 

stream angle u, with respect to atmospheric variable q which is varying in layer n. 

ATMOSWF_DIRECTBOA_STERM_F 

(MAX_PARAMETERS, MAXLAYER,MAX_USER_STREAMS) 

BOA direct-beam source term Fourier component weighting functions B m (q,n,u) for off-quadrature stream 

angle u, with respect to atmospheric variable q which is varying in layer n. 

ALBEDOWF_MSCATBOA_STERM_F (MAX_USER_STREAMS) 

BOA multiple scatter source term Fourier component weighting functions B Rm (u) with respect to surface 

albedo for off-quadrature stream angle u. 

2.2.5 Internal Extended Structures 

LIDORT_L_MULTIPLIERS.VARS 

A number of storage arrays required for the post-processing of the linearized RT solution at arbitrary 

stream angles and optical depths. 

LIDORT_L_SETUPS.VARS 

Storage for local geophysical variables required for the linearization (weighting function) analysis (the single 

scattering albedo, optical depth and phase function moments variational inputs will be changed if the 

Delta-M scaling option is set). All linearizations of the transmittance factors computed in the original setup 

operation and stored in LIDORT_SETUPS.VARS are held in this structure, along with any other help 

arrays that can be pre-computed before the weighting function computation starts. 

LIDORT_L_SOLUTION.VARS 

All variables associated with the linearization of the discrete ordinate solution, including linearized values 

of the homogeneous solution separation constants and eigenvectors, particular integral solution vectors and 

multipliers, and column vectors required for the linearized boundary value problem. 

2.3 Initialization File Example 

In this section, we present a typical initialization file. [We use the “Courier” Font to denote text copied 

from software code]. The file is read in LIDORT_V23E_INPUT, using the FINDPAR module - this looks 

for a character string comprising a prefix (in this case the word “LIDORT”) and a text indication of the 

variable to be assigned (for example Include direct beam?) and then reads the variables specified 

underneath the character string. All strings ending with a question mark indicate the assignment of Bool-


ean variables. The file-read is divided into three parts - one task for the assignment of variables in 

LIDORT_CONTROL.VARS, the second task for variables in LIDORT_MODEL.VARS, and the third task 

assigns variables in LIDORT_L_CONTROL.VARS. The first two parts are mandatory for both versions 

V2.3S and V2.3E, the third part is only required for V2.3E. 

**** PART 1. Variables in LIDORT_CONTROL.VARS **** 

LIDORT - Do full radiance calculation? 

.TRUE. 

LIDORT - Do single scatter correction? 

.TRUE. 

LIDORT - Output multiple scatter layer source functions? 

.FALSE. 

LIDORT - Perform double convergence test? 

.TRUE. 

LIDORT - Rayleigh atmosphere only? 

.FALSE. 

LIDORT - Isotropic atmosphere only? 

.FALSE. 

LIDORT - Include direct beam? 

.TRUE. 

LIDORT - Use classical beam solution? 

.FALSE. 

LIDORT - Quasi-spherical treatment of direct beam? 

.TRUE. 

LIDORT - Perform internal Chapman function calculation? 

.TRUE. 

LIDORT - Beam path with refractive atmosphere? 

.FALSE. 

LIDORT - Lambertian albedo? 

.TRUE. 

LIDORT - Include surface emission? 

.FALSE. 

LIDORT - No azimuth dependence in the solution? 

.TRUE. 

LIDORT - Use full -1 to +1 quadrature? 

.FALSE. 

LIDORT - Compute all Fourier components? 

.FALSE. 

LIDORT - Include Delta-M scaling? 

.TRUE. 

LIDORT - Generate mean-value output additionally? 

.TRUE.


LIDORT - Generate only mean-value output? 

.FALSE. 

LIDORT - Upwelling output? 

.TRUE. 

LIDORT - Downwelling output? 

.TRUE. 

LIDORT - Debug write? 

.FALSE. 

LIDORT - Input control write? 

.TRUE. 

LIDORT - Input scenario write? 

.TRUE. 

LIDORT - Results write? 

.TRUE. 

LIDORT - Results write IDL output? 

.FALSE. 

LIDORT - Fourier component output write? 

.FALSE. 

LIDORT - Include quadrature angles in output? 

.TRUE. 

LIDORT - User-defined stream angles? 

.TRUE. 

LIDORT - User-defined optical depths? 

.FALSE. 

LIDORT - Layer boundary optical depths? 

.TRUE. 

LIDORT - filename for main output 

lidort_myproblem.result 

LIDORT - filename for main IDL output 

lidort_myproblem_IDL.result 

LIDORT - filename for input write 

lidort_myproblem.input 

LIDORT - filename for scenario write 

lidort_myproblem.scenario 

LIDORT - Fourier output filename 

lidort_myproblem.fourier 

**** Part 2. Variables in LIDORT_MODEL.VARS **** 

LIDORT - Number of half-space streams 

10 

LIDORT - Number of atmospheric layers 

43 

LIDORT - Number of Legendre moments


19 

LIDORT - Solar flux constant 

1.0 

LIDORT - Fourier series convergence 

0.001 

LIDORT - Zenith tolerance level 

0.0001 

LIDORT - Solar zenith angle (degrees) 

65.0 

LIDORT - Earth radius (km) 

6371.0 

LIDORT - Number of user-defined relative azimuth angles 

2 

LIDORT - User-defined relative azimuth angles 

60.0 

50.0 

LIDORT - Number of user-defined stream angles 

3 

LIDORT - User-defined stream angles 

0.0 

15.0 

30.0 

LIDORT - Number of user-defined optical depths 

1 

LIDORT - User-defined optical depths 

0.2065 

LIDORT - Number of layer boundary optical depths 

3 

LIDORT - Which layer boundaries for optical depth 

0 

1 

2 

**** Part 3. Variables in LIDORT_L_CONTROL.VARS **** 

LIDORT - Simulation only? 

.FALSE. 

LIDORT - Variation for atmospheric layers? 

.TRUE. 

LIDORT - Variation for surface albedo? 

.FALSE. 

LIDORT - Variation for surface emission?


.FALSE. 

LIDORT - Total number of Layer weighting functions 

1 

LIDORT - Layer weighting function names 

ozone volume mixing ratio------


3. LIDORT Package; Environment and Master Modules 

3.1 Package Organization 

The LIDORT package is organized in six directories: 

src_v23e_master 

src_v23s_master 

src_standard 

src_extension 

include_s 

include_e 

All structures as described in the previous chapter are to be found in the two “include” directories. The 

source code is divided into 4 directories which we list here with remarks. 

Directory src_v23e_master 

This contains 3 high-level master modules which control the execution of the extended model. These are: 

LIDORT_V23E_INPUT. This carries out the initialization file-read as described in the previous Section. It 

is called directly from a user-defined environment; an example is given in the next Section. 

LIDORT_V23E_MASTER. This is the main module, again called directly from a user-defined environment. 

The next Section gives an example of its use. 

LIDORT_V23E_FOURIER. This is the master module for calculation of a single Fourier component of the 

intensity and weighting function output. It is called directly in LIDORT_V23E_MASTER as part of the 

Fourier loop, but the call will not be present in the user-defined environment. In Section 3.3 we illustrate 

the use of this module inside LIDORT_V23E_MASTER. 

Directory src_v23s_master 

Contains 3 high-level master modules which control the execution of the standard model. These are 

LIDORT_V23S_INPUT, LIDORT_V23S_MASTER and LIDORT_V23S_FOURIER. The role of these 

modules is exactly analogous to their equivalents in the extended master directory, and the above remarks 

apply equally. Usage of these modules will be noted in the examples that follow. 

Directory src_standard 

Contains all core code for the standard version; there are 12 modules in all. We list them with a brief note 

on function. The first 9 are called in the order in which they appear in the main and/or the Fourier master 

modules; the remaining 3 are auxiliary modules. 

LIDORT_CHECKINPUT.f 

Checks all control, model and geophysical inputs for consistency. 

LIDORT_DERIVEINPUT.f


This is an internal assignation of model variables that are not declared explicitly as part of the file-read or 

in an external environment. Includes tasks like sorting the stream angles input, sorting and assigning masks 

for the arbitrary optical depth output. 

LIDORT_MISCSETUPS.f 

Only called for Fourier component m = 0. A number of set-up operations including the Delta-M scaling 

and the preparation of all optical depth exponentials that can be pre-calculated. 

LIDORT_RTSOLUTION.f 

Solution of the discrete ordinate radiative transfer equation. Returns the eigensolutions and separation constants 

from the homogeneous equation, plus the particular (beam) solution vectors for both the classical 

solution technique and the Green function method. 

LIDORT_BVPROBLEM.f 

Set-up and solution of the boundary-value problem (constants of integration) in a multi-layer atmosphere. 

This requires the L-U decomposition (matrix inversion). 

LIDORT_INTENSITY.f 

Computation of intensities at user-defined optical depths and stream angles; this is the post-processing 

(source function integration). 

LIDORT_MULTIPLIERS.f 

Computation of multipliers required for the layer source terms that form the heart of the source function 

integration technique. 

LIDORT_CONVERGE.f 

Examines convergence of Fourier series for all intensities; upgrades the intensity Fourier series. 

LIDORT_SSCORRECTION.f 

If flagged by DO_SSCORRECTION, this module calculates the post-processed single scatter intensity and 

will return the corrected total intensity direct to the output. 

LIDORT_WRITEMODULES.f 

Four modules for writing outputs to file. Will produce full intensity results, Fourier components of intensity, 

a scenario description and a summary of the input data (corresponding to the write control flags set in 

LIDORT_CONTROL.VARS). 

LIDORT_AUX.f 

Standard numerical routines for eigenproblem solution (ASYMTX as used in DISORT is preferred, though 

LAPACK software is available), and linear algebra modules (LAPACK band storage and L-U decomposition 

modules). Legendre polynomial and Gauss quadrature evaluation. Includes the input file-read tool 

FINDPAR. Includes also error tracing modules. 

LIDORT_READINPUT.f 

Sequential read of all standard control and model variables in the initialization file. 

Directory src_extension


Contains all internal modules that are additionally required for the extended version; there are 10 modules 

in all, and they are all concerned with aspects of weighting function generation. The first 8 are called in the 

order in which they appear in the main and/or the Fourier master modules; the remaining 2 are auxiliary 

modules. The naming is chosen so that the tasks executed by these additional modules are equivalent to the 

original tasks for the original (intensity-only) modules. 

LIDORT_L_CHECKINPUT.f 

Checks additional control and geophysical inputs for consistency. 

LIDORT_L_MISCSETUPS.f 

Only called for Fourier component m = 0. A number of additional set-up operations including the Delta-M 

scaling for variational inputs and the preparation of all optical depth exponentials that can be pre-calculated. 

LIDORT_L_RTSOLUTION.f 

Linearization analysis of the discrete ordinate radiative transfer equation. Returns the derivatives of the 

eigensolutions and separation constants, plus the particular (beam) solution vectors with respect to atmospheric 

parameter variations. 

LIDORT_L_BVSETUPS.f 

Column vector set-up for the linearized boundary-value problem. The actual matrix back-substitution 

which returns the linearized integration constants is done in LIDORT_V23E_FOURIER. 

LIDORT_L_WFCALC.f 

Computation of weighting functions at user-defined optical depths and stream angles. This is the post-processing 

(source function integration) technique; the routine is modeled along the same lines as 

LIDORT_INTENSITY. 

LIDORT_L_MULTIPLIERS.f 

Computation of derivatives of the multipliers required for the layer source terms that form the heart of the 

source function integration technique. 

LIDORT_L_FOURIERADD.f 

Upgrades the weighting function Fourier series. 

LIDORT_L_SSCORRECTION.f 

Post-processing single scatter correction for the weighting functions. This generates the single scatter 

quantities and returns the corrected weighting functions directly to output. Note that single scatter corrections 

only involve atmospheric scatter - this means that albedo weighting functions do not need to be corrected. 

LIDORT_L_WRITEMODULES.f 

Modules for writing additional (weighting function) outputs to file. Will produce full weighting function 

results, Fourier components, additional scenario descriptions and a summary of supplementary input data. 

LIDORT_L_INPUTREAD.f 

Read of all extended control variables in the initialization file.


3.2 Use of LIDORT Master Module in an External Environment 

This is demonstrated in this schematic computational sequence (pseudo-code), which should be referred to 

in the discussion that follows. Comment lines are prefaced by the letters “/*”. The example deals with a 

simple wavelength loop calculation. LIDORT names and variables are given in upper case letters. 

main user_lidort 

/* Core fixed structure 

include ‘LIDORT.PARS’ 

/* Core input structures 

include ‘LIDORT_CONTROL.VARS’ 

include ‘LIDORT_MODEL.VARS’ 

include ‘LIDORT_GEOPHYS.VARS’ 

/* Core output structure 

include ‘LIDORT_RESULTS.VARS’ 

/* Supplementary fixed structure 

include ‘LIDORT_L.PARS’ 

/* Supplementary input structures 

include ‘LIDORT_L_CONTROL.VARS’ 

include ‘LIDORT_L_GEOPHYS.VARS’ 

/* Supplementary output structure 

include ‘LIDORT_L_RESULTS.VARS’ 

/* status declarations 

INTEGER STATUS_INPUTREAD 

INTEGER STATUS_INPUTCHECK 

INTEGER STATUS_CALCULATION 

/* Define output in the way you want! 

Declare output arrays 

/* File-read variables in the input structures 

call LIDORT_V23E_INPUT 

(‘LIDORT.INP’,’LIDORT.ERR’,STATUS_INPUTREAD) 

if (STATUS_INPUTREAD = LIDORT_SERIOUS) write message & abort 

/* Start wavelength loop 

for i = 1, n_user_wavelengths, begin 

/* Assign variables in LIDORT_GEOPHYS.VARS and LIDORT_L_GEOPHYS.VARS 

call USER_LIDORT_PREPARE 

/* LIDORT master call and error check 

call LIDORT_V23E_MASTER 

(STATUS_INPUTCHECK,STATUS_CALCULATION) 

if (STATUS_INPUTCHECK=LIDORT_SERIOUS) error message & abort 

if (STATUS_CALCULATION=LIDORT_SERIOUS) error message & abort 

copy LIDORT output to user-defined output arrays


/* End wavelength loop 

end for 

/* finish 

write user-defined output arrays 

stop and end of program 

LIDORT execution is controlled by a single module LIDORT_V23E_MASTER for the extended version, 

and LIDORT_V23S_MASTER for version 2.3S (intensity-only). Both these modules must be called from 

a user-defined environment, once for each wavelength (LIDORT is monochromatic). Note that within the 

wavelength loop, the call to one of these master module is preceded by the user-defined preparation module 

which will assign values to input variables in LIDORT_GEOPHYS.VARS (both versions) and 

LIDORT_L_GEOPHYS.VARS (extended version only). 

The main call is preceded by a call to either the LIDORT_V23S_INPUT master module (intensity-only) or 

LIDORT_V23E_INPUT (extended version). These modules will read the appropriate input from the Initialization 

file LIDORT.INP (passed as a subroutine argument). File-read errors will be written to an error 

file name LIDORT.ERR (also passed as a subroutine argument). If the STATUS_INPUTREAD integer 

output is not equal to LIDORT_SUCCESS, the program should stop and the user should examine the error 

file. A similar output (STATUS_INPUTCHECK) is available for the checking of the input data once the 

file-read is complete (checking is internal to LIDORT). This step is optional; it is possible to write down 

the control inputs instead of using file-read methods, but this requires a certain level of confidence! 

After each call to the master module, the STATUS_CALCULATION integer output is examined, and the 

program stopped if there is a failure. Any errors arising in the master modules will be traced and written to 

the error file passed as an argument to the module. 

In most cases, it is sufficient to call LIDORT_V23E_INPUT just once before a wavelength loop, though 

some of the input options may be changed before each call to LIDORT_V23E_MASTER. Note that the 

file-read modules in LIDORT_V23E_INPUT are optional - it is possible for the user to dispense with this 

kind of input set-up and assignment and simply assign input variables explicitly. The output for some 

wavelengths may be suppressed (note that the output write files in LIDORT have “unknown” status - they 

are always overwritten). It is also necessary in most cases for the user to create anew the variables in 

LIDORT_GEOPHYS.VARS and LIDORT_L_GEOPHYS.VARS for each wavelength. 

The geophysical input declared in LIDORT_GEOPHYS.VARS is application-dependent, and must be set 

up by the user (the module USER_LIDORT_PREPARE in the above example). Note that most of the input 

variables in LIDORT_CONTROL.VARS and LIDORT_MODEL.VARS (and for the supplementary file 

LIDORT_L_CONTROL.VARS) are checked for consistency inside the LIDORT master module. In fact 

this is the first task - if problems arise with the inputs, then LIDORT will abort before any calculations are 

done. The geophysical inputs are also checked at this stage for consistency. 

3.3 Computational Sequence in LIDORT_V23E_MASTER 

This is the computational sequence inside LIDORT_V23E_MASTER, written in pseudo-code (conventions 

as above). Note that the three separate calls to output modules (“SCENARIO”, “FOURIER” and


“RESULTS”) are governed by separate flags and output filenames. Note also that the list of “include” 

structures is the same as that required for the user-defined environment defined in the previous Section. 

subroutine LIDORT_V23E_MASTER 

(STATUS_INPUTCHECK,STATUS_CALCULATION) 

/* Core fixed structure 

include ‘LIDORT.PARS’ 

/* Core input structures 

include ‘LIDORT_CONTROL.VARS’ 

include ‘LIDORT_MODEL.VARS’ 

include ‘LIDORT_GEOPHYS.VARS’ 

/* Core output structure 

include ‘LIDORT_RESULTS.VARS’ 

/* Supplementary fixed structure 

include ‘LIDORT_L.PARS’ 

/* Supplementary input structures 

include ‘LIDORT_L_CONTROL.VARS’ 

include ‘LIDORT_L_GEOPHYS.VARS’ 

/* Supplementary output structure 

include ‘LIDORT_L_RESULTS.VARS’ 

/* subroutine output arguments 

INTEGER STATUS_INPUTCHECK 

INTEGER STATUS_CALCULATION 

/* declare local variables (AZMFAC are Azimuth Fourier cosine factors) 

double precision AZMFAC(MAX_USER_RELAZMS); 

integer FOURIER, N_FOURIER; boolean LOCAL_ITERATION; 

integer LOCAL_STATUS_1, LOCAL_STATUS_2 

/* initialize output 

STATUS_INPUTCHECK = LIDORT_SUCCESS 

STATUS_CALCULATION = LIDORT_SUCCESS 

/* Check inputs 

call LIDORT_CHECKINPUT (LOCAL_STATUS_1) 

if (DO_LINEARIZATION) call LIDORT_CHECKINPUT (LOCAL_STATUS_2) 

if (LOCAL_STATUS_1 or LOCAL_STATUS_2 = LIDORT_SERIOUS) then 

STATUS_INPUT_CHECK = LIDORT_SERIOUS 

write message & abort 

end if 

/* If you want LIDORT to calculate TAUTHICK_INPUT, here is the call! 

if (DO_CHAPMAN_FUNCTION) call CHAPMAN_FUNCTION 

/* write input variables 

if (DO_WRITE_INPUT) then 

open input_write file


call LIDORT_WRITEINPUT 

if (DO_LINEARIZATION) call LIDORT_L_WRITEINPUT 

close input_write file 

end if 

/* Get quadrature 

call to Gaussian quadrature module - get X, A, AX and XANG 

/* Get derived input 

/* (assign non-file-read model variables in LIDORT_MODEL.VARS) 

call LIDORT_DERIVE_INPUT 

/* Initialize Fourier loop 

Set number of Fourier components required (N_FOURIER) 

FOURIER = -1 

LOCAL_ITERATION = .true. 

/* Fourier loop (iterative) 

do while (LOCAL_ITERATION.AND.FOURIER.LT.N_FOURIER) 

FOURIER = FOURIER + 1 

Set up azimuth cosine factors AZMFAC in Fourier series 

call LIDORT_V23E_FOURIER (FOURIER,LOCAL_STATUS_1) 

if (LOCAL_STATUS_1 = LIDORT_SERIOUS) then 

STATUS_CALCULATION = LIDORT_SERIOUS 

write message & abort 

end if 

call LIDORT_CONVERGE (AZMFAC, FOURIER, LOCAL_ITERATION) 

if ( DO_LINEARIZATION ) then 

call LIDORT_L_FOURIERADD ( AZMFAC, FOURIER ) 

end if 

if (DO_WRITE_FOURIER) then 

if (FOURIER = 0) open Fourier_write file 

call LIDORT_FOURIER_OUTPUT 

if (DO_LINEARIZATION) call LIDORT_L_WRITEINPUT 

if (.not.LOCAL_ITERATION) close Fourier_write file 

end if 

end do 

/* Single scatter corrections 

if (DO_SSCORRECTION) then 

call LIDORT_SSCORRECTION 

if (DO_LINEARIZATION) call LIDORT_L_SSCORRECTION 

end if 

/* write major result variables 

if (DO_WRITE_RESULTS) then 

open Results_write file 

call LIDORT_WRITERESULTS


if (DO_LINEARIZATION) call LIDORT_L_WRITERESULTS 

close Results_write file 

end if 

/* write scenario variables 

if (DO_WRITE_SCENARIO) then 

open Scenario_write file 

call LIDORT_WRITESCEN 

if (DO_LINEARIZATION) call LIDORT_L_WRITESCEN 

close Scenario_write file 

end if 

/* finish 

end 

Remarks 

The main loop is over the Fourier components; LIDORT_V23E_FOURIER computes all intensities and 

weighting functions for each component. The number N_FOURIER depends on the application, but cannot 

be greater than a pre-defined maximum number MAX_FOURIER defined in LIDORT.PARS. For the isotropic 

case, N_FOURIER = 0; for Rayleigh only, N_FOURIER = 2. Normally the Fourier loop will continue 

until the convergence test (LIDORT_CONVERGE) has indicated that the addition of a Fourier term 

to the intensity has not improved the accuracy beyond that set by the input convergence criterion. 

LIDORT_CONVERGE will set the iteration flag LOCAL_ITERATION. (The additional module 

LIDORT_L_FOURIERADD merely updates the Fourier series for the weighting functions). 

Note the use of the DO_LINEARIZATION flag (see Section 2.2.1 for the definition). If we remove all the 

clauses with this flag and replace LIDORT_V23E_FOURIER with the standard (intensity-only) Fourier 

master module LIDORT_V23S_FOURIER, then we have simply LIDORT_V23S_MASTER, the overall 

master module for the standard version. It’s that simple!


4. Geophysical input preparation 

We will first go through the variables in the core structure LIDORT_GEOPHYS.VARS as they appear in 

Section 2.2.4, showing how they may be assigned for some representative scenarios. Each layer in the 

atmosphere is assumed homogeneous. More details can be found in [1] and [7]. We then go through the 

variational input required for the supplementary structure LIDORT_L_GEOPHYS.VARS. 

4.1 Preparation for LIDORT_GEOPHYS.VARS Core Structure 

LIDORT_GEOPHYS.VARS. Main atmospheric input. 

OMEGA_TOTAL_INPUT(MAXLAYER). Total single scattering albedo ω p for layer p. If there is just one 

scatterer s with absorption coefficient is α ps and scattering coefficient σ ps , then ω p = ω ps = σ ps /(σ ps + α ps 

). For aerosols and clouds we are usually given σ ps and the extinction coefficient ε ps = σ ps + α ps .For 

molecular scattering σ ps = ρ(T p ,P p )σ Ray (λ) which is the air density ρ(T p ,P p ) for layer temperature and 

pressure T p and P p , multiplied by the Rayleigh scattering coefficient σ Ray (λ). Also for molecular absorption 

due to trace gas g, α psg = ρ(T p ,P p )V gp α gp (λ) where V gp is the volume mixing ratio of trace gas g in 

layer p, and α gp (λ) the trace gas absorption coefficient. In general, we have ω ps = σ ps / ε p where ε p = 

Σ s (ε ps ) is the total layer extinction (sum over all particulates), and ω p = Σ s ω ps . 

TAUGRID_INPUT(0:MAXLAYER). Using the above notation, and if the layer height thickness is h p , then 

the layer optical thickness is τ p −τ p-1 = ∆ p = ε p h p with τ 0 = 0. This is enough to define the vertical optical 

depth grid. Slant path thickness values TAUTHICK_INPUT(MAXLAYER,MAXLAYER) are obtained in 

a similar fashion, but vertical height thicknesses h p must be replaced by slant-path distances s p . For a 

plane-parallel atmosphere, s p = h p / µ 0 . 

PHASMOMS_TOTAL_INPUT(0:MAXMOMENT,MAXLAYER). We write (2l+1) β lps for the l th Legendre 

moment of the phase function of scatterer s in layer p. The first moment (indexed by 0) β 0ps =1 

(phase function normalization). For the isotropic case, there is only this value. For Rayleigh scattering, β 1p 

=0,β 2p = (1−δ)/(2+δ), where δ(λ) is the (wavelength-dependent) depolarization ratio, and β lp = 0 for 

l > 2. For aerosols with a Henyey-Greenstein phase function with asymmetry parameter γ p , β lp = (γ p ) l . For 

spherical scattering particles, Mie theory must be used to generate the required coefficients. The total 

moment is a sum-weighted value: ω p β lp = Σ s ω ps β lps . 

LIDORT_GEOPHYS.VARS. Surface input. 

ALBEDO, symbol R. For Lambertian case, this is to be specified explicitly. For the bidirectional case, it is 

1 1 1 

defined as R = -- µµ′ρ , where is the azimuth-independent component of 

4 

0 

∗ 

∫ ( µµ′ , ) d µ dµ′ 

ρ 

0∫ 

0 

∗ ( µµ′ , ) 

0 

the Fourier-expanded bidirectional reflectance. One can use quadrature to evaluate this integral: this 

quadrature need not be the same as the one specified for LIDORT computation. However, once you have 

calculated R, you will have to prepare the bidirectional coefficients at the LIDORT angles (quadrature and 

other polar angles) before use in the model (see entries BIREFELEC and USER_BIREFLEC below).


BIREFLEC(0:MAXMOMENT,MAXSTRM,MAXSTRM). Albedo-normalized bidirectional reflection 

Fourier expansion coefficients ρ m (−µ i ,µ j ) for scattering from incident direction −µ i to reflection direction 

µ j . This is required for the reflection of the downward radiation. For the Lambertian case, ρ 0 (−µ i ,µ j )=1 

and ρ m (−µ i ,µ j ) = 1 for m > 0 (true for all directions). 

BIREFLEC_0(0:MAXMOMENT,MAXSTRM). Albedo-normalized bidirectional reflection Fourier 

expansion coefficients, for scattering from incident direction −µ 0 to reflection direction µ j . This is required 

for the reflection of the direct beam at the surface. For the Lambertian case, ρ 0 (−µ 0 ,µ j )=1andρ m (−µ 0 ,µ j ) 

= 1 for m > 0, for all outward directions. 

EMISSIVITY(MAXSTRM). The directional emissivity κ(µ j ) is determined from Kirchhoff’s law: 

∫ 

1 

κµ ( j 

) = 1– 

2R µ′ρ 0 

( µ j 

, µ′ ) dµ′ 

, where the symbols on the right hand side are defined as above. This 

0 

may be evaluated off-line using a suitable quadrature scheme. For an assumed Lambertian albedo R, the 

emissivity is 1−R for all directions. 

USER_BIREFLEC(0:MAXMOMENT,MAX_USER_STREAMS,MAXSTRM). Albedo-normalized bidirectional 

reflection Fourier expansion coefficients, for scattering from incident direction −µ i to reflection 

direction µ α (one of the user-defined stream angles).This is required for the reflection of the field at the surface. 

For the Lambertian case, ρ 0 (−µ i ,µ α ) = 1 and ρ m (−µ i ,µ α ) = 1 for m > 0, for all directions. 

USER_BIREFLEC_0(0:MAXMOMENT,MAX_USER_STREAMS). Albedo-normalized bidirectional 

reflection Fourier expansion coefficients, for scattering from incident direction −µ 0 to reflection direction 

µ α (one of the user-defined off-quadrature stream angles). For the Lambertian case, ρ 0 (−µ 0 ,µ α )=1and 

ρ m (−µ 0 ,µ α ) = 1 for m > 0, for all directions. 

USER_EMISSIVITY(MAX_USER_STREAMS). Directional emissivity at user-defined stream angles). 

Again follows from Kirchhoff’s law. For Lambertian albedo R, the emissivity is 1−R for all directions. 

4.2 Preparation for LIDORT_L_GEOPHYS Supplementary Structure 

OMEGA_VARS_TOTAL_INPUT(MAX_PARAMETERS,MAXLAYER). We use the symbol u xps for the 

relative variation in single scattering albedo ω ps for scatterer s in layer p, with respect to variable x in that 

layer. We write u xp similarly for the relative variation in total single scatter albedo ω p ; this is the desired 

input quantity OMEGA_VARS_TOTAL_INPUT. By definition: 

x ∂ω 

u xps 

-------- ps 

x ∂ω 

= ⋅ ----------- and u . 

ω ps ∂x xp 

------ p 

= ⋅ --------- 

ω p ∂x 

Thus for the case of just one scatterer with absorption and scattering coefficients α ps and σ ps respectively, 

and ω ps = σ ps /(σ ps + α ps ), then if x = σ ps we have u xps = α ps /(σ ps + α ps ). Similarly if x = α ps then u xps 

= −α ps /(σ ps + α ps ). For molecular scattering with σ ps = ρ(T p ,P p )σ Ray (λ) and molecular absorption due 

to a single trace gas such that α ps = ρ(T p ,P p )M p α p (λ) where M p is the volume mixing ratio in layer p, and 

α p (λ) the trace gas absorption coefficient, then u xps = −ρ(T p ,P p ) α p (λ) /(σ ps + α ps ) for x = M p (result for 

volume mixing ratio weighting function output). Since we know that ω p = Σ s ω ps , then we have:


ω p 

u xp 

= 

∑ 

s 

ω ps 

u xps 

. 

EXT_VARS_INPUT(MAX_PARAMETERS, MAXLAYER). We use the symbol v xp for the relative variation 

in layer extinction coefficient e p for layer p, with respect to variable x. The definition is: 

v xp 

= 

x 

---- 

e p 

∂e p 

⋅ -------- 

∂x 

. 

For one scatterer in layer p with layer absorption and scattering coefficients α p and σ p respectively, we 

have e p = σ p + α p .Ifx = σ p then v xp = σ p /e p Note that this definition is different from that used in the 

User’s Guide to Version 1. 

PHASMOM_VARS_TOTAL_INPUT(MAX_PARAMETERS,0:MAXMOMENT,MAXLAYER). We use 

the symbol z xpls for the relative variation in l-th phase function moment β lps for scatterer s in layer p, with 

respect to variable x in that layer. We write z xpl similarly for the relative variation in the total l-th phase 

function moment β lp ; this is the desired input quantity PHASMOM_VARS_TOTAL_INPUT. By definition: 

x ∂β 

z xpls 

-------- lps 

x ∂β 

= ⋅ ----------- . and z . 

β lps ∂x xpl 

------ lp 

= ⋅ --------- 

β lp ∂x 

In general the derivatives z xpls must be derived from electromagnetic scattering theory, but for some simple 

cases, we can find them directly. For example, with the Henyey-Greenstein phase function with asymmetry 

parameter γ ps , β lps = (γ ps ) l . and hence z xpls = l if the parameter to be varied x = γ ps . Since we know that 

ω p β lp = Σ s ω ps β lps , then in general we have by differentiation: 

∑ 

ω p 

β lp 

( u xp 

+ z xpl 

) = ω ps 

β lps 

( u xps 

+ z xpls 

). 

s 

where u xps and u xp have already been determined for OMEGA_VARS_TOTAL_INPUT. Note - this input 

should be used with caution!


5. Installation, Makefile and Test Example 

5.1 Installation Procedure 

The LIDORT package may be downloaded from the SAO’s anonymous ftp site: 

ftp://cfa-ftp.harvard.edu/ 

Change directory to 

cd pub/lidort/v2 

There are six LIDORT source code “tar” files, which contain all the master modules, internal modules and 

include files as outlined in Chapters 2 and 3: 

src_V23e_master.tar, 

src_V23s_master.tar 

src_standard.tar, 

src_extension.tar 

include_s.tar, 

include_e.tar 

The two additional files 

V23E_RELEASE.tar, 

V23S_RELEASE.tar 

contain user-defined environments for an extended calculation (with weighting functions) and a standard 

calculation (intensity-only) respectively, for an example involving backscatter radiation in a terrestrial 60- 

layer atmosphere with Rayleigh scattering, aerosol scattering and extinction, and ozone absorption for a 

number of wavelengths in the UV atmosphere. This application is important for the ozone profile retrieval 

algorithms. More details on this test case are given below in Section 5.3. 8 separate eponymous directories 

should be created for all these tar files. 

The “RELEASE” directories each contain a makefile, some input files and some test data results, and a 

“README.TXT” file which explains the installation procedure. Before compilation, the user should enter 

each makefile and alter the absolute path variable $(LIDORT_PATH) to the desired directory on the home 

computer. The user should then examine the compilation options and make changes as necessary (the list 

of compilation flags is by no means complete - the original work was done on a Sun unix workstation, and 

there are compilation options satisfying the “gnu” compiler). Executing the makefile creates all the object 

files and an executable (lidort_runs_V23e_o3 in the example below). 

5.2 Makefile Example in Directory V23E_RELEASE 

We now examine the makefile in the V23E_RELEASE directory. This will generate an executable for an 

extended calculation of intensities and weighting functions. The corresponding makefile for the standard 

calculation is very similar, being in essence just a subset of the extended file. We leave out details of the 

compilations; one example will suffice. 

# Makefile for LIDORT Version 2.3E 

# general notes 

############### 

# Name of the compiler and the LINK.f statement should be supplied.


# Ordering of modules follows that required for GNU - the linker 

# needs the modules in a certain order (a called module must be 

# already compiled and linked before). 

# GNU compilation constructed by W. Thomas, DLR, 01 December 1999 

# Solaris compiler instructions (original) 

############################### 

#LIDORT_COMPILE = f77 -c -u -C 

LIDORT_COMPILE = f77 -c -u -fast 

F77 = f77 

# The debug flag is -g 

# For speed, use LIDORT_COMPILE = f77 -c -u -fast 

# Link definition 

################# 

LINK.f = $(F77) 

# Path variables 

################ 

# Define a variable LIDORT_PATH and users must change only this! 

LIDORT_PATH = /home/rspurr/LIDORT/v2p3 

MPATH_S = $(LIDORT_PATH)/src_V23s_master/ 

MPATH_E = $(LIDORT_PATH)/src_V23e_master/ 

SPATH_S = $(LIDORT_PATH)/src_standard/ 

IPATH_S = $(LIDORT_PATH)/include_s/ 

SPATH_E = $(LIDORT_PATH)/src_extension/ 

IPATH_E = $(LIDORT_PATH)/include_e/ 

# OBJECT MODULES 

################ 

# LIDORT modules in directory src_standard 

OBJECTS_LIDORT_SOLUTIONS = LIDORT_RTSOLUTION.o \ 

LIDORT_MULTIPLIERS.o 

OBJECTS_LIDORT_SETUPS = LIDORT_MISCSETUPS.o \ 

LIDORT_DERIVEINPUT.o \ 

LIDORT_CHECKINPUT.o 

OBJECTS_LIDORT_INTENSITY = LIDORT_BVPROBLEM.o \ 

LIDORT_INTENSITY.o \ 

LIDORT_CONVERGE.o \


LIDORT_SSCORRECTION.o 

OBJECTS_LIDORT_AUX = LIDORT_AUX.o \ 

LIDORT_READINPUT.o \ 

LIDORT_WRITEMODULES.o 

# LIDORT modules in directory src_extension 

OBJECTS_LIDORT_L_SOLUTIONS = LIDORT_L_RTSOLUTION.o \ 

LIDORT_L_MULTIPLIERS.o 

OBJECTS_LIDORT_L_SETUPS = LIDORT_L_MISCSETUPS.o \ 

LIDORT_L_CHECKINPUT.o 

OBJECTS_LIDORT_L_WGHTFNS = LIDORT_L_BVSETUPS.o \ 

LIDORT_L_FOURIERADD.o \ 

LIDORT_L_WFCALC.o \ 

LIDORT_SSCORRECTION.o 

OBJECTS_LIDORT_L_AUX = LIDORT_L_INPUTREAD.o \ 

LIDORT_L_WRITEMODULES.o 

# LIDORT top level modules in directory src_V23e_master 

OBJECTS_LIDORT_MASTERS = LIDORT_V23E_MASTER.o \ 

LIDORT_V23E_INPUT.o \ 

LIDORT_V23E_FOURIER.o 

# environment & interface modules external to the main package 

OBJECTS_LIDORT_ENVIRON = user_testenv_o3.o \ 

user_prepare_o3.o 

# EXECUTABLE 

############ 

APPLICATION 

= lidort_run_V23e_o3 

all: $(APPLICATION) 

$(APPLICATION): $(OBJECTS_LIDORT_ENVIRON) \ 

$(OBJECTS_LIDORT_MASTERS) \ 

$(OBJECTS_LIDORT_SETUPS) \ 

$(OBJECTS_LIDORT_SOLUTIONS) \ 

$(OBJECTS_LIDORT_INTENSITY) \ 

$(OBJECTS_LIDORT_AUX) \ 

$(OBJECTS_LIDORT_L_SETUPS) \ 

$(OBJECTS_LIDORT_L_SOLUTIONS) \ 

$(OBJECTS_LIDORT_L_WGHTFNS) \ 

$(OBJECTS_LIDORT_L_AUX) 

$(LINK.f) -o $(APPLICATION) \ 

$(OBJECTS_LIDORT_ENVIRON) \


# COMPILATIONS 

############## 

$(OBJECTS_LIDORT_MASTERS) \ 

$(OBJECTS_LIDORT_SETUPS) \ 

$(OBJECTS_LIDORT_SOLUTIONS) \ 

$(OBJECTS_LIDORT_INTENSITY) \ 

$(OBJECTS_LIDORT_AUX) \ 

$(OBJECTS_LIDORT_L_SETUPS) \ 

$(OBJECTS_LIDORT_L_SOLUTIONS) \ 

$(OBJECTS_LIDORT_L_WGHTFNS) \ 

$(OBJECTS_LIDORT_L_AUX) \ 

$(FLIBRARIES) 

# (one example) Environment modules 

user_testenv_o3.o: user_testenv_o3.f \ 

$(IPATH_S)LIDORT.PARS \ 

$(IPATH_E)LIDORT_L.PARS \ 

$(IPATH_S)LIDORT_CONTROL.VARS \ 

$(IPATH_S)LIDORT_SETUPS.VARS \ 

$(IPATH_S)LIDORT_MODEL.VARS \ 

$(IPATH_S)LIDORT_RESULTS.VARS \ 

$(LIDORT_COMPILE) user_testenv_o3.f 

etc.... 

5.3 Test-case Example in Directory V23E_RELEASE 

This Directory contains the following files: 

user_testenv_o3.f, 

user_prepare_o3.f 

input_o3.inp 

scenario_o3.inp 

make_o3_atm_1.dat 

makefile 

O3_results.out_save 

O3_input.out_save 

O3_scenario.out_save 

For the test example listed here, user_testenv_o3.f is the MAIN program which calls 

LIDORT_V23E_INPUT and LIDORT_V23E_MASTER, while user_prepare_o3.f is a subroutine 

to generate the atmosphere required for this test. There are 3 input files: input_o3.inp is the initialization 

file to be read in LIDORT_V23E_INPUT for the control and model variables; scenario_o3.inp 

contains the height grid to be used in this test, and is read in user_prepare_o3.f; 

make_o3_atm_1.dat is a file of atmospheric data (pressure, temperature, volume mixing ratio, aerosol 

loading), also to be read by the preparation module. makefile will create the desired executable, which 

can then be run for this test case. The result will be three files ending in ‘*.out’, and these should be iden-


tical to the existing three files ending in ‘*.out_save’. The layer weighting functions are with respect to 

ozone volume mixing ratio and temperature. This completes the installation test. 

In order to create your own application, the following steps should be carried out. 

1. Create a new sub-directory for the application (equivalent to V23E_RELEASE). 

2. Copy the user-defined environment source code files for the test case user_testenv_o3.f and 

user_prepare_o3.f into this new directory and rename them. 

3. Go inside these renamed files and alter them according to your application. The preparation module 

should be created using the guidelines in Section 4. 

4. Copy makefile and the input file input_o3.inp. Rename the input file and alter the contents 

according to your specifications. You may need to create some additional files for reading in your 

atmospheric scenario. 

5. Go into the makefile and make necessary name changes to the two environment source files and 

give the application a new name. Don’t forget to alter the absolute path variable! You do not need to 

alter anything else in the makefile.


6. Error Handling, Utilities, References 

6.1 Error Handling 

The main modules LIDORT_V23S_MASTER and LIDORT_V23E_MASTER have two status arguments 

- the integer variables STATUS_CHECKINPUT and STATUS_CALCULATION. They which can take one 

of the two values indicated in the LIDORT.PARS structure (see error indices). The error handling internal 

to LIDORT works as follows. If an error has been returned from one of the internal routines, then a message 

about the error is generated along with a trace for that error; these quantities are then written to the 

output error file. If an internal error has occurred this will be traced through LIDORT and an error value 

will be given to one of the above two STATUS variables upon output from the master module. 

The same applies to LIDORT_V23E_INPUT, which has three arguments - the integer variable 

STATUS_INPUTREAD, the name of the initialization file, and the name of the error output file. See the 

pseudo-code sequence in Section 3.1 for an example. 

Apart from the use of standard numerical routines to solve the eigensystem and a number of linear algebra 

problems, LIDORT is entirely analytical. With this in mind, errors are confined to the following situations. 

File-read errors. 

These will be generated by incorrect key words in the input file. These errors are detected by the FINDPAR 

package. 

Checking errors. 

Most of the LIDORT control, model and geophysical inputs are checked internally before actual computations 

start. All such consistency checks are collected in the modules LIDORT_CHECKINPUT and 

LIDORT_L_CHECKINPUT. All errors produced in this way will be due to incorrect or inconsistent input 

specifications. In addition to each error message and trace, an additional character string is generated for 

the error file - this string provides a hint for the user to correct the appropriate input. 

Numerical errors. 

Only in exceptional circumstances should an error condition be returned from the eigenroutine ASYMTX 

or one of the LAPACK [6] linear algebra modules. One possibility to watch out for is degeneracy caused 

by two layers having identical optical properties. Errors of this kind should be referred to the author for his 

perusal. 

Note on the use of L’Hopital’s Rule. Occasionally, analytic expressions used in LIDORT break down when 

there is a coincidence of stream angles. The commonest example of this occurs for downwelling radiation 

in a plane-parallel atmosphere when one of the user stream angles is identical to the solar zenith angle. The 

corresponding layer source term multiplier must be replaced by a limit value found by the use of L’Hopital’s 

Rule. The example noted here has already been dealt with in this manner, but similar breakdowns 

could in theory happen say when one of the separation constants is vanishingly close to one of the user 

stream angles. Additional applications of L’Hopital’s Rule will be incorporated into the code.


6.2 Utilities 

LIDORT utility routines are collected together in the source code file LIDORT_AUX.f. They include a 

selection of modules from the LAPACK library, plus a number of other standard numerical modules, and 

some file-read and error handling modules. The most important modules in the LAPACK selection are: 

DGBTRF 

DGBTRS 

DGETRF 

DGETRS 

DGBTF2 

DLASWP 

XERBLA 

DGETF2 

DGEMM 

DGEMV 

DGER 

DTBSV 

DTRSM 

These LAPACK modules are not performance-optimized for the LIDORT package (there is in particular a 

lot of redundancy in the linear algebra problems). In the near future, it is expected that the whole linear 

algebra package will be replaced by dedicated routines based in part on the above, but with enhanced performance 

in terms of run-time and efficiency. 

Additional numerical modules are: 

ASYMTX (eigenfunction problem solution module from DISORT); 

GAULEG (Gauss-Legendre quadrature determination; Numerical Recipes); 

CFPLGARR (Legendre-polynomial generator). 

The FINDPAR tool for reading the initialization file was developed by J. Lavagnino (SAO) and is also 

found here. 

6.3 A Note on Validation 

6.3.1 Plane-parallel validation against DISORT 

Since DISORT and LIDORT are discrete ordinate codes, it is possible to validate LIDORT intensity output 

against DISORT in a precise fashion. As far as intensity goes, both models take identical inputs, so this is 

really an “apples” and “apples” comparison. There is none of the difficulties inherent in comparing RT 

models which solve the RT equations in different ways, or with the “layers versus levels” issues that 

bedevil some of the comparisons. Further, DISORT is the most widely used and tested plane-parallel RT 

model currently in use in the atmospheric community; it has been a benchmark against which other models 

have been tested. Version 2.0 of DISORT has an automatic correction for the single scatter using the Nakajima-Tanaka 

algorithm, and we now present some examples of this validation. 

We take a 60-layer atmosphere with ozone as the trace gas absorber, and with Rayleigh and aerosol scattering, 

and with surface albedo 0.3. Optical properties are defined at wavelength 335.4579 nm. The solar 

zenith angle is 55 o , with a relative azimuth of 0 o , and variable line-of-sight zenith angle. We look at the


TOA upwelling intensity, comparing the original (without single scatter correction) value with the NT-corrected 

value, and also making a comparison of the single scatter terms needed to make the NT-correction. 

Note that the uncorrected intensity is the value generated by earlier versions of DISORT. Table 1 summarizes 

these comparisons. 

. 

Table 1: Plane-parallel DISORT/LIDORT intensity comparisons 

Angle ( o ) Stream DISORT LIDORT 

number 

1. Original intensity before correction I(original): 

0.0 6 7.83513E-02 7.835137117005225E-002 

10.0 6 7.64205E-02 7.642061517245054E-002 

20.0 6 7.66746E-02 7.667478191066619E-002 

30.0 6 7.92090E-02 7.920910104336290E-002 

40.0 6 8.41525E-02 8.415260726598663E-002 

25.0 8 7.72878E-02 7.728777647817973E-002 

2. Nakajima-Tanaka single scatter term I(ss_exact): 

0.0 6 2.36675E-02 2.366747046434831E-002 

10.0 6 2.15248E-02 2.152485001476636E-002 

20.0 6 2.04918E-02 2.049181517129264E-002 

30.0 6 2.08425E-02 2.084253500622847E-002 

40.0 6 2.29805E-02 2.298054353823100E-002 

25.0 8 2.03631E-02 2.036306163710890E-002 

3. Removed single scatter term I(ss_removed): 

0.0 6 2.34218E-02 2.342176693498307E-002 

10.0 6 2.12635E-02 2.126346281984243E-002 

20.0 6 2.05236E-02 2.052363098791028E-002 

30.0 6 2.12063E-02 2.120627412852576E-002 

40.0 6 2.33098E-02 2.330982413854123E-002 

25.0 8 2.03020E-02 2.030196554124376E-002 

4. NT-corrected Intensity I(corrected): 

0.0 6 7.85970E-02 7.859707469941750E-002 

10.0 6 7.66818E-02 7.668200236737446E-002 

20.0 6 7.66428E-02 7.664296609404855E-002 

30.0 6 7.88453E-02 7.884536192106562E-002 

40.0 6 8.38232E-02 8.382332666567639E-002 

25.0 8 7.73489E-02 7.734887257404488E-002 

Remarks on Table 1: 

1. DISORT is single precision (apart from the eigenproblem), with LIDORT double precision throughout. 

This explains differences in the last decimal place. 

2. These results were obtained by ensuring that all possible Fourier components were calculated (that is, 

with 8 streams, a maximum of 7 Fourier components can be included in the intensity sum). This is the only 

way to get agreement between the models, since the computation of the single scatter term I(ss_removed) 

is only correct in DISORT when all Fourier components are included.


6.3.2 Pseudo-spherical validation against SDISORT 

The situation regarding pseudo-spherical validation is similar. A pseudo-spherical version SDISORT 

exists, but is part of a large suite of RT programs [10]. However it is possible to extract SDISORT from this 

infrastructure, and carry out some comparisons. SDISORT does not have the Nakajima-Tanaka correction, 

so the results presented in the next table are not corrected in this way. In Table 2, results are for upwelling 

intensities at TOA for line-of-sight viewing zenith angles as indicated. The solar angle was 82 o , with relative 

azimuth angle 60 o . 16 streams in the complete polar direction space were used. Atmosphere as in previous 

table. The figures apply to a 60-layer atmosphere with 8 discrete ordinate streams in the half-space. 

Table 2: SDISORT and LIDORT comparisons, 16 stream 

Angle SDISORT LIDORT 

0.0 1.74379E-02 1.743787E-02 

1.0 1.74259E-02 1.742588E-02 

2.0 1.74194E-02 1.741939E-02 

5.0 1.74333E-02 1.743328E-02 

10.0 1.75709E-02 1.757091E-02 

15.0 1.78613E-02 1.786122E-02 

20.0 1.83187E-02 1.831864E-02 

25.0 1.89573E-02 1.895727E-02 

6.3.3 Weighting Function Validation 

For weighting function validation, we use a finite-difference estimate of the weighting function (FDWF): 

FDWF I ( ξ 1) – I( ξ 2 

) 

≈ ------------------------------- 

2ε 

This requires two separate calls to the intensity-only model, once with atmospheric property ξ perturbed to 

ξ 1 = ξ(1+ε) to give result Ι(ξ 1 ), and then again with property ξ perturbed to ξ 2 = ξ(1−ε) to give result Ι(ξ 2 ). 

The two results are subtracted and divided by twice the relative perturbation ε. In the limit as ε tends to 

zero, the finite difference result should reproduce the exact weighting function. This kind of validation 

works well for variables ξ such as O 3 volume mixing ratio, but is less satisfactory for weighting functions 

with respect to phase function quantities such as asymmetry parameter. The atmospheric conditions for 

Table 3 results are the same as those for Table 1. All results were taken at line-of-sight zenith angle 25 o ; the 

layers are numbered from TOA (thus layer 35 is between 25 and 26 km). 

Table 3: Weighting Function Validation 

1. Original Weighting Function before correction:


parameter angle layer WF_exact FDWF % diff ε 

O3 VMR 25.0 60 -1.4825540E-05 -1.4825540E-05 4.4795101E-08 0.01 

Temperature 25.0 60 -2.2132436E-03 -2.2134503E-03 9.3373358E-03 0.01 

Aer_extinction 25.0 60 -2.1818227E-02 -2.1819131E-02 4.1391573E-03 0.01 

Pressure 25.0 60 2.1600948E-03 2.1600946E-03 1.1995438E-05 0.01 

O3 VMR 25.0 35 -1.5466107E-04 -1.5461694E-04 2.8528781E-02 0.01 

O3 VMR 25.0 35 -1.5466107E-04 -1.5466062E-04 2.8526293E-04 0.001 

O3 VMR 25.0 35 -1.5466107E-04 -1.5466106E-04 3.0789991E-06 0.0001 

2. NT-corrected Weighting Functions 

Remarks on Table 3 

1. Weighting functions with respect to 4 parameters (O 3 VMR, temperature, pressure and aerosol extinction 

coefficient) are presented in the table. Two layers were selected; in particular layer 35 (at 25 km) is 

close to the peak concentration of O 3 , and this is where the weighting functions are largest. 

2. The last three rows in each section illustrate the finite-differencing; by making ε really small, one can get 

arbitrarily close to the exact result, which of course LIDORT produces as a matter of course. 

3. It is possible for LIDORT to generate any sort of weighting function - there are no limits here, as the 

entire discrete ordinate solution is differentiable. Finite-differencing verification is generally OK for those 

variables which do not affect the phase function itself. Using discrete ordinates always involves some sort 

of truncation of the phase function (with or without the delta-M scaling) and the discarded phase function 

contributions can be very sensitive to small changes in the asymmetry parameter. For this reason it is not 

recommended to generate weighting functions with respect to parameters which affect phase functions 

unless you are certain that there are enough phase function moments in the Legendre expansion to ensure 

that the phase function is accurately represented. 

6.4 References 

Table 3: Weighting Function Validation 

O3 VMR 25.0 60 -1.4827224E-05 -1.4827224E-05 1.0904033E-08 0.01 

Temperature 25.0 60 -2.2122634E-03 -2.2124700E-03 9.3376304E-03 0.01 

Aer_extinction 25.0 60 -2.1820707E-02 2.1821610E-02 4.1387162E-03 0.01 

Pressure 25.0 60 2.1591086E-03 2.1591083E-03 1.2019646E-05 0.01 

O3 VMR 25.0 35 -1.5425527E-04 -1.5421106E-04 2.8658387E-02 0.01 

O3 VMR 25.0 35 -1.5425527E-04 -1.5425483E-04 2.8655009E-04 0.001 

O3 VMR 25.0 35 -1.5425527E-04 -1.5425526E-04 3.4798750E-06 0.0001 

[1] Spurr, R. J. D., T. P. Kurosu and K. V. Chance, A Linearized Discrete Ordinate Radiative Transfer 

Model for Atmospheric Remote Sensing Retrieval, J. Quant. Spectrosc. Radiat. Transfer, in press 

(2000).


[2] Stamnes, K., S.-C. Tsay, W. Wiscombe, and K. Jayaweera, Numerically stable algorithm for discrete 

ordinate method radiative transfer in multiple scattering and emitting layered media, Applied 

Optics, 27, 2502-2509 (1988) 

[3] Chandrasekhar, S., Radiative Transfer, Dover, New York, (1960). 

[4] Siewert, C. E., A concise and accurate solution to Chandrasekhar basic problem in radiative transfer, 

J. Quant. Spectrosc. Radiat. Transfer, 64, 109-130 (2000). 

[5] Wiscombe, W., J. Atmos. Sci., 34, 1408-22 (1977). 

[6] Anderson, E., et. al., LAPACK user’s guide, Second edition, Society for Industrial and Applied 

Mathematics, Philadelphia (1995) 

[7] Spurr, R. J. D., Simultaneous derivation of intensities and weighting functions in a general pseudospherical 

radiative transfer treatment, submitted to JQSRT, (2000). 

[8] Nakajima, T., and M. Tanaka, Algorithms for radiative intensity calculations in moderately thick 

atmospheres using a truncation approximation, J. Quant. Spectrosc. Radiat. Transfer, 40, 51-69, 

(1988). 

[9] van Oss, R. F., and R. J. D. Spurr, Fast and accurate 4 and 6 stream linearized discrete ordinate radiative 

transfer models for ozone profile retrieval, paper in preparation, (2000). 

[10] Kylling, A. and K. Stamnes, Efficient yet Accurate Solution of the Linear Transport Equation in the 

Presence of Internal sources: The Exponential-linear-in-depth Approximation, J. Comp. Phys., 102, 

265-276, (1992). 

Acknowledgments 

R. Spurr was funded mainly through: 

SCIAMACHY Data Processor Development, Contract number 332/60570480 

Deutsches Forschungszentrum für Luft und Raumfahrt (DLR) 

Postfach 1116, D82230 Wessling, Germany. 

Ozone SAF Visiting Scientist Grants, P-4799-2-00 and P-4799-2-01 

KNMI (Royal Dutch Meteorological Institute) 

P.O. Box 201, 3730 AE De Bilt, The Netherlands 

with additional internal funds from the Smithsonian Astrophysical Observatory. 

Biggest thanks go to my colleague Roeland van Oss for numerous stimulating discussions on the theory 

and applications of LIDORT; we are working together on faster LIDORT versions for the Ozone profile 

retrieval problem. Special thanks also to Piet Stamnes for careful reviewing of the papers, and Knut 

Stamnes for a number of interesting discussions on the LIDORT model. Thanks also to Thomas Kurosu for 

assistance with paper preparation, and Werner Thomas for help with software checks. The author is grateful 

for the enthusiasm and support provided by his Ph.D. thesis team (sponsor Hennie Kelder, co-sponsors 

Piet Stammes and Roeland van Oss, external advisor Knut Stamnes).

User's Guide

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?