DISAMAR (determining instrument specifications and analysing methods for atmospheric retrieval) is a computer model developed to simulate retrievals of properties of atmospheric trace gases, aerosols, clouds, and the ground surface from passive remote sensing observations in a wavelength range from 270 to 2400 nm. It is being used for the TROPOMI/Sentinel-5P and Sentinel-4/5 missions to derive Level-1b product specifications. DISAMAR uses the doubling–adding method and the layer-based orders of scattering method for radiative transfer calculations. It can perform retrievals using three different approaches: optimal estimation (OE), differential optical absorption spectroscopy (DOAS), and the combination of DOAS and OE, called DISMAS (differential and smooth absorption separated). The derivatives, which are needed in the OE and DISMAS retrievals, are derived in a semi-analytical way from the adding formulae. DISAMAR uses plane-parallel homogeneous atmospheric layers with a pseudo-spherical correction for large solar zenith angles. DISAMAR has various novel features and diverse retrieval possibilities, such as retrieving aerosol layer heights and ozone vertical profiles. This paper provides an overview of the DISAMAR model version 4.1.5 without treating all the details. We focus on the principle of the layer-based orders of scattering method, the calculation of the semi-analytical derivatives, and the DISMAS retrieval method, and it is to our knowledge the first time that these methods are described. We demonstrate some applications of DISMAS and the derivatives.

DISAMAR stands for determining instrument specifications and analysing methods for atmospheric retrieval.
It is a computer code written in Fortran 90 to simulate retrievals of atmospheric components like
trace gases, aerosols, and clouds and properties of the ground surface from passive remote sensing observations of the Earth. The wavelength range considered is from 270 to 2400 nm.
The development of DISAMAR started around the year 2000
but made use of the heritage of
doubling–adding code from the 1980s

DISAMAR contains a radiative transfer module to simulate
radiance, irradiance, and sun-normalized radiance (or reflectance).
The radiative transfer is based on the doubling–adding method and
includes a more efficient variant, called layer-based
orders of scattering (LABOS). Polarization (Stokes four-vectors and

A cloud or aerosol
layer can be modelled by a simple reflecting Lambertian surface or by a layer of scattering particles with a Henyey–Greenstein phase function or a phase function using expansion coefficients
that are read from file (e.g. Mie scattering particles).
The surface below the atmosphere is a Lambertian reflector
with an albedo that can vary with wavelength.
In addition, surface emissions can be included to simulate sun-induced fluorescence

The applications of DISAMAR are retrievals of the ozone profile from the ultra-violet (UV),
the total column of

Due to the time-consuming line-by-line calculations, DISAMAR is not suitable for fast computation or applications in numerical weather prediction (NWP) models. We would recommend RTTOV

The advantage of DISAMAR is that it includes diverse instrument features, simulations, and retrieval algorithms in one code. For such a complex model, it is not possible to cover all details in one paper. We therefore highlight the novel and unique features of DISAMAR (version 4.1.5), which are listed in Sect. 2. Section 3 presents the forward simulations. The retrieval part is described in Sect. 4. Some applications are shown in Sect. 5 as examples. Section 6 presents a discussion and conclusions.

DISAMAR uses a separate altitude grid for the radiative transfer calculations, which is independent of the grid used for specifying the atmospheric properties. This makes it possible to deal with strong vertical gradients in the radiation field, e.g. near the top of clouds.

The altitude grid, wavelength grid, and polar angles are all defined using Gauss–Legendre division points. This makes the integration over altitude, wavelength, or polar angle more accurate than the integration with an equidistant grid using a similar number of grid points.

The number of streams used for integration over polar angles when multiple scattering is involved can be arbitrarily large. The number of coefficients used for the expansion of the phase function in Legendre functions can also be arbitrarily large. All of these features make the DISAMAR calculations more accurate.

DISAMAR provides not just the radiance spectrum of the backscattered sunlight
but also the derivatives with respect to the elements of the state vector.
These derivatives are essential in
OE and are used to find the solution in an iterative manner.
They are also used to determine the error covariance matrix,
gain vectors, and the averaging kernel

Because DISAMAR was originally developed to determine instrument specifications and
analyse methods for atmospheric retrieval, special emphasis is given to the
error information of the retrieved products and instrumental features.
The main retrieval algorithm is optimal estimation (OE)

By using the newly developed DISMAS approach, which combines the principle of the DOAS retrieval method with OE, the number of wavelengths at which forward calculations have to be performed can be reduced significantly.

The forward model of DISAMAR provides simulated radiance (or sun-normalized radiance) and irradiance and the derivatives of the radiance with respect to the state vector elements. The semi-analytical approach computes the forward-mode (tangent-linear) derivatives of the radiance spectrum. Here, attention is given to the calculation of these derivatives using a semi-analytical approach because this important aspect for retrievals has not been published before.

The simulated radiance and irradiance can be convoluted separately with the instrument spectral response function (ISRF).
The ISRF is defined in terms of a convolution operator.
A Gaussian ISRF and a flat-topped ISRF are included in DISAMAR.
A tabulated ISRF, e.g. measured for a specific instrument, can also be used as an external file. During the convolution, the wavelength grid of the ISRF is interpolated to a high-resolution wavelength grid
(see Sect.

Gas absorptions include those from line absorbers and non-line absorbers.
In the wavelength region considered here (270 to 2400 nm), line absorbers
are

The standard database for line-absorbing molecules is the HITRAN 2008 database

The gas absorptions from line absorbers have to be calculated line-by-line at a high-resolution wavelength grid, and then convolved with the ISRF. Gaussian division points are used to define the wavelength grid, which improves the accuracy of the integration during the convolution of the ISRF.

There are two steps to construct the high-resolution wavelength grid: (1) determining wavelength intervals and (2) dividing each interval with proper Gaussian division points. The full width at half maximum (FWHM), the number of Gaussian division points for one FWHM (

We start from the shortest wavelength with an interval of one FWHM of the ISRF. If there is a strong absorption line in the interval, the boundary of the interval is set to the position of the strong line, making this interval smaller than one FWHM. The next interval starts from the position of the strong line with one FWHM interval again. This process is repeated until the end of the wavelength range.

If the interval is one FWHM, the interval is divided by the number of Gaussian points. If the interval is smaller than one FWHM, the number of Gaussian points is scaled with the size of the interval. For example, if the interval is half of one FWHM, the number of Gaussian points is

Illustration of the wavelength grid used in DISAMAR.

Figure

Vertical profiles of absorbing gases are specified using the volume mixing ratio (VMR, in parts per million by volume, ppmv) at pressure grid levels. The volume mixing ratio has the advantage that it remains constant if the temperature changes. The number density of gas molecules is calculated internally using hydrostatic equilibrium. The pressure profile and temperature profile grid can be different for simulation and retrieval: for example, trace gas profile retrievals may be using 50 grid points for the simulation to have high forward model accuracy but only 12 or 18 grid points for the retrieval to represent the limited information content of the measurements.

Two different types of cloud and aerosol layers are distinguished in DISAMAR: an opaque Lambertian reflector and a layer of scattering particles. For the Lambertian type, the cloud and aerosol reflector is located at the top of a pressure interval.
For the scattering particles layer, a Henyey–Greenstein phase function or a Mie phase function (or phase matrix in case of polarization) can be used for aerosols and clouds in DISAMAR.
If the Henyey–Greenstein phase function is used, the wavelength dependence of the optical thickness is modelled using the
Ångstrøm exponent and the absorption is controlled by the
single-scattering albedo of the aerosol and cloud particles.
A Mie phase function is provided in the form of expansion coefficients

In DISAMAR the atmosphere is vertically divided into pressure intervals. The surface pressure (e.g. 1000 hPa) defines the lower boundary of the first interval.
The intervals are further specified by the pressures of the top of the intervals. For example, if the pressure levels are specified at 700, 600, and 0.1 hPa, the atmosphere has three intervals: from the surface to 700 hPa, from 700 to 600 hPa, and from 600 to 0.1 hPa. The top of the atmosphere (TOA) is at 0.1 hPa.
This division is suited to model a cloud layer with boundaries at 700 and 600 hPa. Internally, the pressure levels are translated into altitude levels assuming hydrostatic equilibrium. The formulae are provided in Appendix

In order to perform radiative transfer calculations, each interval is divided into a number of homogeneous layers using Gaussian division points for the altitude. For example, the three intervals from the surface to TOA may be divided into layers using 12, 10, and 28 Gaussian division points, respectively. The standard atmospheres can be used as input for the pressure, temperature, and gas mixing ratio profiles.

Schematic illustration of the altitude grid used in DISAMAR.

Figure

The surface below the atmosphere is a Lambertian, i.e. isotropically reflecting, surface. In DISAMAR one can choose between a wavelength-independent surface albedo or a surface albedo that has a polynomial wavelength dependence for each spectral band.

Sun-induced fluorescence from vegetation due to photosynthesis can be observed
in the near-infrared spectrum

The doubling-and-adding method (also called adding method) is described in many books and papers on
radiative transfer in planetary atmospheres

DISAMAR uses the doubling method for the calculation of the
scattering properties of the individual atmospheric layers. The adding method is used to add different atmospheric layers, and thus it is used to construct the entire atmosphere.
As discussed by

Definition of directions in DISAMAR.

Radiative transfer matrices are denoted in bold font, and they operate on incident radiation from right to left. A beam of polarized radiation is represented by the Stokes four-vector

The doubling method is started with an optically thin layer, with an optical thickness
of, e.g.

Attenuation of an incident beam of light due to extinction (sum of scattering and absorption) by the thin layer is given by the direct transmission matrix

We now consider adding two layers on top of each other.
Let

The adding scheme for illumination at the top of the two layers is presented in Eqs. (

Radiation fields after adding two layers,

The scheme of adding layer

The matrices

The formula for repeated reflections at the interface, Eq. (

Retrieval algorithms that are designed to derive quantitative information about properties of the atmosphere
from measured spectra require not only an atmospheric model
but also derivatives of the reflectance with respect to the optical properties of the atmosphere, sometimes called Jacobians or weighting functions (see, e.g.

The derivatives can be found by calculating the change in reflectance at top-of-atmosphere using the changes in atmospheric properties at each
height,

We divide the atmosphere into two parts, a top part and a bottom part.
Let

Using the adding formulae, Eqs. (

Inserting Eq. (

First, replace

If polarization is ignored, the partial derivative of the reflectance with respect to some specific
quantity

The first example is the altitude-resolved air mass factor (also called block air mass factor) at wavelength

The second example is the derivative of the reflectance to the altitude of a Lambertian cloud below an absorbing and scattering gas layer:

As noted before, the derivatives can also be calculated numerically. Thus,

In the adding method as implemented in DISAMAR, the reflection matrix is calculated for a set of solar directions equal to the number of Gaussian division points used for integration over the polar angle plus at least two additional directions, namely the actual viewing direction and the actual solar position. Thus, for satellite retrievals the adding method provides more results than required.

The layer-based orders of scattering method (LABOS) has been developed to obtain the reflection matrix for
the actual solar position or, when derivatives are required, for the actual solar position
and a solar position corresponding to the
viewing direction. The reflection and transmission properties of the individual homogeneous layers are still
calculated with the doubling method; only the adding of different
layers and the subsequent calculation of the internal field is replaced by the successive orders of scattering method (see Eqs.

In LABOS we deal with the upward and downward internal radiation fields, i.e.

Illustration of the principle of the LABOS method, showing the local upward

Let

The local upward first-order radiation at level

From the local radiation fields, we can now derive the total radiation fields. The first-order-scattered
upward radiation at level

The following equations are used to determine the local radiation fields for the higher orders of scattering, where

One can continue calculating the orders of scattering and summing them up until the contribution of higher orders can be ignored. At the end of the calculation, the internal radiation fields at all levels are known. The Fourier coefficients of the upward radiation at the top of the atmosphere provide the reflection, and the internal radiation fields are used to calculate the derivatives. In DISAMAR users can choose to use either the doubling–adding method or the doubling–LABOS method.

The upward radiation at the top of the atmosphere is calculated by
assuming that the atmosphere is consisting of homogeneous layers, both in the doubling–adding method and in the doubling–LABOS method.
The calculation will converge to the correct value if many homogeneous
layers are used to represent the true atmospheric scattering and absorption profile.
In DISAMAR the reflection can also be calculated from
numerical integration over the source function

Three retrieval methods are implemented in DISAMAR: the optimal estimation (OE), differential optical absorption spectroscopy (DOAS), and differential and smooth absorption separated (DISMAS) methods.

The OE method is implemented in DISAMAR
following

The DOAS method
is described in the literature

Illustration of the principle of DISMAS using the

The DISMAS method uses the principle of DOAS for the forward simulation of spectra and the OE method for the retrieval.
Therefore, the DISMAS method reduces the forward simulation time, making it faster than a full forward simulation, but still provides the full error estimation as in OE.
DISMAS can be used for the retrieval of total columns
of weakly absorbing gases, like

For the reflectance spectrum around a weak atmospheric absorption
line, e.g. from

To simulate a measured spectrum in DISMAS, the modelled reflectance spectrum on the high-resolution grid is multiplied with a high-resolution solar irradiance spectrum and convoluted with the instrument's spectral response function (or slit function). This provides a simulated radiance spectrum of the instrument that can be compared to an actually measured radiance spectrum. In practice, the simulated radiance spectrum is divided by the simulated irradiance spectrum to eliminate common errors, and the logarithm of the reflectance is used for fitting.

The air mass factor

If there is more than one absorbing trace gas in a fit window,
one can use the slant absorption
optical thickness,

For fitting the modelled reflectance to the measured reflectance and determining the errors in the retrieved parameter values, the derivatives are needed. Here we give the derivatives with respect to the total column of a trace gas and to the cloud fraction for a partly cloudy pixel.

For a partly cloudy pixel with

Comparison of DISAMAR and DAK simulations of TOA reflection spectra between 325 and 335 nm:

Altitude-resolved air mass factor of

As an example, we show a comparison of DISAMAR with the DAK radiative transfer model

Figure

The accuracy of the semi-analytical formulae for the derivatives based on the adding method is checked here by comparison with the numerically calculated derivatives.
We show an example of the
derivative of the cloud height,

Comparison between DISMAS and full optimal estimation (OE) retrievals of the total

The altitude-resolved AMF,

Figure

The DISAMAR code can simulate the retrieval of total columns of trace gases, ozone profile,
cloud layer height, aerosol layer height, cloud (or aerosol) optical thickness,
surface albedo, and surface pressure.
Some OE retrieval applications were mentioned in the introduction of Sect.

The parameters were accurately retrieved by both methods.
The error estimates for the retrieved surface albedo and cloud
fraction differ only little, but DISMAS underestimates the error
in the retrieved

If the cloud fraction is fixed to its true value (0.20) so that
the

From our results, we find that DISMAS and OE are functionally equivalent for weak absorption, except that the error in the retrieved column may differ in some cases. The main advantage is that
DISMAS is much faster than full OE, which might make operational use of DISMAS attractive. This would make it possible to derive the surface albedo and the cloud fraction directly from the measured radiance in the

The aim of the DISAMAR code is to determine instrument specifications
and analyse methods for retrievals of atmospheric composition from satellite observations in the range of 270–2400 nm.
We have described the principles of the DISAMAR radiative transfer model and its innovative retrieval capabilities. The novel features in DISAMAR are the semi-analytical derivatives of the reflectance using the internal radiation field, the layer-based orders of scattering method LABOS, which speeds up the calculations, and the DISMAS retrieval method, which reduces the number of radiative transfer calculations in DOAS-type retrievals. The semi-analytical derivatives are based on the linearization of the adding method, which is detailed in Appendix

DISAMAR can be used as an accurate radiative transfer model for simulations and as a retrieval algorithm for several applications.
We have given a few examples of applications, namely forward modelling of the polarized reflectance in the UV, the derivative of cloud height in the

In the past few years DISAMAR has been used extensively in trace gas, aerosol, and cloud retrieval studies for OMI, TROPOMI, Sentinel-4, and Sentinel-5.
DISAMAR is currently being used for operational retrievals of ozone profiles from UV spectra for OMI

Currently, DISAMAR is still under development and has certain limitations:
Raman scattering in water

Here we derive the formulae for the derivative of the reflection,

We now add the upper partial atmosphere to this combined (“thin

Inserting Eq. (

The pseudo-spherical correction in DISAMAR corrects for
the curvature of the atmosphere for incident sunlight.
For the other light paths the atmosphere remains plane-parallel.
In order to calculate the pseudo-spherical correction, the adding scheme for illumination of the layers from below is implemented. These equations are as follows:

The partial derivative of the reflectance to the absorption coefficient, Eq. (

The partial derivative of the reflectance to the scattering coefficient, Eq. (

The pressure grid in DISAMAR is converted to an altitude grid using the hydrostatic equation (assuming hydrostatic equilibrium):

An approximate expression for the gravitational acceleration as a function of the altitude above the mean sea level,

The DISAMAR code v4.1.5 and data are available from the following Zenodo DOI:

JFdH developed the DISAMAR codes and wrote the algorithm document and manual. PW wrote the paper based on the algorithm document, performed the calculations, and made the figures. MS and JPV contributed to the DISAMAR code. MS and PW maintain the code. PS contributed to the writing of the paper. All co-authors commented on the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Over the years many people have contributed to the testing of the code and development of the applications. We want to thank Mark ter Linden (Science and Technology B.V., Delft, the Netherlands) for building the software interface around DISAMAR.

The development of DISAMAR was funded by the Netherlands Space Office (NSO), under the OMI and TROPOMI science contracts, and by ESA under various future mission contracts.

This paper was edited by Volker Grewe and reviewed by Minzheng Duan, Patrick Stegmann, and Pengwang Zhai.