% This file defines the general math macros. % Mathematical symbols used only once, or for a particular purpose, % should not be included here. Note that the scalar quantities exist % in a subscript version, and it is not necessary to define macros for % all possible subscripts of a variable. % A lot of macros have been defined for the Rodgers formalism is it % used extensively. % If you add new definitions, please try to follow the rules below to % get a naming scheme as consistent as possible. Just check how a % similar macro is defined and use it as an example. % Patrick Eriksson 2001-03-13 %----------------------------------------------------------------------------- % Most of the macros are named by putting 3-letters acronyms together. % The acronyms are mainly formed by taking the first letter and the two % first following consonants. The list below shows the used acronyms. % Please, if you introduce a new acronym, add it to this list. % % altitude Alt % angel Ang % average Avr % azimuthal Azm % constant Cns % contribution Cnt % covariance Cvr % derivative Drv % error Err % frequency Frq % forward Frw % function Fnc % identity Idn % inverse Inv % kernel Krn % latitude Lat (exception from general naming scheme) % length Lng % longitude Lon (exception from general naming scheme) % matrix Mtr % measurement Msr % model Mdl % partial Prt % population Ppl % pressure Prs % radius Rds % retrieval/ed Rtr % sensor Sns % size Sze % space Spc % state Stt % style Stl % symbol Smb % temperature Tmp % transfer Trf % transpose Trp % vector Vct % zenith Znt % % a priori Apr % monochromatic pencil beam intensity Mpi % weighting function Wfn % For other terms or features, use as far as possible complete strings to get % macros with clear names. %--- General math ------------------------------------------------------------ % Vector style \newcommand{\VctStl}[1] {\ensuremath{\mathbf{#1}}} % Matrix style \newcommand{\MtrStl}[1] {\ensuremath{\mathbf{#1}}} % The identity matrix \newcommand{\IdnMtr} {\MtrStl{I}} % Scalar or matrix inverse \newcommand{\Inv} {^{-1}} % Vector or matrix transpose \newcommand{\Trp} {^T} % Size symbol \newcommand{\SzeSmb} {\ensuremath{\in}} % Vector space \newcommand{\VctSpc}[1] {\ensuremath{\mathbf{R}^#1}} % Matrix space \newcommand{\MtrSpc}[2] {\ensuremath{\mathbf{R}^{#1 \times #2}}} % Vector length (simple) \newcommand{\VctLng} {\ensuremath{n}} \newcommand{\aVctLng}[1] {\ensuremath{n_#1}} % Index (to vector, matrix ...) \newcommand{\Ind} {\ensuremath{i}} \newcommand{\aInd}[1] {\ensuremath{i_#1}} % Partial derivative symbol \newcommand{\PrtDrv} {\ensuremath{\mathrm{d}}} %--- Powers, degrees and other special units, etc. % *10^#1 \newcommand{\topowerten}[1] {\ensuremath{\cdot10^{#1}}} % degrees \newcommand{\degree} {\ensuremath{^\circ}} %--- Plotting line styles ---------------------------------------------------- \def \lsolid {\mbox{------}} \def \ldashed {\mbox{--~--~--}} \def \ldashdot {\mbox{--~$\cdot$~--}} \def \ldotted {\mbox{$\cdot~\cdot~\cdot$}} %--- Physical constants ------------------------------------------------------ % Speed of light in vaccum [m/s] \newcommand{\speedoflight} {\ensuremath{c}} % Planck constant [Js] \newcommand{\planckCns} {\ensuremath{h}} % Boltzmann constant [J/K] \newcommand{\boltzmannCns} {\ensuremath{k_b}} % Avogadro's number [molec/kg] \newcommand{\avogadrosCns} {\ensuremath{N_a}} %--- The Rodgers formalism --------------------------------------------------- % True (natural) forward model \newcommand{\trueFrwMdl} {\ensuremath{F}} % Discrete forward model \newcommand{\FrwMdl} {\ensuremath{\mathcal{F}}} % Inverse model \newcommand{\InvMdl} {\ensuremath{\mathcal{I}}} % Transfer model \newcommand{\TrfMdl} {\ensuremath{\mathcal{T}}} % A priori symbol \newcommand{\AprSmb} {\ensuremath{_a}} % Measurement vector \newcommand{\MsrVct} {\VctStl{y}} % Vector of monochromatic pencil beam intensities \newcommand{\MpiVct} {\VctStl{i}} % Measurement error vector \newcommand{\MsrErrVct} {\ensuremath{\varepsilon}} % State vector \newcommand{\SttVct} {\VctStl{x}} % A priori state vector \newcommand{\AprSttVct} {\SttVct\AprSmb} % Retrieved state vector \newcommand{\RtrVct} {\ensuremath{\hat{\SttVct}}} % A state vector with subscript \newcommand{\aSttVct}[1] {\SttVct\ensuremath{_#1}} % A state vector with subscript and transpose \newcommand{\aSttVctTrp}[1] {\SttVct\ensuremath{_#1\Trp}} % Forward model parameter vector \newcommand{\FrwMdlVct} {\VctStl{b}} % A priori forward model parameter vector \newcommand{\AprFrwMdlVct} {\FrwMdlVct\AprSmb} % A forward model parameters vector with subscript \newcommand{\aFrwMdlVct}[1] {\FrwMdlVct\ensuremath{_#1}} % A forward model parameters vector with subscript and transpose \newcommand{\aFrwMdlVctTrp}[1] {\FrwMdlVct\ensuremath{_#1\Trp}} % Inverse model parameters \newcommand{\InvMdlVct} {\VctStl{c}} % Weighting function matrix \newcommand{\WfnMtr} {\MtrStl{K}} % A weighting function matrix with a subscript \newcommand{\aWfnMtr}[1] {\WfnMtr\ensuremath{_#1}} % A weighting function matrix with a subscript and transpose \newcommand{\aWfnMtrTrp}[1] {\WfnMtr\ensuremath{_#1\Trp}} % Contribution function matrix \newcommand{\CtrFncMtr} {\MtrStl{D_y}} % Averaging kernel matrix \newcommand{\AvrKrnMtr} {\MtrStl{A}} % A averaging kernel matrix with subscript \newcommand{\aAvrKrnMtr}[1] {\AvrKrnMtr\ensuremath{_#1}} % Sensor (and data reduction) matrix \newcommand{\SnsMtr} {\MtrStl{H}} % Sensor (and data reduction) matrix with subscript. \newcommand{\aSnsMtr}[1] {\SnsMtr\ensuremath{_#1}} % Transformation between vector spaces \newcommand{\VctTrfMtr} {\MtrStl{B}} % Population matrix \newcommand{\PplMtr} {\MtrStl{\Sigma}} % A population matrix with a subscript \newcommand{\aPplMtr}[1] {\PplMtr\ensuremath{_#1}} % A population matrix with a subscript and invererse \newcommand{\aPplMtrTrp}[1] {\PplMtr\ensuremath{_#1\Inv}} % Covariance matrix \newcommand{\CvrMtr} {\MtrStl{S}} % A covariance matrix with a subscript \newcommand{\aCvrMtr}[1] {\CvrMtr\ensuremath{_#1}} % A covariance matrix with a subscript and invererse \newcommand{\aCvrMtrTrp}[1] {\CvrMtr\ensuremath{_#1\Inv}} % --- Special functions ------------------------------------------------------ % % The Planck function \newcommand{\Planck} {\ensuremath{B}} % --- General scalar quantities ---------------------------------------------- % % All quantities shall have a subscript version named as aXxx. % Altitude (above geoid) \newcommand{\Alt} {\ensuremath{z}} \newcommand{\aAlt}[1] {\ensuremath{z_#1}} % Azimuthal angle \newcommand{\AzmAng} {\ensuremath{\omega}} \newcommand{\aAzmAng}[1] {\ensuremath{\omega_#1}} % Frequency \newcommand{\Frq} {\ensuremath{\nu}} \newcommand{\aFrq}[1] {\ensuremath{\nu_#1}} % Latitude \newcommand{\Lat} {\ensuremath{\alpha}} \newcommand{\aLat}[1] {\ensuremath{\alpha_#1}} % Longitude \newcommand{\Lon} {\ensuremath{\beta}} \newcommand{\aLon}[1] {\ensuremath{\beta_#1}} % Monochromatic pencil beam intensity \newcommand{\Mpi} {\ensuremath{I}} \newcommand{\aMpi}[1] {\ensuremath{I_#1}} % Pressure \newcommand{\Prs} {\ensuremath{P}} \newcommand{\aPrs}[1] {\ensuremath{P_#1}} % Pressure altitude \newcommand{\PrsAlt} {\ensuremath{\zeta}} \newcommand{\aPrsAlt}[1] {\ensuremath{\zeta_#1}} % Radius \newcommand{\Rds} {\ensuremath{r}} \newcommand{\aRds}[1] {\ensuremath{r_#1}} % Temperature \newcommand{\Tmp} {\ensuremath{T}} \newcommand{\aTmp}[1] {\ensuremath{T_#1}} % Zenith angle \newcommand{\ZntAng} {\ensuremath{\psi}} \newcommand{\aZntAng}[1] {\ensuremath{\psi_#1}}