We present an empirical model for nitric oxide (

Our model relates the daily (longitudinally) averaged

It has been recognized in the past decades that the mesosphere and
stratosphere are coupled in various ways

Previously the role of

Different instruments have been measuring

Chemistry–climate models struggle to simulate the

The

Early work to parametrize

The paper is organized as follows: we present the data used
in this work in Sect.

We use the SCIAMACHY nitric oxide data set version 6.2.1

The data were retrieved for the whole Envisat period
(August 2002–April 2012).
This satellite was orbiting in a sun-synchronous orbit
at around 800 km altitude, with Equator crossing times
of 10:00 and 22:00 local time.
The

We averaged the individual orbital data longitudinally on a daily basis
according to their geomagnetic latitude within 10

The measurement sensitivity is taken into account via the averaging kernel diagonal elements, and days where its binned average was below 0.002 were excluded from the time series. Considering this criterion, each bin (geomagnetic latitude and altitude) contains about 3400 data points.

We use two proxies to model the

In the same manner as for the irradiance variations, the “right”
geomagnetic index to model particle-induced variations of

It should be noted that tests using Kp (or its linear equivalent, Ap) instead of AE
and using

We denote the number density by

In the (multi-)linear case, we relate the nitric oxide
number densities

We determine the coefficients via least squares, minimizing the squared differences of the modelled number densities to the measured ones.

In contrast to the linear model above,
we modify the AE index by a finite lifetime

The lifetime-corrected

The parameters are usually estimated by maximizing the likelihood, or, in the case of additional prior constraints, by maximizing the posterior probability. In the linear case and in the case of independently identically distributed Gaussian measurement uncertainties, the maximum likelihood solutions are given by the usual linear least squares solutions. Estimating the parameters in the non-linear case is more involved. Various methods exist, for example, conjugate gradient, random (Monte Carlo) sampling, or exhaustive search methods. The assessment and selection of the method to estimate the parameters in the non-linear case are given below.

Because of the complicated structure of the model function in Eq. (

The likelihood

Note that the normalization constant

The prior distribution

Parameter search space for the non-linear model and uncertainty estimation.

In the simple case, the measurement covariance matrix

Both problems can be addressed by adding a
covariance kernel

The approximately

We demonstrate the parameter estimates using example time series

The fitted densities of the linear model Eq. (

Time series data and linear model values and residuals
at 70 km for 65

For the sample time series (65

Past and recent research in model selection provides a number of
choices on how to compare models objectively.
The results are so-called information criteria which aim to provide a
consistent way of how to compare models,
most notably the “Akaike information criterion” (AIC;

Same as Fig.

Using the non-linear model, we show the latitude–altitude
distributions of the medians of the sampled
Lyman-

Latitude–altitude distributions of the fitted solar index parameter
(Lyman-

The Lyman-

The geomagnetic influence is largest at high latitudes
between 50 and 75

The latitude–altitude distributions of the lifetime parameters
are shown in Fig.

Latitude–altitude distributions of the fitted
base lifetime

The constant part of the lifetime,

For three selected latitude bins in the Northern Hemisphere
(5, 35, and 65

Coefficient profiles of the solar index parameter (Lyman-

For the same latitude bins in the Southern Hemisphere (5, 35, and 65

Coefficient profiles of the solar index parameter (Lyman-

The distribution of the parameters confirms our understanding
of the processes producing

The AE coefficients are largest at auroral latitudes, as
expected for the particle nature of the associated

The associated constant part

The annual variation of the lifetime is largest at high
northern latitudes with a nearly constant amplitude of
10 days between 70 and 85 km.
An empirical lifetime of 10 days in winter was used by

We propose an empirical model to estimate the

The parameter distributions indicate in which regions the different processes
are significant.
We find that these distributions match our current understanding of
the processes producing and depleting

A potential improvement would be to use actual measurements of precipitating particles
instead of the AE index.
Using measured fluxes could help to confirm our current understanding
of how those fluxes relate to ionization

The SCIAMACHY

SB developed the model, prepared and performed the
data analysis, and set up the paper. MS provided input on the model
and the idea of a variable

The authors declare that they have no conflict of interest.

Stefan Bender and Miriam Sinnhuber thank the Helmholtz Association for funding part of this project
under grant number VH-NG-624.
Stefan Bender and Patrick J. Espy acknowledge support from the Birkeland Center
for Space Sciences (BCSS), supported by the Research Council of
Norway under grant number 223252/F50.
The SCIAMACHY project was a national contribution to ESA Envisat,
funded by German Aerospace (DLR),
the Dutch Space Agency, SNO, and the Belgium Federal Science Policy Office, BELSPO.
The University of Bremen as principal investigator has led the scientific
support and development of the SCIAMACHY instrument as well as the scientific exploitation of its
data products.
This work was performed on the Abel Cluster,
owned by the University of Oslo and Uninett/Sigma2,
and operated by the Department for Research Computing at USIT,
the University of Oslo IT department (