The continuous equation of motion for the tracer, χ (a mixing ratio), in 3D is nominally ∂χ(t)∂t+u(t)⋅∇χ(t)=Sρ(t)+κh∇h2χ(t)1+κv∂2χ(t)∂z2, where u is the prescribed 3D wind vector (u=uvw⊺), ρ is the 2D surface flux field, the operator Sρ constructs a 3D field of zero apart from the surface where it assigns the value ρ/(Δz0d0) (surface layer thickness Δz0, surface air density d0=1.225kgm-3), κh and κv are the horizontal and vertical diffusion coefficients, ∇ is the 3D gradient operator, and ∇h2 and ∂2/∂z2 are the horizontal and vertical Laplacian operators. In practice, Eq. () is solved on a discrete grid in space and time (see below). In EnviFlux, χ has units of µgkg-1 (although the units in many applications is moles per mole of dry air), and ρ has units of µgm-2s-1. Since the purpose of this work is to study the inverse problem in its simplest form, the tracer in Eq. () has no sub-grid influences, like convection and turbulence, nor chemical reactions that would apply to methane e.g. like hydroxyl or photochemical oxidation. Stable tracers like carbon dioxide or sulphur hexafluoride would be the most suitable species for EnviFlux, but this paper focuses on a hypothetical tracer (although in the second example application shown in this paper, Sect. , the tracer has a methane-like flux pattern).

The core component of EnviFlux is the state vector, x, which comprises the initial and boundary conditions needed to make predictions of the atmospheric tracer mixing ratio, χ(t). The state vector is written 2x=χ(0)ρ(0)ρ(τρ)⋮ρ(Tτρ), where χ(0) is the initial 3D tracer field, and ρ(nτρ) is the surface flux field at a set of equally spaced time-steps n=0,1,…,T with τρ the time interval between flux fields. Bold symbols indicate the variables are part of the state vector. In this paper sometimes only the χ(0) part of x is referred to and other times only the ρ(0),…,ρ(Tτρ) part. The notation xχ and xρ shall refer to these parts respectively.

The spatial grid is shown in Fig. , which has nx longitudes, ny latitudes, and nz vertical levels. The time-stepping is done in three stages, namely (i) the semi-Lagrangian (SL) advection stage, (ii) the diffusion stage, and (iii) the flux stage, which are described below. EnviFlux uses four kinds of time step, which are as follows (see Fig. ).

The minor time step, δt, is the interval used to divide up the back-trajectory for the solver needed for the SL advection scheme.

The second caveat concerns the adjoint operator, i.e. χ^[k-1]Δt:=Mk-1→kx^ρ⊺χ^kΔt. The action of the adjoint propagates gradient

Gradient of the observation term of the cost function, see Eqs. () and ().

EnviFlux uses an incremental formulation of the cost function that is minimised during the 4D-Var. In this formulation, x=xb+δx, where xb is the background field (a priori), and δx is the increment, found by minimising the 4D-Var cost function. Instead of varying δx directly, a control variable, δv is varied, where δx and δv are related via δx=B12δv, where B is the background error covariance matrix (Sect. ). This change of variable removes the B-matrix in explicit form from the cost function, and helps improve the conditioning (see e.g. or Sect. 3.1 of ). In terms of δv the cost function is Jδv(δv)=12δv⊺δv+12∑k=0NTyk-HkM0→kxb+B12δvρχb(0)+B12δvχ⊺5×Rk-1•, where the notation xχ and xρ is explained in Sect. , NT is the number of major time-steps in the problem, yk is the set of observations made at time-step k, Hk is the observation operator, Rk is the observation error covariance matrix, and χb(0)=xbχ is the background tracer at t=0. The “a⊺A-1•” notation is shorthand for “a⊺A-1a”, for brevity. Even though the notation in Eq. () implies that observations are at major time-steps only, EnviFlux does allow observations to be made at intermediate times, which are treated with linear interpolation (for details see Sect. 3.2.1 of ).

The gradient of the cost function is ∇δvJ(δv)=δv-B⊺2∑k=0NTM0→kxb+B12δvρ⊺Hk⊺Rk-16×yk-HkM0→kxb+B12δvρχb(0)+B12δvχ, where B⊺2=B12⊺, which is fed to a conjugate gradient-based descent algorithm to iteratively minimise the cost function. A version of the gradient calculation that is more efficient to compute than Eq. () is used by EnviFlux (see Sect. 3.2.2 of ).

The block partitioning of δv is similar to that of x in Eq. () in the sense that the first part concerns the initial conditions and the remainder concerns the surface fluxes: 7δv=initial conditionssurface fluxes. The size of each block in δv is the same as in x, but the elements have different meanings. The elements represent weights of different spectral components, and vertical or temporal modes. This is discussed further in Sect. .

EnviFlux allows three types of observation to be assimilated, namely in situ tracer observations, integrated total column amount (TCA) observations, and direct flux observations. Observations can be at any geographical location, and tracer observations can additionally be made at any height above the surface within the domain. The model's version of the observations, Hkχ(kΔt), are found by linear interpolation in space and time. The TCA observations are found from a discretisation of the vertical integral ∫χ(z)dzdz, where z is height above the surface, dz is the air density (this has a similar mathematical structure as the Jacobian of the radiative transfer models used to assimilate some satellite observations), and χ(z) is the tracer field interpolated to the geographical location and time of each observation.

Currently in EnviFlux, observation errors are assumed uncorrelated. The observation error standard deviations, σo, can be set arbitrarily for each observation, and the diagonal elements of the R-matrix is made up of values σo2. EnviFlux uses these observation operators in the assimilation, and the same operators are used to make synthetic observations as part of the twin experiment method.

The background error covariance matrix, or B for short, is a fundamental component of a Gaussian-based Bayesian problem, such as that described by Eq. (). As in most assimilation systems, the probability density function of background errors is assumed to be Gaussian, where the mean is xb and the error covariance is B. The B-matrix specifies, statistically, how large background errors are and how background errors are correlated, e.g. . The background term of the unprecondioned cost function – when written in terms of δx, see Eq. (6) of – is proportional to minus the logarithm of this PDF.

In the preconditioned cost function, Eq. (), the new variable, δv, is chosen to have unit background error covariances, which means that the background errors in this δv-representation are uncorrelated and have unit variance. The B-matrix appears in this preconditioned system via a square-root in the variable transform B12, i.e. δx=B12δv (for more details about this kind of variable transform, see e.g. or ). B12 is a large matrix, and so instead of storing it explicitly, it is formulated with a set of component operators, called a background error covariance model. In practice these are compactly encoded as subroutines. In EnviFlux it has the following form:

8B12=ΣχbFvχΛvχ12Fvχ⊺RhShΛhχ12|0|0|ΣρbFtρΛtρ12Ftρ⊺RhShΛhρ12|, which acts on δv of the form of Eq. (). The top-left block operates on the initial condition part of δv and the bottom-right block operates on the flux part of δv, and there is no coupling between them in this choice of model. Alternative B-matrix models may be coded instead, but the parts of this B12, associated with the initial conditions and fluxes, are now explained in turn.

The posterior (analysis) states that follow from a run of EnviFlux with near perfect specifications (meaning the B-matrix is a near perfect reflection of the actual errors

Apart from the flux field due to the nature of this experiment.

We now study the effect of model error on the results. It is already known that transport uncertainties in the assimilation can influence the position and strength of sources, , and we study this using the simple source/sink pair set-up. There are a number of ways of simulating model transport error, but it is simulated here by multiplying the vertical wind, w, in the assimilation by a number, η, close to unity (0.9 and 1.1), while the observations assimilated are still generated from samples from a run using the true winds (actually they are identical to those experiments in Sect. ). Changing the vertical winds, but keeping the same horizontal winds, will not directly affect how the tracer is circulated around the globe, although there will be an indirect consequence, e.g. by affecting which levels (including jet levels) the tracer can reach. Maintaining the true horizontal winds though means that the tracer still has the same potential to reach distant points as the truth. This choice of scheme does, however, create anomalous sources and sinks in the atmosphere (Appendix ), so it could, e.g. represent an erroneous sub-grid transport parametrisation of convection, which the assimilation will compensate for by creating anomalous surface sources and sinks. The reason for choosing this particular method is that it is simple to implement, and vertical transport is likely to be an important deficiency of real-world models. No attempt has been made here to account for the model error (e.g. by inflating σo as is done in other works, ).

Figure  shows analysis increments of χ(0) (identical to the respective analysis fields in these experiments) with these two η values for the two observation types. The impact of these (rather modest) changes to vertical motion is greater in the assimilation of TCA than SIS observations. For the TCA observations, the ability of the system to identify the positions of the two flux features is almost completely destroyed for η=0.9 (panel a). We refer to the property of an inferred flux field having a localised source and sink, such as in Fig. b as coherent; Fig. a is an example of incoherence in this field. This may be due to a weakening of the coupling between the above-surface tracer contributions to the TCA measurements, thus meaning that this part of the TCA is attributed to a whole range of diffuse locations around the globe (thinking in terms of back-trajectories from points along the vertical columns associated with these observations). The source location remains well identified in the η=1.1 case (although shifted slightly south), but there is more than one sink location identified, although one is still in the right place (panel b). For the SIS observations, the ability of the assimilation to identify the two positions is more robust. The most noticeable effect is the shift of the centre of the source region eastwards by ∼5° in panel c, and the interpretation of some other regions as weak sources and sinks, such as Kazakhstan (an anomalous sink in panel c) and Greenland (an anomalous sink in c and source in d). The better results for SIS are expected, as these observations are more closely/directly influenced by the fluxes than the TCA observations are, and so are less affected by model error.

We also studied the effect of observation bias on the flux analysis. The results proved that the inferred positions of the flux features is robust against observation bias in the ranges studied (for biases up to just over half the value of σ° for each observation type, analysis maps not shown). The flux amounts though are influenced by the bias, which is studied in the next section.

A summary of the analysed fluxes found for the source/sink location experiments, while varying η is shown in Fig. . Panels (a) and (c) show how the global total and peak source/sink values vary with η (see the caption for definitions of the total and peak values). Panel (a) assimilates TCA and (c) assimilates SIS observations. For η>1, there is a tendency for the global total and peak source amounts (red lines) to increase with η. For sources, increasing η generally distributes the sources to more distant points downstream and so prevents the build-up of tracer at geographical locations close to the emissions. Hence larger fluxes are required to match the observations, which is found in the results for η>1. A mirror argument may be made for the sinks (blue lines). It is harder to explain why the results show a strengthening of the total and peak sources and sinks as η gets smaller than particular values that depend on the experiment, e.g. η<0.8 in (a). A possible explanation is that the DA is compensating for the increasing relative importance of numerical diffusion in the SL transport scheme as the winds get weaker, although this is just a supposition.

Further inspection of Fig. a and c reveals the following. The variations of the flux values are smaller for the SIS than for the TCA observations, which is not surprising as the SIS observations are less influenced by the global circulation. The relative variations of the global total flux values are much greater than for the peak flux values (total and peak values have different units – see the caption – which is why we refer to relative variations). The analysed global total values are mostly larger than the true global total values (compare solid lines with dashed lines) with the exception of η≤1 for the global total source (red solid line in a), but this will change with different values of Σρb. The analysed peak values are always smaller than the true peak values (compare dotted lines with dot-dashed lines).

A further robust finding of these experiments is that the posterior errors in the fluxes are always smaller than the prior errors. This is not surprising because the prior flux is a zero field. The posterior errors in the initial conditions, however, are larger than the prior errors, except when η=0 (not shown). This is an example of model error aliasing onto another component of the system, in this case the initial conditions.

Figure b and d shows how the positions of the peak source (red dots) and sink (blue dots) change with η for the TCA (b) and SIS (d) observations. The true positions of the source and sink are shown as squares. Even though maximum and minimum flux positions are always found in ρa(0), the field patterns are not always coherent (e.g. compare Fig. b with Fig. a).

The inferred peak positions of the source form a “cloud” of locations inside SE Asia for both observation systems, with the η=0.95 and 1.0 being closest for both observation systems. Even though ρa(0) for the η=0.9 and TCA observations case loses its coherence (Fig. a), the η=0.95 case (not shown) maintains a coherent a set of correctly positioned source and sink features. The positions inferred from the SIS observations are slightly more robust against deviations of η from 1.0.

As η deviates from unity, the inferred sink positions become more variable than the source positions. The ρa(0) sinks tend to be diagnosed at a more diverse range of locations, often on different continents to the true location. Often the analysed peak locations are close to the true locations, but these are usually just the strongest of many local peaks which the assimilation has constructed.

Figure  shows how the global total, and the peak source and sink values vary with observation bias (there is no model error in these experiments). Panel (a) is for the TCA observations and (b) is for the SIS observations. Each different experiment assimilates similar observations discussed previously, but where an observation bias has been added. The amount of bias varies up to ± just over half of the value of σ° for each observation type (see bottom row, middle column of Table ). The peak source and sink values (red and blue dotted lines) are almost completely insensitive to the observation bias, but the total source and sink fluxes (solid lines) do show variation in the way expected (i.e. the total source fluxes increase with the values of the biases, and the total sink values decrease). This is a similar property of the system (mentioned but not shown in Sect. ), that the analysed total flux values are sensitive to the choice of Σρb, but the analysed peak values are not. Further, the locations and coherency of the source and sink features are unaffected by the range of biases studied (not shown).

Figure  shows analyses and analysis errors when the specifications of the DA system are perfect (i.e. the correct background and observation errors, no model error, and no observation biases). The assimilation of TCA and SIS observations each effectively correct the background errors, panels (a) and (c). On the colour scale of the plots, it is not possible to see differences between these experiments, so the analysis errors, ρa(0)-ρt(0) have been plotted in panels (b) and (d). The analysis errors following the assimilation of the different observation types are comparable, with the SIS observations leading (marginally) to the lower errors, consistent with the findings of Sect. .

The effect of the model error proxy on these assimilation experiments is smaller than in the source/sink location experiments in Sect. . Figure  shows the effect of using η=0.7 and 1.3 in the DA. Compared to the η values used in Fig. , values of η further from unity are required here to see noticeable effects in the analysis; this is probably because of the presence of the meaningful a priori state here. The first point to note is that all experiments are still able to reduce the error from the background despite the presence of model error. Assimilating SIS observations is more robust against model error than for TCA observations, and it is hard to see differences on the colour scale between the bottom rows of Figs.  and c. The most noticeable exception is for η=0.7 in panel (c) where there is still some error in the sink field over Indonesia. This anomaly is also present for the η=0.7 TCA experiment, panel (a), but exists together with anomalous sinks over Patagonia, New Zealand, north central Russia, north western Canada, and Greenland. Some of these anomalies (e.g. over Patagonia) are not due to persistent background errors, but result from unaccounted for model error in the DA. The Patagonia anomaly changes sign from an anomalous sink for η=0.7 in (a) to a source for η=1.3 in (b).

A systematic analysis of the way that model error affects the total flux amounts is shown in Fig. a (TCA, solid lines; SIS dotted lines, truth dashed line; sources are red and sinks are blue). There is a general tendency of the analysed total fluxes to increase with η for both observation systems (this is straightforward to see for the sources for η>0.9; in the case of the sinks, the absolute value decreases with η for η<1.2, but the actual total sink is minus this value, which increases). This is consistent with the idea that, as the global circulation anomalously speeds up in the forward model used inside the DA (i.e. as η increases), tracer can no longer accumulate near the surface (where the SIS observations are located, and where the highest contribution to the TCA observations is made) and so higher emissions are required to compensate. As found in the flux location experiments (Sect. ) and here, assimilating the SIS observations systematically gives results closer to the truth than for the TCA observations (this can be seen in panel (a) where the dotted red line (SIS) is closer to the dashed line (truth) than the solid red line (TCA), and the dotted blue line (SIS) is closer to zero (truth) than the solid blue line (TCA)).

Interestingly, this result appears to be the reverse of the findings of , who found that TCA observations over land allowed more robust flux analyses than in situ observations. Their explanation concerned unaccounted for convective transport of tracer, which could affect in situ observations more than TCA observations. This is a compelling argument, but it may also be due to any suboptimalities due to incorrectly assigned error characteristics of each observation network, e.g. by incorrectly assigning the representativeness error for in-situ observations. This issue is not studied further here, but could be by introducing scale mis-matches between the in-situ data and the assimilation (e.g. by generating SIS observations from a much higher resolution version of the model, but assimilating them with the lower resolution model).

The effect of model error on the root mean squared analysis errors (RMSEs) is quantified in Fig. b. The RMSEs for flux (green lines) reassuringly minimise when η=1 and are smaller for the assimilated SIS observations (dotted line) than for the TCA observations (solid), consistent with previous results. The finding is different for the RMSEs for the initial tracer (black lines). Here the experiment with the smallest RMSE for the TCA observation network is at η=0.95 and for the SIS network is at η=1.05. Further work would be required to decide if these deviations from η=1 are due to the particular background or set of observations.

The final results in this demonstration study shows how the total flux amounts and analysis errors in the quasi-real-world situation vary with observation bias. We find that the presence of bias in this well-observed system does not change the pattern of inferred fluxes from that of the unbiased observation system in the left panels of Fig.  (not shown), but does change the flux amounts.

A systematic analysis of the way that observation bias affects the total flux amounts is shown in Fig.  for TCA observations (a) and SIS observations (c), where solid lines are the analysed total fluxes (sources are red and sinks are blue), and the red dashed lines represent the true total source (the true total sink is zero in these experiments). The inferred total sources and sinks increase in value as the observation bias increases (the sinks need to be multiplied by -1 to find the actual sink values). This is the same conclusion as for the location experiments in Sect. . The absolute values of the total sources and sinks are overestimated in all experiments conducted, but as the amounts are found to increase with observation bias, the sources get further from the truth, but the sinks get closer to the truth (the true total sink is zero). This may not be a universal effect as it will probably change with different values of background error standard deviations. The effect on the TCA-observed system is larger than for the SIS-observed system. Recall that the observation biases are varied as a proportion of the random error up to ± just over half of σ° for each observation type, so it is a broadly fair comparison between the two observation networks.

The RMSE analysis errors in Fig. (b) and (d) show minima at the observation bias values indicated with the arrows (these are added as an aid owing to the shallow curves on the logarithmic scale). With the exception of the black line in (b) (for χa(0) errors for the TCA observations), the minima in the RMSEs do not align with zero bias. In the case of flux analysis errors, the “optimal” observation biases are negative. This has the effect of reducing the fluxes to values closer to the truth. This is an example of one type of error partially correcting for another (systematic biases compensating for the general overestimation of fluxes in the system), and can make the process of tuning such a DA system a complex task.

This paper introduces a simplified tool (EnviFlux), to allow investigations into inverse modelling techniques to infer surface fluxes. These techniques have difficulties and shortcomings, mainly because of the indirectness of the observations in order to estimate surface flux information. EnviFlux represents a “back to basics” approach to facilitate a deeper understand of the basic inverse problem without the complexities and expense of other systems, and with no need for high performance computing resources. The tool uses a semi-Lagrangian (SL) transport model, a spectrally modelled B-matrix (which implies homogeneous background error correlations), and supports in situ tracer and flux, and total column amount observations. EnviFlux is a 4D-Var-based inverse model, which estimates the flux field on the same grid as the model, and it does this simultaneously with the initial tracer field. The tool can be used “as is” or can be expanded as required. The software is a suite of programs, mainly written in C++, with some Python visualisation tools. Due to its intentional simplicity, there are some issues that EnviFlux cannot currently explore, such as those specific to satellite measurements (such as cloud and aerosol effects), and with unresolved transport, but it is designed to look at the general problem of extracting information from indirect observations.

EnviFlux is demonstrated in “weakly Bayesian” (where there is missing a priori information, Sect. ) and Bayesian (where the mean of the a priori PDF is specified, Sect. ) scenarios. The weakly Bayesian scenario comprises tests to infer the locations and strengths of a source/sink pair at (a priori) unknown locations from a time sequence of observations of the tracer only. The observation network comprises either total column amount (TCA) or surface in situ (SIS) observations, each following the path of a polar orbiting satellite. The former observation network is meant to resemble an actual instrument such as GOSAT, and the latter is a control for comparison. As long as there is no model error (see below) in the data assimilation, the system can successfully determine the locations of the flux features, but with peak values that are too small, and a spatial extent that is too large (the latter property is likely due to the particular lengthscales used to model the B-matrix, and the fundamental smoothing effect of the inversion). A very simple proxy for model error is used by multiplying the vertical wind by a factor η in the inverse system, where η=1 represents a perfect model. This first set of experiments is very sensitive to even small deviations of η from unity (e.g. 0.9 and 1.1), where the ability of the tool to infer even the correct locations of the flux features is compromised. Even though the horizontal wind components are not multiplied by η, the circulation of the tracer can be affected indirectly (the tracer may, e.g. be advected to vertical levels that are affected more or less by jet streams). Observation biases, however (in the ranges tested from -0.57σ° to 0.57σ°, where σ° is the observation error standard deviation in each set of experiments – itself set to be 10% of the variability of the modelled observations in each case), do not affect the inferred locations, nor the peak values, but do affect the global total source and sink values. These results confirm other work that real-world inverse models need to have minimal imperfections in the forward model.

The Bayesian scenario (also referred to as the quasi-real-world case) uses similar observation networks, but uses methane-like spatial flux distributions as the observed true flux field. In the perfect model run, the choice of observation network has little effect on the ability of the assimilation to infer the correct qualitative structure of the true fluxes, but the in situ observations do offer advantages. Introducing model error gives rise to false sources and sinks. This becomes evident though at higher deviations from η=1 than for the weakly Bayesian experiments, probably owing to the a priori information available to the quasi-real case. Despite this, the assimilation still improves the flux estimates from the a priori. Model error does also quantitatively change the total tracer released and absorbed and for most of the η values tested there is a tendency for the total sources and sinks to increase with η, which is may be due to the assimilation compensating for the faster transport of tracer in space away from the places where the observation operators are most sensitive. Separately, introducing observation bias in the same way as for the weakly Bayesian experiments, has the same effect on the total source and sink. By varying the bias, we find that the lowest RMSE analyses are not found for the experiments when the biases are zero (negative observation biases give the best results). A compelling argument for this is that it is compensating for the total analysed sources, which are too high in these experiments.

It is anticipated that this simplified system as a base can be used and developed further as a low-cost tool to test new ideas in inverse models for research projects and even for student projects. As it stands, EnviFlux is configured for global flux inversion, but could be modified to study regional flux inversion by constructing a suitable limited area grid, transport model, and B-matrix, and introducing lateral boundary conditions, which could themselves become part of the state vector. It is also possible to swap the SL advection scheme with another, e.g. one that is designed to strictly conserve tracer amount, and to include some chemical processes, such as tropospheric hydroxyl sinks, e.g. , and stratospheric methane oxidation, e.g. . Where they are found to exist, correlations between observation errors can profoundly affect assimilation, e.g. , . By design, the observation networks used in this paper impose no such correlations, but these can be introduced into the observation generation and the Rk matrices. Providing error correlations between observations within the same major time step only is easiest to consider as this would require least development effort in EnviFlux. The way the B-matrix is modelled in this work dictates the horizontal grid to have dimensions (2L+1)×(L+1) (see Appendices –). Other than that, the choice of B-matrix model in the cost function can be replaced with alternatives, via the square-root (control variable transform) formulation. As long as the resolution is relatively low, an explicit form of the horizontal forecast error covariances could be used, which could e.g. have different lengthscales over the land to ocean to study their effect.

EnviFlux can be simplified further (e.g. by reducing the horizontal and vertical resolution) to allow large ensembles of inversions for tuning purposes, or as mentioned previously, can be made more complex. Adding chemical loss and production would allow closer comparison with real observations (especially for short-lived tracer species and longer transport times), but would add cost, complexity, and potentially non-linearity. The problem would no longer be a pure transport inverse problem and parcel trajectories would be subject to modification of the mixing ratio, potentially dependent on sunlight, cloud cover, temperature, and concentration of other constituents. Should the chemical modification be erroneous, and where the process is observed (directly or downstream), the upstream surface fluxes in the model would likely be spuriously modified to compensate.

In addition to the issues related to the above developments, some ideas and research questions that EnviFlux can contribute to are as follows. How sparse can the observation network be to infer detailed information on the location of new sources and sinks? How can EnviFlux be used to understand which observations contribute to which surface regions (e.g. by estimating the singular vectors of the forward model; example observation footprints are shown in )? How well can an inverse model determine the timing of transient fluxes (e.g. )? How precise does the initial condition field need to be to effectively estimate fluxes (the expectation is that a precise initial condition field is needed for flux estimation to be effective ). How long does an experiment have to run to virtually lose memory of the initial conditions (e.g. )? How is it possible to tune the background error covariances in real-world cases when they are not well known (e.g. )? How can model error be accounted for by developing techniques such as inflating the observation error variances , or by including a model error term throughout the atmosphere? How do different methods of estimating posterior error covariances compare, e.g. Lanczos or quasi-Newton-based, , , ? Can the system be used to test the ability of machine learning-based tools to emulate the forward part of the problem (e.g. )? By adding a simple chemical or land surface processes parametrisation (e.g. ), can the inversion be modified to estimate parameters in those models?