transport out of the volume) and fully turbulent conditions with negligible storage. In urban environments during stable atmospheric conditions the $\Delta C_S$ term will be non-negligible \cite{Helfter_2011}; for example, $\Delta C_S$ was found to be five times the magnitude of the turbulent vertical flux term ($F_{CO_2}$) close to dawn and dusk in suburban Vancouver, Canada \cite{Crawford_Christen_2014}. Other urban studies found $\Delta C_S$ to be smaller, but still significant, with maximum $\Delta C_S$ values 11\% and 22\% of the magnitude of $F_{CO_2}$ in Edinburgh, Scotland \cite{Nemitz_2002} and Basel, Switzerland \cite{Feigenwinter_2012}, respectively. Similarly, in rural environments horizontal advection may be non-negligible (e.g., \cite{Aubinet_2003}). This paper focuses on methodological considerations when assessing $\Delta C_S$ from a vertical profile in an urban environment. Future manuscripts will address horizontal variation and advection.

Of three studies discussed above (Vancouver, Edinburgh, Basel) with reported urban CO$_2$ storage values, only one (Vancouver, \cite{Crawford_Christen_2014}) presented values derived from a data-set of longer than one month; however, both these values and those reported for Edinburgh by \cite{Nemitz_2002} were calculated based on the assumption of a constant relation between carbon dioxide concentration ([CO$_2$]) measured above the blending height and the concentration in the street canyon. In contrast, for Basel, \cite{Feigenwinter_2012} did not make this assumption and reported $\Delta C_S$ calculated from [CO$_2$] at ten levels; however, the results are only for one month (15th June to 15th July 2002). There is therefore scope to improve not only the understanding of the processes affecting CO$_2$ storage over a greater range of meteorological and anthropogenic conditions, but also to develop recommendations for future measurement programmes.

The objective of this paper is to evaluate potential approaches for such studies, illustrated with examples from, and analysis of, high temporal resolution data collected at 10 locations from 6.5 to 46.4 m above ground level between 2011 and 2014 at King's College London, in Central London, UK. The paper is organised as follows. In the rest of this Section 1 we provide a brief background of how CO$_2$ storage is calculated and then (Section 2) a discussion of the methods used in this paper. This is followed by an exploration of the temporal variation of CO$_2$ storage (Section 3) and the relation between measured CO$_2$ storage and anthropogenic and natural factors in a highly urbanised environment (Section 4). The required number and placement of sample points for CO$_2$ storage measurements in a deep urban street canyon is addressed (Section 5) and the effect of sensor response and sampling interval on calculated CO$_2$ storage is tested (Section 6). At the processing stage, three different temporal and spatial interpolation methods are evaluated against measured data (Section 7). Finally, the impact of CO$_2$ storage calculated by two different methods on the turbulent vertical CO$_2$ flux is assessed. The Supplementary material (noted by S.1 to S.9) includes the notation with corresponding units (S.1) used in the text, further information on CO$_2$ storage calculation (S.2), previous CO$_2$ storage studies (S.3), equipment (S.4), meteorological characteristics (S.5), example time series (S.6), variations of CO$_2$ with friction velocity, wind direction and height above ground level (S.7, S.8) and further references (S.9). The online version of this paper provides the figures in colour.

\subsection{1.1. Calculation of CO$_2$ storage}

There are two main approaches to calculating CO$_2$ storage flux density, i.e., the rate of change of CO$_2$ per unit area below the Eddy Covariance (EC) measurement height. Here they are referred to as the `single height' and the `profile' approaches. For a brief discussion of the theory and some reported results, see Supplementary material S.2 and S.3, respectively. For a more in depth discussion of the theoretical considerations regarding CO$_2$ storage the reader is referred to \cite{Finnigan_2006} and subsequent discussion \cite{Kowalski_2008, Finnigan_2009}.

The approach to calculate CO$_2$ storage depends upon the number of vertical locations at which CO$_2$ concentration ([CO$_2$], the symbol [ ] is used to indicate concentration) data are collected. In the first approach, `single height' CO$_2$ storage ($\Delta C_{SS}$) is calculated from [CO$_2$] data at one location, usually by eddy covariance equipment in the inertial sub-layer \cite{Nemitz_2002, Crawford_Christen_2014}. In the second approach, the `profile' method, $\Delta C_S$ is calculated from data collected at multiple heights ($\Delta C_{SP}$). The profile method uses a vertical [CO$_2$] profile at heights $z_i$, which is generally measured by cycling through all the sample locations within a set time period with a data-logger controlled valve array \cite{Xu_1999, Molder_2000, Vogt_2006, Hutyra_2008}. This cycle period may not be the same as the averaging period used in the $\Delta C_{SP}$ calculation. For example, measurements collected with a sampling interval, $t_s$, of 2 Hz for 75 s at 8 heights, giving a full profile cycle every 10 min, may be used to calculate $\Delta C_{SP}$ with an averaging period ($T$) of 30 min. The storage is calculated as the sum of the changes in time averaged concentration ([CO$_2$]$_i$) between time $t=-T/2$ and $t=T/2$ for each location ($i$) in the profile, weighted by the vertical span, $\Delta z_i$, over which each profile measurement is considered representative and divided by the averaging period ($T$), which can be expressed as (modified from \cite{Aubinet_2005}):

\begin{equation}
\Delta C_{SP} = \frac{1}{T} \sum_i \left( [CO_2]_{i,t=T/2} - [CO_2]_{i,t=-T/2} \right) \Delta z_i
\end{equation}

If the measurements at each height are not made concurrently, [CO$_2$] at each height may first be interpolated in time to generate instantaneous profiles from which $\Delta C_{SP}$ can be calculated, though this is neglected in some cases \cite{Iwata_2005}. The impact of interpolation on calculated $\Delta C_{SP}$ is discussed further in Section 7.

The single height method is a simplification of the profile method to one height, which is usually the height of the eddy covariance equipment. The change in [CO$_2$] with time at $z_h$ ($\Delta[CO_2]/\Delta t$) is weighted by the vertical distance from the ground to the measurement point ($z_h$). The single height CO$_2$ storage ($\Delta C_{SS}$) is given by (modified from \cite{Nemitz_2002}):

\begin{equation}
\Delta C_{SS} = \frac{\Delta [CO_2]}{\Delta t} z_h
\end{equation}

As the data are continuous, the change in the instantaneous CO$_2$ concentration with time ($\Delta[CO_2]/\Delta t$) can be used instead of the change in the time averaged CO$_2$ concentration with time ([CO$_2$]/$\Delta t$), though it may still be advisable to average in time to reduce measurement noise \cite{Finnigan_2006}.

The $\Delta C_{SS}$ calculation assumes any change in [CO$_2$] below the measurement height results in a change of equivalent magnitude at the measurement height. This assumption appears not to be supported by any evidence in the literature; reported diurnal cycles of CO$_2$ mixing ratios in the roughness sub-layer over both rural (e.g., \cite{Xu_1999}) and urban (e.g., \cite{Litke_Vogt_2013}) surfaces are known to vary with height. This problem is particularly acute during periods of low turbulence, such as at night or during cold weather, where measurements above the surface layer may become decoupled from processes near ground level \cite{Helfter_2011}.

If temporal variability is large compared to the spatial variability, the single height method may provide a more accurate measure of storage than the profile method as the maximum data availability at each sample location for the latter may be $1/k$ of the total time