(10.4)
The first term on the right-hand side is the mean advective flux representing the contribution from the mean vertical wind, and the second term is the eddy correlation flux representing the contribution from turbulent eddies. The mean vertical wind close to the surface is near zero, so that the mean advective flux is generally much smaller than the eddy correlation flux.
Eddy correlation flux measurements must resolve eddies of all sizes, making a significant contribution to the mean flux. This requires fast instrumentation with a measurement frequency of 1–10 Hz. Such instrumentation is often not available. If a fast measurement of the vertical velocity is available, an alternative is to use the eddy accumulation method. In this method, air is collected in two different storage reservoirs, one for upward flow and one for downward flow. The collected air is then analyzed and the flux is computed from the difference in mass between the reservoirs.
Another approach to estimate the vertical flux that does not require high-speed instrumentation is the flux-gradient method. As shown in Section 9.3, vertical transport in the surface layer can usually be parameterized as an eddy diffusion process in which the vertical flux is proportional to the mean vertical gradient in concentration and the eddy diffusion coefficient is Kz = ku*z. The friction velocity u* can be inferred from the slope of a ln z vs. u plot by assuming the log law for the wind (Section 9.3.2). Alternatively, if the surface flux Fz,Ψ of a reference variable Ψ is known, one can use the similarity assumption to infer the flux of any species i by comparing the mean vertical gradients:
(10.5)
Here, Ψ may represent sensible heat, water vapor, or any chemical variable for which the surface flux can be measured by eddy correlation or is otherwise known. The similarity assumption operates in both directions, so that it is possible to infer the flux of a species for which the surface is a sink (Fz,i < 0) from the flux of a variable for which the surface is a source (Fz,Ψ > 0) or vice versa.
10.2.5 Observation Platforms
Atmospheric measurements are conducted from a wide range of platforms including ground-based stations, vehicles, ships, balloons, aircraft, and satellites (Figure 10.7). These different platforms have advantages and disadvantages that often make them complementary (Table 10.2). Some can carry extensive payloads to measure a wide range of species while others are more limited. Addressing a particular scientific problem may call for a carefully designed observing system involving an ensemble of platforms each with a different role to play. Such an observing system must generally include models to place into context the measurements taken from different platforms with different payloads, schedules, and locations. Figure 10.8 gives some general considerations for the design of such an observing system. We elaborate below on the roles of surface sites, aircraft, and satellites.
Figure 10.7 Platforms for measuring atmospheric composition. (a) Surface station installed for a field campaign in Texas (University of Houston); (b) NASA unmanned Global Hawk aircraft during the Airborne Tropical Tropopause Experiment (ATTREX) in California (2014); (c) instrumentation aboard a C-130 aircraft during the Front Range Air Pollution & Photochemistry Experiment in Colorado (FRAPPE) (2014); (d) constellation of satellites called the A-Train flying in formation (NASA).
Table 10.2 Advantages and disadvantages of observation platforms for atmospheric composition
Surface sites Vehicles, ships Balloons Aircraft Satellites
Temporal coverage Good Limited Limited Limited Good
Horizontal coverage Limited Good Poor Good Good
Vertical resolution Poor Poor Good Good Limited
Payload Good Good Limited Good Limited
Figure 10.8 Observing system for atmospheric composition illustrating some applications of such a system and the role of different observing system components (ground based, aircraft, satellites, models).
Surface sites provide local data, generally with very high accuracy and extended temporal coverage. They provide the basis for analyzing long-term trends in atmospheric chemistry as well as interannual, seasonal, and diurnal variations. Some stations record the concentrations of an ensemble of species, and the observed relationships between species can then provide constraints on their sources and chemical evolution. Other measurements that may be taken at surface sites include total columns (such as from a Dobson spectrophotometer or FTIR instrument), vertical profiles (lidar, ozonesondes), and surface fluxes (eddy correlation). Although one generally regards surface sites as serving monitoring purposes (often involving networks of similarly configured sites), they are also often used in field campaigns and provide temporal continuity.
Aircraft provide vertical coverage, and horizontal coverage beyond what ground-based stations allow. Large research aircraft with comprehensive payloads allow detailed measurements of atmospheric composition with great flexibility in operations. Observations by research aircraft provide information for relatively short periods of time (typically 1–2 months, the practical length of a field campaign). A few commercial long-range aircraft with automated instrumentation provide routine data along their flight routes at cruising altitude (upper troposphere/lower stratosphere) as well as vertical profiles during take-off and landing. Remotely piloted aircraft offer the possibility of long-endurance flights with small payloads. Vertical ranges of most aircraft do not extend above 12 km altitude but some specialized aircraft can operate up to 20 km altitude. In-situ measurements at higher altitudes require balloons.
Satellites provide global continuous coverage to varying degrees depending on their observation schedule, orbit track, cross-track sampling, and viewing geometry. Typical horizontal pixel resolution is of the order of 10 km for nadir view. Figure 10.9 illustrates different viewing strategies for satellites in low Earth orbit (LEO), 500–2000 km above the surface and with an orbital period of 1–2 hours. Solar backscatter instruments detect solar radiation backscattered by the Earth surface and its atmosphere. They generally provide information on total atmospheric columns with little or no vertical resolution. Thermal IR/microwave instruments detect radiation emitted by the Earth’s surface and its atmosphere, and can operate either in nadir or limb mode. Nadir viewing affords better horizontal resolution and vertical penetration, but detection of the lower troposphere is limited by the need for thermal contrast with the surface. Limb viewing can achieve vertical resolution of order 1 km with horizontal resolution of order 100 km, but has little sensitivity below the upper troposphere due to interference by clouds and water vapor in the line of sight. Lidar instruments can achieve high vertical resolution but have no cross-track capability so their horizontal coverage is very limited. Solar occultation instruments detect the direct radiation from the Sun passing through the atmosphere as the satellite experiences sunrise and sunset over its orbital period. The strong signal from the Sun enables detection of species for which other methods would not achieve sufficient signal, with high vertical resolution down to cloud level. However, the measurements are sparse (twice per orbit) and the geographical coverage is limited.
Figure 10.9 Observing strategies for atmospheric composition from low Earth orbit. Distances are not to scale.
Figure 10.10 shows different possible orbits for satellite observations. Observations of atmospheric composition have so far mainly been from LEO. The polar Sun-synchronous orbit is the most common and provides global observations at the same local time of day everywhere. Cross-track viewing can achieve global daily coverage. Inclined orbits provide a higher frequency of observations at low latitudes but sacrifice high latitudes. Geostationary orbits, where the satellite is in an equatorial plane 36 000 km away from the Earth with a 24-hour orbiting period, provide continuous data over a limited geographical domain (up to 1/3 of Earth’s surface, though smaller domains are typically used); spatial resolution is limited poleward of 60° latitude. Other orbits that have been proposed for measurements of atmospheric composition include the Molniya orbit (high-latitude observations several times per day), the Lagrange
L1 orbit (continuous global view of the sunlit Earth), and the Lagrange L2 orbit (continuous solar occultation).
Figure 10.10 Satellite orbits and their distances from earth. Distances are not to scale.
The determination of atmospheric concentrations from space is considerably more complex than for in-situ observations. Retrievals of concentrations from the radiance spectra must account for interferences from the surface and clouds. In the case of gas retrievals, they must also account for interferences from aerosols and from other gases. In the case of aerosol retrievals, they must also account for variations in aerosol optical properties. The retrieval is generally underconstrained, which means that external prior information on atmospheric composition must be assumed. The prior information often comes from models, which may lead to an incestuous relationship between the satellite observations and the models that they are supposed to evaluate. This will be discussed further in Chapter 11 in the context of inverse modeling and data assimilation.
10.3 Characterization of Errors
10.3.1 Errors in Observations
Observations are characterized by systematic and random errors (Taylor, 1996; Hughes and Hase, 2010). Systematic errors are consistent biases that repeat themselves every time the measurement is made by the same instrument under identical conditions. They cause the measured quantity to be shifted away from its true value due to factors that reproducibly affect the measurement, such as inaccurate calibration of the instrument or, when the measurement is indirect, inaccuracy in the retrieval model. The magnitude of the systematic error determines the accuracy of a measurement. Random errors are caused by factors that affect the measurements erratically, such as photon counting noise. They determine the precision of the measurement. The best estimate of a measured value is the mean of individual measurements, and the random error is the distribution around this mean. When the random error distribution is near Gaussian it can be characterized by its standard deviation (see Appendix E).
The derivation of atmospheric quantities from remote sensing observations requires that a retrieval calculation be performed (see example in Box 11.5). The retrieval involves inversion of a radiative transfer model to infer the atmospheric concentrations of interest from the radiances measured by the instrument. The model represents the physics of the measurement and often requires prior information to provide the best statistical fit to the observed radiances, accounting for errors in the measurements and the model. The instrument sensitivity may have a dependence on altitude, so that the retrieved concentration profiles reflect different altitude weightings and dependences on the prior information. Satellite observations have complicated error budgets with contributions from the measured radiances and the retrieval model, and including smoothing as well as random errors. Errors on vertical profiles are usually correlated across different altitudes, so that the error statistics must be represented by an error covariance matrix (see Chapter 11).
10.3.2 Errors in Models
Error in complex models is difficult to characterize. Chemical transport models provide a continuous 3-D simulation of atmospheric composition evolving with time, but accurate observations to evaluate the model are sparse and do not cover the full range of conditions over which the model is to be applied. Statistical metrics for comparing models to observations, and from there estimating model error, are presented in Section 10.5. Model error characterization for purposes of inverse modeling can be done with the residual error variance method that lumps model and instrument error into an overall observational error (Box 11.2). We discuss here some more specific approaches for estimating model error.
Model errors originate from four sources: (1) the model equations and underlying scientific understanding; (2) the model parameters input to these equations; (3) the numerical approximations in solving the equations; and (4) coding errors. Coding errors are generally revealed by computational failure of the model under some conditions or by unexpected model behavior. Detailed output diagnostics are important for detecting coding errors, including statistical distributions that allow detection of anomalies. Benchmarking of successive model versions is essential to detect coding errors introduced during a version update. Model output should always make sense to the modeler in terms of the underlying processes. If it does not, then a bug is probably lurking and must be chased without complacency.
Errors in the numerical approximations used to solve the equations are discussed in Chapters 6 and 7. Ideally, the numerical approximations should be benchmarked against exact analytical solutions, but such analytical solutions are available only in idealized cases that are generally not relevant to the atmosphere. For example, numerical advection of a given shape in a uniform flow can be compared to the shape-preserving analytical solution, but this greatly underestimates the error in the divergent flow typical of the atmosphere (Section 8.13). In the absence of an exact calibration standard, one can still estimate numerical errors by conducting model sensitivity simulations using objectively better, higher-order numerical methods. For example, one can compare a fast chemical solver of relatively low accuracy used in standard simulations to a more accurate solver applied to the same chemical mechanism.
Errors caused by model grid resolution can be estimated by conducting sensitivity simulations at finer resolution. Extrapolation is possible with Aitken’s convergence method (Wild and Prather, 2006). Let Co be the exact solution at infinitely fine resolution for a model variable of interest and C(h) the solution computed with the model at grid resolution h. We can write
Co = C(h) + [C(h/2) − C(h)] + [C(h/4) − C(h/2)] + [C(h/8) − C(h/4)] + …
(10.6)
Assume now that the model converges geometrically to the exact solution as the grid resolution increases, with a scale-independent geometric convergence factor k < 1 such that
(10.7)
Replacing into (10.6) we obtain the error estimate
(10.8)
We can calculate a value for k from (10.7) by conducting simulations at three different resolutions h, h/2, h/4, and we can further check the quality of the geometric convergence assumption by conducting a fourth simulation at resolution h/8. The assumption of a scale-independent geometric factor is unlikely to hold down to infinitely small resolutions. However, if k is sufficiently small, the first terms in the series may provide a good approximation of the error.
Errors due to model input parameters can be estimated in principle by conducting an ensemble of simulations where these different parameters are varied over their ranges of uncertainty, using for example a Monte Carlo method. This can be practically done for chemical mechanisms where error estimates for individual chemical rate constants are available from compilations of kinetic data. Errors in other model parameters, such as winds or emissions, are not as well quantified, and a Monte Carlo analysis over the full range of parameter space in a 3-D model would be impractical anyway. Errors can be estimated in a limited way by conducting sensitivity simulations with different meteorological fields, emission inventories, etc. Model error may be dominated by a small number of parameters, and it is part of the modeler’s skill to recognize which model parameters are important and to focus error characterization accordingly.
Errors in the model formulations of chemical and physical processes can be estimated to the extent that there are objectively better or equally valid formulations to apply. For example, we can estimate the error associated with using a reduced chemical mechanism for faster computation by conducting a sensitivity simulation with the complete mechanism. Errors associated with subgrid parameterizations of processes (Chapter 8) can be estimated by comparing different choices of parameterizations, or by conducting a test simulation at high resolution where subgrid parameterization is not needed. For example, it is useful to assess the sensitivity of model results to the choice of boundary layer mixing and convective transport parameterizations.
We discussed in Chapter 4 the noise in climate models caused by chaos in the solution to the Navier–Sto
kes equation for momentum. There is no such chaos in the solution to the chemical continuity equations under practical atmospheric conditions. In fact, numerical errors in the solutions to chemical systems tend to dissipate with time following Le Chatelier’s principle. Offline chemical transport models driven by input meteorological variables thus do not show the chaotic behavior found in climate models. However, in the case of an online chemical transport model built within a free-running climate model, the chemical concentrations develop noise driven by the noise in meteorological variables. This noise has physical basis in the internal variability of climate and needs to be characterized in the model for comparison to observations. This can be done by conducting a number of simulation years and/or by repeating the model simulations a number of times with slightly different initial meteorological conditions or physical parameters. From this ensemble of realizations we can construct probability density functions (PDFs) of concentrations to compare to the corresponding multi-year PDFs in long-term observations. Such a statistical comparison is called a climatological evaluation, with the PDFs representing the climatologies of the model and of the observations.
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