the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
The first application of a numerically exact, higherorder sensitivity analysis approach for atmospheric modelling: implementation of the hyperdualstep method in the Community Multiscale Air Quality Model (CMAQ) version 5.3.2
Jiachen Liu
Eric Chen
Shannon L. Capps
Sensitivity analysis in chemical transport models quantifies the response of output variables to changes in input parameters. This information is valuable for researchers engaged in data assimilation and model development. Additionally, environmental decisionmakers depend upon these expected responses of concentrations to emissions when designing and justifying air pollution control strategies. Existing sensitivity analysis methods include the finitedifference method, the direct decoupled method (DDM), the complex variable method, and the adjoint method. These methods are either prone to significant numerical errors when applied to nonlinear models with complex components (e.g. finite difference and complex step methods) or difficult to maintain when the original model is updated (e.g. direct decoupled and adjoint methods). Here, we present the implementation of the hyperdualstep method in the Community Multiscale Air Quality Model (CMAQ) version 5.3.2 as CMAQhyd. CMAQhyd can be applied to compute numerically exact first and secondorder sensitivities of species concentrations with respect to emissions or concentrations. Compared to CMAQDDM and CMAQadjoint, CMAQhyd is more straightforward to update and maintain, while it remains free of subtractive cancellation and truncation errors, just as those augmented models do. To evaluate the accuracy of the implementation, the sensitivities computed by CMAQhyd are compared with those calculated with other traditional methods or a hybrid of the traditional and advanced methods. We demonstrate the capability of CMAQhyd with the newly implemented gasphase chemistry and biogenic aerosol formation mechanism in CMAQ. We also explore the crosssensitivity of monoterpene nitrate aerosol formation to its anthropogenic and biogenic precursors to show the additional sensitivity information computed by CMAQhyd. Compared with the traditional finite difference method, CMAQhyd consumes fewer computational resources when the same sensitivity coefficients are calculated. This novel method implemented in CMAQ is also computationally competitive with other existing methods and could be further optimized to reduce memory and computational time overheads.
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Ambient air pollution, including particulate matter (PM), poses significant threats to human health. According to the Global Burden of Disease study, among all risk factors, ambient PM pollution accounted for 4.7 % of the disabilityadjusted life years (Murray et al., 2020) and over 4.1 million deaths (Fuller et al., 2022) in 2019. Therefore, understanding the complex physicochemical and atmospheric transport processes that lead to PM formation is essential to reducing PM and other harmful secondary atmospheric pollutants. Amongst atmospheric scientists, chemical transport models (CTMs) have become essential tools for interpreting observations of and examining inferences about formation processes of atmospheric pollutants. By solving the mass conservation equation for different species based on atmospheric dispersion and transport, photochemical processes, atmospheric chemistry, and aerosol processes, CTMs can provide estimates of primary and secondary air pollutants (Seinfeld and Pandis, 2016). Environmental decisionmakers and researchers rely on CTMs to determine appropriate policies to control air pollution and predict atmospheric pollutant concentrations. Experimental studies (e.g. Ng et al., 2008) and measurement campaigns (e.g. Sareen et al., 2016) provide researchers with more insights about the anthropogenic and biogenic aerosol formation processes. These studies ultimately lead to developments and updates to the gasphase chemistry and aerosol formation mechanisms in CTMs. For these newly added species and mechanisms in CTMs, understanding the sensitivity of aerosol species concentrations to emissions of their precursors is crucial for determining the priority of primary pollutant emission reductions to achieve atmospheric pollutant reduction objectives.
Sensitivity analysis methods have become invaluable for evaluating uncertainties, understanding concentration–emission relationships in CTMs, and assimilating observations of atmospheric pollutants to improve model parameters. Specifically, the kind of sensitivities described here are the partial derivative of one or more model outputs with respect to one or more model inputs. For instance, if the model has input variables X and output variables Y, the nthorder sensitivity coefficient of one output variable Y_{i} to one input variable X_{i} can be represented as the nthorder partial derivative of Y_{i} with respect to X_{i}, $\frac{{\partial}^{n}{Y}_{i}}{\partial {X}_{i}^{n}}$ (Cohan and Napelenok, 2011). Most sensitivity analysis techniques are formulations of the tangent linear model, which provides sourceoriented sensitivities or, mathematically, one column of the Jacobian or Hessian at the model state. In contrast, the model adjoint provides receptororiented sensitivities or, mathematically, one row of the Jacobian at the model state. Two distinct approaches to developing these models are the continuous approach, in which the derivative of the underlying equation is formulated and then implemented numerically, and the discrete approach, in which the derivative of the numerical solution of the model is formulated (Sandu et al., 2005). Since the model adjoint provides sourceoriented sensitivities and is not directly comparable with other methods (including the hyperdualstep method, which is the focus of this work), it will not be further discussed in the following sections. Other augmented model methods, including the Integrated Source Apportionment Method (Kwok et al., 2013, 2015), are based on a different approach and are thus also not discussed further in the following paragraphs.
The firstorder sensitivity coefficient is usually the most useful for CTM applications because it describes the linear relationship between X_{i} and Y_{i}. Higherorder sensitivities can be helpful when assessing the nonlinear relationships or dynamics among multiple input variables. Previous studies found that highly nonlinear concentration–emission responses commonly exist in CTMs, particularly for the formation process of PM (Hakami, 2004; Xu et al., 2018; Tian et al., 2010). Therefore, accurately determining the firstorder and secondorder relationships is useful for understanding concentration–emission responses in CTMs. Practically, the characteristics of an ideal sensitivity analysis method are numerical accuracy, computational efficiency, and minimal development (Lantoine et al., 2012).
Because analytical sensitivities are impractical for these models, researchers have employed a few numerical methods to calculate the first and higherorder sensitivities in CTMs. One such method is the finite difference method (FDM), which is often designated the bruteforce method. The FDM is based on the first or higherorder approximation of the Taylor series expansion from a small perturbation (Boole, 1960). The sensitivities are calculated by running the model multiple times with incrementally different values for the input variables of interest and taking the difference in the resulting concentration fields. While this method is simple to understand and implement, truncation and subtractive cancellation errors can substantially reduce the accuracy of the calculated sensitivity coefficients, particularly for nonlinear input–output relationships (Fornberg, 1981). Truncation errors originate from neglected higherorder terms in the Taylor series expansion. For instance, suppose a policymaker is interested in calculating the effects of reducing SO_{2} emission on the total PM_{2.5} concentration. The sensitivity analysis indicates that the firstorder partial derivative is positive and the secondorder partial derivative is negative. In that case, only considering the firstorder FDM approximation will overestimate the effect of reducing the SO_{2} emission on the total PM_{2.5} concentration. The truncation error can be minimized by taking a small perturbation step, thus eliminating the impact of higherorder sensitivity terms on the firstorder result. However, smaller perturbation steps might lead to subtractive cancellation errors, which stem from the fact that computers cannot distinguish two numbers close to each other. If the perturbation size is within the numerical noise of the model, the numerator difference sometimes approaches zero or the sensitivity information might be meaningless, which causes an inherent tension between reducing the truncation error and the subtractive cancellation error for the FDM. Determining ideal perturbation sizes for different variables is challenging because the ideal perturbation size depends on the input species of interest and other parameters in the model. The necessity of selecting the proper perturbation size for each input variable of interest and running the model multiple times makes the FDM a computationally expensive method to obtain sufficiently accurate sensitivities from CTMs.
As a continuous, sourceoriented approach, the decoupled direct method (DDM) eliminates the numerical accuracy issues of the FDM and improves the computational efficiency of calculating sourceoriented sensitivities but only with a hefty development cost. Dunker (1981) introduced to atmospheric modelling the direct method, which involves formulating new sensitivity equations from the original model and solving both sets simultaneously. The direct method has been proven numerically unstable for solving stiff equations, which exist in many chemical transport models. On the other hand, DDM formulates sensitivity equations like the direct method but separately solves the original and sensitivity equations. This approach improves the computational efficiency and stability compared to the direct method. Yang et al. (1997) was the first to apply the DDM3D method in a threedimensional chemical transport model. Hakami et al. (2003) extended the method to a higherorder DDM (HDDM) in the gas phase of the Community Multiscale Air Quality Model (CMAQ), while Zhang et al. (2012) augmented CMAQHDDM to include the secondorder sensitivities of PM_{2.5} concentration to NO_{x} and SO_{2} emissions. Unlike the FDM, the DDM does not incur truncation or subtractive cancellation errors since separate equations are solved for the sensitivities. The DDM also allows the computation of sensitivities of many outputs to more than one input simultaneously, saving significant computation resources. The major disadvantage of DDM for CMAQ and other CTMs is the difficulty of codevelopment alongside ongoing scientific model development, which is one purpose of CTMs. The implementation of DDM requires writing sensitivity equations for nonlinear steps, which commonly exist in the chemistry and advection parts of CTMs (Fike and Alonso, 2011). New sensitivity equations must be written when CTMs are updated, reducing the ease of maintenance of DDM in complex CTMs and eliminating the opportunity for sensitivities to be used for evaluation in the process of developing new scientific modules within the CTMs.
The complex variable method (CVM) and the multicomplex step approach (MCX) are the methods most comparable to the hyperdual step method. Lyness and Moler (1967) introduced the concept of using imaginary space to propagate derivatives for functions in real space based on Cauchy integrals. Squire and Trapp (1998) made the idea practical through an elegant truncation of the Taylor series expansion of complex numbers which allows nearly exact firstorder sensitivities if the imaginary perturbation is small enough. Constantin and Barrett (2014) applied the CVM on the adjoint method of the global CTM GEOSChem to compute nearexact sensitivities with receptor orientation in one order and source orientation in the other. Lantoine et al. (2012) developed a multicomplex number system to allow higherorder sensitivities to be calculated to machine precision for functions in real space. Berman et al. (2023) implemented MCX in the inorganic aerosol thermodynamic model ISORROPIA, which is used in CMAQ and other air quality models. These methods require the inclusion of a library of overloaded operators to treat the types of numbers required and the conversion of the model from real to complex space. The accuracy of these approaches is only limited by ensuring that the imaginary perturbation is small enough, which may require tuning depending on the complexity of the model. CVM does not require as much memory and computational time as MCX, but both contribute overhead. Both are very easily updated with new scientific modules. The main limitation of CVM and MCX is that sensitivities cannot be propagated through models, like CMAQ, that originally include calculations in imaginary space.
Finally, the method of interest in this work is the hyperdualstep method (HYD), which computes sourceoriented first and secondorder sensitivities to machine precision. HYD relies on hyperdual numbers, which are a specific type of generalized complex number developed particularly for first and secondorder derivative calculations (Fike and Alonso, 2011). The HYD, like the CVM or MCX, is an approach based on a Taylor series expansion in a nonreal number space. The unique mathematical properties of hyperdual numbers lead to an elegant calculation of first, second, and potentially higherorder sensitivities to machine precision without truncation or cancellation errors. Hyperdual numbers have been applied in numerical models in different fields of study to calculate exact first and secondorder derivatives. Cohen and Shoham (2015) applied hyperdual numbers to compute secondorder derivatives in multibody kinetics problems. Tanaka et al. (2015) utilized hyperdual numbers to automatically differentiate hyperelastic material models. Rehner and Bauer (2021) applied hyperdual numbers to equation of state modelling and the calculation of critical points. This method, which is applicable to models with calculations in imaginary space, is accurate to machine precision, reasonably computationally expensive, and quite straightforward to update.
Here, we implement the HYD in CMAQ version 5.3.2 to develop the novel augmented model CMAQhyd, and we apply it to calculate the sensitivities of both inorganic and organic aerosol concentrations to their precursor emissions. To our best knowledge, this work represents the first implementation of hyperdual numbers to calculate first and secondorder sensitivities in a CTM. In Sect. 2, hyperdual numbers and the hyperdualstep method are introduced, as is the process of implementing and evaluating HYD in CMAQ. In Sect. 3, the evaluation of first and secondorder sensitivities from CMAQhyd is conducted, including the computational costs. In Sect. 4, CMAQhyd is applied to understand the influences of anthropogenic and biogenic emissions on select secondary organic aerosol (SOA) concentrations. This work provides an accurate and easily manageable method to compute first and secondorder sensitivities implemented in CMAQ version 5.3.2 and an example of its potential application in other complex models where the sensitivities are of interest.
2.1 Hyperdual numbers and the hyperdualstep method
A hyperdual number (Fike and Alonso, 2011) has four components and is characterized by
where a_{0}, a_{1}, a_{2}, and a_{12} are real numbers and ϵ_{1}, ϵ_{2}, and ϵ_{12} are nonreal parts. The three crucial properties which enable numerically exact first and secondorder sensitivity calculations are
The squares of the nonreal individual parts equal zero (Eq. 2). The nonreal parts themselves do not equal anything in real space (Eq. 3). The multiplication of ϵ_{1} and ϵ_{2} is equal to the third nonreal component ϵ_{12} (Eq. 4). The addition and multiplication of hyperdual numbers are commutative, and the definitions help eliminate the higherorder terms in a Taylor series expansion. A demonstration of several basic operations is provided in the Supplement, while a more detailed discussion of the mathematical properties of hyperdual numbers is given by Fike and Alonso (2011).
Akin to the Taylor series expansion about the real value of x in the finite difference method, the method of ascertaining sensitivities through a perturbation in hyperdual space is based on a Taylor series expansion in an orthogonal dimension of the number. Specifically, a hyperdual number with unity in a_{0} and unity in one of a_{1} or a_{2} is multiplied by the independent variable of interest. After model execution, a Taylor series expansion is applied to extract sensitivities. For instance, the hyperdualstep method is applied to a scalar function f(x) by multiplying x by the hyperdual number ${H}_{\mathrm{h}}=\mathrm{1.0}+{h}_{\mathrm{1}}{\mathit{\u03f5}}_{\mathrm{1}}+{h}_{\mathrm{2}}{\mathit{\u03f5}}_{\mathrm{2}}$, which results in
where “…” represents higherorder terms in the series. Eliminating all terms that are zero due to the definition of hyperdual numbers (Eq. 2) leads to
where f(xH_{h}) is a hyperdual number.
The properties of hyperdual numbers (Eqs. 2–4) lead to two significant results. First, all terms in the Taylor series expansion with derivatives higher than second order become zero because all values include ${\mathit{\u03f5}}_{\mathrm{1}}^{\mathrm{2}}$, ${\mathit{\u03f5}}_{\mathrm{2}}^{\mathrm{2}}$, or ${\mathit{\u03f5}}_{\mathrm{12}}^{\mathrm{2}}$. Second, the real component is unchanged. A more detailed expansion of terms can be found in Eq. (S7) in the Supplement or the original development of hyperdual numbers; it follows the multiplication rule between a hyperdual and a real number (Fike and Alonso, 2011). Finally, the first and secondorder derivatives are separated into different parts of the hyperdual number. The firstorder derivative exists in either the ϵ_{1} or the ϵ_{2} term, while the secondorder derivatives only exist in the ϵ_{12} term. The first and secondorder derivatives are
where ϵ_{1}part[], ϵ_{2}part[], and ϵ_{12}part[] represent functions that extract the a_{1}, a_{2}, or a_{12}, respectively. Since the derivative computation process (Eqs. 6–9) does not involve subtractions or higherorder sensitivities, the first and secondorder sensitivities calculated by the hyperdualstep method are free from subtractive cancellation and truncation errors. This method (Eqs. 8 and 9) extends to vector operations to compute arrays of numerically exact derivatives. For instance, the partial first and secondorder derivatives for f(x), where $\mathit{x}=[{x}_{\mathrm{1}},{x}_{\mathrm{2}},\mathrm{\dots},{x}_{n}]$, with respect to x_{1} through a hyperdualstep perturbation to x_{1} is
Similarly, two different independent variables x_{1} and x_{2} may be perturbed simultaneously. In this case, two arrays of firstorder sensitivity and one array of crosssensitivity result:
Therefore, the two variations of the hyperdualstep method will generate either one or two arrays of firstorder sensitivities and one array of secondorder or cross sensitivities with a single run of the model.
2.2 Community Multiscale Air Quality Model and the implementation of the hyperdualstep method
The Community Multiscale Air Quality Model (CMAQ), developed by the U.S. Environmental Protection Agency (EPA), is an Eulerian CTM which can predict air pollutant concentrations at regional and hemispheric scales (Byun and Schere, 2006). CMAQ represents advection, diffusion, gasphase chemistry, aerosol processes, cloud processes, and photolysis. CMAQ has been applied to predict pollutant concentrations in the atmosphere (Liu et al., 2010; Sayeed et al., 2021), understand fundamental atmospheric chemistry and aerosol formation mechanisms (Zhu et al., 2018; Li et al., 2019), and guide policymaking processes (Chemel et al., 2014; Che et al., 2011; Li et al., 2019; Ring et al., 2018). CMAQ is used in the regulatory process of the US EPA when states, tribes, or local jurisdictions demonstrate how they will attain the National Ambient Air Quality Standard (NAAQS) and or comply with the Regional Haze Rule (Mebust et al., 2003). CMAQ solves the atmospheric diffusion equation shown in Eq. (14) to calculate the concentrations of gaseous and aerosol species in the atmosphere:
where c_{i}, u, K, R_{i}, and E_{i} are the concentration of species i, the wind velocity vector, the diffusivity tensor, the change in concentration due to chemical reactions of species i, and the emission rate of species i, respectively. Species concentrations are stored in a multidimensional array and propagated through different scientific modules within the model. For this work, CMAQ was run with 12 km by 12 km horizontal resolution and with 35 vertical layers, 100 columns, and 80 rows over the Southeast US on 1 July 2016, GMT (U.S. Environmental Protection Agency, 2019). The gasphase chemistry mechanism used is Carbon Bond 6 (Luecken et al., 2019).
In CMAQ, the hyperdualstep method was implemented by strategically converting the model to use hyperdual numbers (Fig. 1). First, the operators were overloaded by translating a Cbased library from Fike and Alonso (2011) to Fortran (“HDMod”) and augmenting it to treat multidimensional data as required by CMAQ. HDMod, which defines a hyperdual version of all possible calculations related to the chemical concentration array, was developed. The library includes basic arithmetic operations, such as addition and subtraction, as well as more advanced functions like trigonometric functions. Before being applied to CMAQ, the operator overloading library was separately validated by comparing it against analytical derivatives using a testing framework developed by Pellegrini and Russell (2016). Secondly, the CMAQ variable containing species concentration information and all other variables that depend on it were converted from real numbers to the newly defined hyperdual number type. The source code was carefully analysed to select only the necessary variables for conversion. Many variables in CMAQ do not need to be altered because they do not influence the main concentration array. This highly detailed process helped minimize the additional computational burden of the model since calculations with hyperdual numbers are more computationally intensive than those with real numbers. For instance, one hyperdual multiplication operation shown in Eq. (5) results in five more additions and nine more multiplications than an operation with real numbers. According to Fike and Alonso (2011), the computational cost of a hyperdual calculation ranges from 4 to 14 times higher than the original operation. Applying hyperdual numbers to all the variables in a CFD model results in an approximately 10 times higher computational cost (Fike and Alonso, 2011). Thirdly, the first and secondorder sensitivities of the species concentrations to perturbed emissions are included in the new hyperdual array, which is then saved to additional output files using the same structure as the output of the original concentration array. As a result, first and secondorder sensitivities can be propagated through the model without significantly modifying the source code. The modification efforts mainly focused on determining the variables that must be converted to the hyperdual type. Consequently, updating CMAQhyd when there are changes to the original model is a simple process that involves converting only the newly added variables to the hyperdual type. This simplicity is an advantage over other computational techniques, such as the DDM and adjoint method, which compute numerically exact sensitivities but require more complex and timeconsuming update procedures, including writing new equations.
Several source code alterations were made to reduce the complexity of development and overcome the numerical instabilities related to hyperdual calculations in CMAQ's aerosol module, specifically within the inorganic thermodynamic module ISORROPIA (Fountoukis and Nenes, 2007; Nenes et al., 1998). For the simplicity of development, we applied a Fortran90compilant version of ISORROPIA to replace the original Fortran 77 version of ISORROPIA in CMAQ.
ISORROPIA, a key component of the aerosol module in CMAQ, is called in either the forward or the reverse mode. The forward mode of ISORROPIA takes the sum of gas and aerosol species concentrations, along with the relative humidity and temperature, to determine the partitioning of Aitken and accumulationmode species across the gas and aerosol phases in CMAQ. In the original CMAQ model, ISORROPIA is run in the forward mode without limiting the temperature and pressure of the simulation. The determination process for species concentrations involves an iterative method which is sometimes numerically unstable during iterations for upper layer cells with low temperature and pressure for sensitivity computations with the HYD. To increase the numerical stability of CMAQhyd, we implemented temperature and pressure constraints so that the forwardmode ISORROPIA is only called when the cell temperature exceeds 260 K and the cell pressure exceeds 20 000 Pa. A similar set of temperature and pressure limits were applied to the call of ISORROPIA in the adjoint of CMAQ (Zhao et al., 2020). These changes do not affect the species concentrations computed by CMAQ while ensuring that the sensitivity computation process is stable.
To calculate the dynamic equilibrium of coarsemode aerosol species with the gas phase (Pilinis et al., 2000; Capaldo et al., 2000), CMAQ employs the reverse mode of ISORROPIA. The input to reversemode ISORROPIA includes concentrations of aerosol species, relative humidity, and temperature, and it results in partitioned concentrations in the solid, liquid, and gas phases for coarsemode inorganic aerosols. The reversemode solution leads to unrealistic sensitivities calculated by the HYD when the aerosol pH is close to neutral. One previous study found that the reverse ISORROPIA fails to capture the actual behaviour of inorganic aerosol when the pH is close to 7 (Hennigan et al., 2015). To ensure the stability of the sensitivity calculations, the changes to the hyperdual components in the coarsemode dynamic equilibrium are ignored when the pH of coarsemode aerosol is close to neutral, which ensures that the real components are identical to the original model.
2.3 Evaluating sensitivities from CMAQhyd
CMAQhyd produces sensitivities that can be semi or fully normalized for concentrations from any range of grid cells and times with respect to emissions or concentrations from any range of grid cells and times. Here, for the sake of illustration, we consider the seminormalized sensitivities of timeaveraged output concentrations of groundlevel PM_{2.5} concentrations (${C}_{{\mathrm{PM}}_{\mathrm{2.5}},c,r,l=\mathrm{0},t}$) to input NO_{x} (NO + NO_{2}) emissions (${E}_{{\mathrm{NO}}_{x},c,r,l,t}$) averaged over time, t, for any given cell as indicated by the column c and row r. The firstorder seminormalized sensitivities, ${s}_{{\mathrm{NO}}_{x}}^{{\mathrm{PM}}_{\mathrm{2.5}}}$, and secondorder seminormalized sensitivities, ${s}_{{\mathrm{NO}}_{x}}^{\left(\mathrm{2}\right){\mathrm{PM}}_{\mathrm{2.5}}}$, exemplify sensitivities relevant to environmental decision makers (Eqs. 15 and 16).
Seminormalized sensitivities reduce the complexity of interpretation by providing sensitivities in the units of concentration per percent change of emissions. The seminormalized sensitivities also scale down the impact from cells with low emission rates, which is consistent with the concentration reduction that is realistic to expect. Similarly, the timeaveraged, seminormalized crosssensitivity of PM_{2.5} to both NO_{x} and monoterpene is denoted as ${s}_{{\mathrm{NO}}_{x},\mathrm{TERP}}^{\left(\mathrm{2}\right){\mathrm{PM}}_{\mathrm{2.5}}}$, with E_{TERP} representing the emission of monoterpenes (Eq. 17).
The evaluation of the first order sensitivities of CMAQhyd is done by performing a comparison of the sensitivities calculated by the hyperdualstep method against those from the FDM (Eq. 18). The comparison is illustrated with an example of calculating cellspecific sensitivities of PM_{2.5} concentration to NO_{x} emissions. The firstorder sensitivity of PM_{2.5} concentration at the end of a 24 h simulation to a cumulative NO_{x} emission perturbation is given by
where the subscripts c, r, and l represent the column, row, and layer; the subscript t represents the time from the start of the model run; and the superscripts “inc”, “dec”, and “orig” represent the initial perturbation direction (i.e. increased, decreased, and original emissions, respectively). For instance, ${C}_{{\mathrm{PM}}_{\mathrm{2.5}},c,r,l,t=\mathrm{24}}^{\text{inc}}$ is the concentration of PM_{2.5} for each column, row, and layer 24 h into the run when there is an increase in NO_{x} emissions throughout the model run. Unless otherwise noted, the relative perturbation sizes for firstorder FDM calculations are 125 % and 75 % for domainwide emissions. The average groundlayer sensitivities for the 24 h simulation time are computed. Previous studies have found that smaller perturbation sizes for inorganic aerosol sensitivity calculations in CMAQ using FDM are more prone to numerical noise (Zhang et al., 2012). The seminormalized sensitivity of each cell is computed with the central difference method and is only an approximation of the actual sensitivity due to subtractive cancellation and truncation errors. The numerator is the difference between PM_{2.5} concentrations with persistent increases or decreases in NO_{x} emissions. The denominator is the total emission perturbation of NO_{x} emission. The sensitivities are seminormalized by the sum of the NO_{x} emissions in the base model run. The calculated firstorder seminormalized sensitivities will have units of µg m^{−3}.
The seminormalized sensitivity of PM_{2.5} concentrations with respect to NO_{x} emissions using a hyperdual perturbation of ${H}_{\mathrm{a}}=\mathrm{1}+{a}_{\mathrm{1}}{\mathit{\u03f5}}_{\mathrm{1}}+{a}_{\mathrm{2}}{\mathit{\u03f5}}_{\mathrm{2}}$ is computed by the hyperdualstep method as
The firstorder seminormalized sensitivity can be computed with either the ϵ_{1} or the ϵ_{2} part. The ϵ_{1} or the ϵ_{2} part of the PM_{2.5} concentration is divided by the initial perturbation in the ϵ_{1} or ϵ_{2} space, respectively. The emissions in the denominator will cancel out with the seminormalized emissions.
Although the FDM can be applied to compute secondorder sensitivities in CMAQ, previous studies have shown that the results are noisy and highly dependent on the perturbation sizes (Zhao et al., 2020; Zhang et al., 2012). In order to evaluate the secondorder sensitivities computed by the HYD method, we adopted a hybrid hyperdual–finite difference method (HYDFDM). The HYDFDM sensitivity calculation is given by
where two separate simulations were run: one with increased and another with decreased initial NO_{x} emissions. The emission perturbations for the two runs are ${H}_{\mathrm{a}}=\mathrm{1}+{a}_{\mathrm{1}}{\mathit{\u03f5}}_{\mathrm{1}}+{a}_{\mathrm{2}}{\mathit{\u03f5}}_{\mathrm{2}}$ for the run with increased initial NO_{x} emission and ${H}_{\mathrm{b}}=\mathrm{1}+{b}_{\mathrm{1}}{\mathit{\u03f5}}_{\mathrm{1}}+{b}_{\mathrm{2}}{\mathit{\u03f5}}_{\mathrm{2}}$ for the run with decreased initial emission of NO_{x}. HYDFDM uses the regular finite difference on the difference between firstorder sensitivities calculated by using the ϵ_{1} part of the hyperdualstep results. The sensitivity in this equation is an estimate and subject to numerical errors because it includes the usage of FDM.
The secondorder sensitivity calculated by the hyperdualstep method is shown in Eq. (21) below:
The hyperdualstep method uses the ϵ_{12} part of the output variable, dividing it by the multiplication of a_{1} and a_{2}. The secondorder sensitivities calculated only by the HYD method are numerically exact. All the sensitivities are computed for each cell, and comparisons between the finite difference and the hyperdualstep method are performed on a celltocell basis.
3.1 Evaluation of the first and secondorder sensitivities
We evaluated the implementation of CMAQhyd by comparing the firstorder sensitivities of various species in CMAQ calculated by HYD with a hyperdualstep perturbation as described in Sect. 2.3 (HYD sensitivities) and FDM with a domainwide emission perturbation (FD sensitivities). The FD sensitivities were computed with the difference between a 25 % increase and a 25 % decrease in domainwide emissions using the central finite difference method. Overall, the different HYD and FD sensitivities agree well, as evidenced by the close alignment of the points on the blue identity line, which represents perfect agreement, in most of the panels of Fig. 2. The slope and R^{2} values for all comparisons are provided in Table 1, and additional slope and R^{2} values are provided in Tables S1 and S2 in the Supplement. Specifically, the slopes and R^{2} values for gasphase species concentrations on the ground layer with respect to their emissions on the ground layer (Fig. 2a–d) are all 1.00 (Table 1), indicating minimal nonlinearity in these relationships, as expected.
Secondary aerosol formation is a more nonlinear process, which is explored by using inorganic or organic aerosol concentrations with respect to select precursors (Fig. 2e–h). Nonlinearities in the modelled processes lead to discrepancies between HYD and FD sensitivities without tuning the FD sensitivity to capture the slope about the model state more exactly. The slopes and R^{2} values of the trendline between these HYD and FD sensitivities range from 0.99 to 1.00 and from 1.00 to 1.04 (Table 1), respectively. The comparisons between the HYD and FD sensitivities of ${s}_{{\mathrm{NO}}_{x}}^{{\mathrm{ANO}}_{\mathrm{3}}}$ and ${s}_{{\mathrm{SO}}_{\mathrm{2}}}^{{\mathrm{ASO}}_{\mathrm{4}}}$ show slight deviations from the identity line, indicating some nonlinearity in their formation processes (Fig. 2e and f). Most points representing the HYD and FD sensitivities of total monoterpene photooxidation products to monoterpenes (${s}_{\mathrm{TERP}}^{{\mathrm{AMTNO}}_{\mathrm{3}}}$) and αpinene (${s}_{\mathrm{APIN}}^{{\mathrm{AMTNO}}_{\mathrm{3}}}$) remain on the identity line (Fig. 2g and h).
Regulatory models are often used to evaluate the response of total PM_{2.5} to emissions changes, so the sensitivities of the total PM_{2.5} concentration to the four different precursor emissions are evaluated (Fig. 2i–n). The HYD and FD sensitivities of ${s}_{{\mathrm{NO}}_{x}}^{{\mathrm{PM}}_{\mathrm{2.5}}}$, ${s}_{\mathrm{TERP}}^{{\mathrm{PM}}_{\mathrm{2.5}}}$, and ${s}_{\mathrm{APIN}}^{{\mathrm{PM}}_{\mathrm{2.5}}}$ (Fig. 2i, k, and n) agree well, with slope and R^{2} values ranging from 0.96 to 1.03 and from 0.99 to 1.00, respectively (Table 1). However, the agreement between the HYD and FD sensitivities of ${s}_{{\mathrm{SO}}_{\mathrm{2}}}^{{\mathrm{PM}}_{\mathrm{2.5}}}$ (Fig. 2j) is much lower, with a slope of 0.65 and an R^{2} value of 0.63 (Table 1). Notably, although ${s}_{{\mathrm{SO}}_{\mathrm{2}}}^{{\mathrm{PM}}_{\mathrm{2.5}}}$ is usually positive, as evidenced by most of the points on the identity line, the ${s}_{{\mathrm{SO}}_{\mathrm{2}}}^{{\mathrm{PM}}_{\mathrm{2.5}}}$ calculated by FDM sensitivities has a few negative values where the HYD and FD sensitivities disagree. Because it is highly unlikely that an 25 % increase in SO_{2} emission leads to a decrease in PM_{2.5} concentration, the negative FD sensitivities likely arise from truncation errors inherent to the FDM since the perturbation sizes are large (i.e. a 25 % emission perturbation). Though it is possible to refine the perturbation size to one more suitable for this particular relationship of emissions to concentration, as demonstrated in the next section, this difference in one of 12 comparisons shows one of the strengths of HYD, which is the irrelevance of the perturbation size to the exactness of the resulting sensitivity.
To further illustrate the impact of nonlinear relationships between emissions and concentrations on FD sensitivities, we explored the sensitivity of groundlevel aerosol nitrate to emissions of sulfur dioxide, ${s}_{{\mathrm{SO}}_{\mathrm{2}}}^{{\mathrm{ANO}}_{\mathrm{3}}}$, calculated with different perturbation sizes using the FDM. Our analysis revealed a low level of agreement between the FD and HYD sensitivities in the base case scenario where the domainwide SO_{2} emission was perturbed by 125 % and 75 %, with a slope of 0.10 and an R^{2} of 0.30 (Table S1). The FD sensitivities calculated with the base case perturbation (125 %, 75 %) and two other perturbation size pairs (110 %, 90 %; 105 %, 95 %) are shown in Fig. 3. The FDM sensitivities calculated with different perturbation sizes are plotted in Fig. 3a–c, respectively. The FDM sensitivities exhibit similar behaviour to the HYD sensitivities over the continents. However, the inconsistency among the sensitivities calculated by FDM with different perturbation sizes over the ocean (Fig. 3) suggests that the FD sensitivities heavily depend on the perturbation sizes. This result demonstrates the relatively low credibility of FD sensitivities, particularly for highly nonlinear relationships where the truncation errors could be large. Notably, reducing perturbation sizes in the FDM did not lead to convergence with hyperdual sensitivities. This divergence may be attributed to the propagation of numerical noise from the model run to the calculated sensitivities as perturbation sizes decrease. This finding is consistent with the results in Zhang et al. (2012). Our findings highlight the importance of using other methods, including the HYD, which are not prone to truncation or cancellation errors for probing nonlinear relationships in CTMs.
We also compared the spatial distribution of HYD sensitivities (Fig. 4a) against the average (Fig. 4b) and the range (Fig. 4c) of the FD sensitivities with three different perturbation sizes. Differences are evident between the HYD and the average FD sensitivities in central North Carolina and Tennessee as well as off the coasts of Georgia and South Carolina. The HYD predicts slightly negative sensitivities in North Carolina and Tennessee, while the FDM predicts slightly positive values. The average FDM sensitivities off the coast of Georgia and South Carolina were noisy, with alternating positive and negative sensitivities, while the HYD sensitivities were much less noisy. In addition, the range of FDM sensitivities with different perturbation sizes was large (Fig. 4c), especially off the coast of Georgia and South Carolina. The results shown in Fig. 4b and c illustrate the dependence of FDM sensitivities on the perturbation sizes, especially for highly nonlinear relationships.
We also compared the secondorder HYD sensitivities with those calculated from the hybrid HYDFDM method (hybrid secondorder sensitivities) described in Sect. 2.3 using onetoone plots with identity lines for each panel (Fig. 4) along with the slope and R^{2} values (Table 2). Additional slopes and R^{2} values for secondorder sensitivities can be found in Tables S1 and S2. Overall, the agreement between HYD and the hybrid secondorder sensitivities is good except for those to SO_{2} emissions. This result can be attributed to the numerical errors in the firstorder sensitivities to SO_{2}, as illustrated in Figs. 2j and 3. Computing secondorder sensitivities with the hybrid method, which includes FDM, is expected to add numerical noise. Except for ${s}_{{\mathrm{SO}}_{\mathrm{2}}}^{\left(\mathrm{2}\right){\mathrm{SO}}_{\mathrm{2}}}$, the hyperdual and hybrid secondorder sensitivities of gasphase species concentrations to emissions of the same species exhibit good agreement, with slopes and R^{2} values ranging from 0.84 to 0.85 and 0.84 to 0.86, respectively (Table 2). The hybrid secondorder sensitivities are sometimes large, while HYD predicts closetozero sensitivities. This result is especially evident in ${s}_{\mathrm{TERP}}^{\left(\mathrm{2}\right)\mathrm{TERP}}$ (Fig. 5c) and ${s}_{\mathrm{APIN}}^{\left(\mathrm{2}\right)\mathrm{APIN}}$ (Fig. 5d). This spread in the hybrid sensitivities likely originates from the FDM step, which is subject to numerical errors. Figure 5e–h show the HYD and HYDFDM sensitivities of aerosolphase product concentrations to the precursor emissions for this modelling period. Except for ${s}_{{\mathrm{SO}}_{\mathrm{2}}}^{{\mathrm{ASO}}_{\mathrm{4}}}$, the slope and R^{2} values range from 0.61 to 0.82 and 0.38 to 0.71, respectively (Table 2). The degree of agreement for ${s}_{{\mathrm{NO}}_{x}}^{{\mathrm{ANO}}_{\mathrm{3}}}$ is slightly lower than those for ${s}_{\mathrm{TERP}}^{\mathrm{\Sigma}\mathrm{AMT}}$ and ${s}_{\mathrm{APIN}}^{\mathrm{\Sigma}\mathrm{AMT}}$, indicating more nonlinearity in the formation process from NO_{x} to aerosol nitrate. The secondorder sensitivities of total PM_{2.5} to different precursors demonstrate excellent agreement, with slope and R^{2} values ranging from 0.95 to 0.99 and from 0.96 to 0.99 (Table 2), again excluding the one to SO_{2}. The secondorder sensitivities of PM_{2.5} to NO_{x} and αpinene are primarily negative, while those to monoterpenes are positive. These findings have important implications for the formation process of PM_{2.5} from monoterpenes and αpinene, which will be discussed in detail in the next section.
3.2 Sensitivities of biogenic aerosol formation in the Southeast US computed by CMAQhyd
In this section, the first and secondorder sensitivities of several biogenic aerosols to both anthropogenic and biogenic aerosol precursors in the Southeast US are explored. The importance of calculating secondorder sensitivities is demonstrated through the spatial distributions of the first and secondorder sensitivities of total aerosolphase monoterpene photooxidation product (ΣAMT) and PM_{2.5} concentrations (Fig. 6). The firstorder sensitivities (Fig. 6a–d) are predominantly positive, indicating that an increase in either TERP or APIN emissions will lead to an increase in groundlayer ΣAMT and PM_{2.5} concentrations. While ${s}_{\mathrm{TERP}}^{\mathrm{\Sigma}\mathrm{AMT}}$ (Fig. 6a) and ${s}_{\mathrm{APIN}}^{\mathrm{\Sigma}\mathrm{AMT}}$ (Fig. 6b) have similar values, ${s}_{\mathrm{TERP}}^{{\mathrm{PM}}_{\mathrm{2.5}}}$ (Fig. 6c) is slightly larger than ${s}_{\mathrm{APIN}}^{{\mathrm{PM}}_{\mathrm{2.5}}}$ (Fig. 6d) due to the formation of other species, such as aerosolphase monoterpene nitrate products (AMTNO_{3}). The secondorder sensitivities (Fig. 6e–h) provide additional information about how ΣAMT and PM_{2.5} concentrations respond to changes in APIN and TERP emissions. The ${s}_{\mathrm{TERP}}^{\left(\mathrm{2}\right)\mathrm{\Sigma}\mathrm{AMT}}$ (Fig. 6e) and ${s}_{\mathrm{APIN}}^{\left(\mathrm{2}\right)\mathrm{\Sigma}\mathrm{AMT}}$ (Fig. 6f) are generally positive, indicating that the response of the concentration of ΣAMT to monoterpene and αpinene emissions is convex. An increase in either monoterpene or αpinene emissions will lead to increases in ${s}_{\mathrm{TERP}}^{\mathrm{\Sigma}\mathrm{AMT}}$ and ${s}_{\mathrm{APIN}}^{\mathrm{\Sigma}\mathrm{AMT}}$. If we only consider firstorder sensitivities, the effect of changes in TERP or APIN emissions on ΣAMT concentrations will be underestimated. On the other hand, ${s}_{\mathrm{TERP}}^{\left(\mathrm{2}\right){\mathrm{PM}}_{\mathrm{2.5}}}$ (Fig. 6g) is mostly negative, while ${s}_{\mathrm{APIN}}^{\left(\mathrm{2}\right){\mathrm{PM}}_{\mathrm{2.5}}}$ (Fig. 6h) is mostly positive. The distinct behaviour of the secondorder sensitivities of PM_{2.5} concentration to either TERP or APIN emissions exemplifies the importance of considering secondorder sensitivities for these nonlinear formation processes. Only considering the firstorder sensitivities often leads to the overestimation or underestimation of the effects. The accurate secondorder sensitivity information can help researchers understand the relationships of concentration to emissions more thoroughly and develop emission control strategies for specific aerosol precursor emissions.
We used the formation of aerosol monoterpene nitrate, AMTNO_{3}, as an example of the importance of computing the crosssensitivity, especially for complex anthropogenic–biogenic aerosol formation processes. The formation of AMTNO_{3} is influenced primarily by two precursors: NO_{x} and monoterpenes. NO_{x} is primarily emitted anthropogenically, while monoterpenes primarily originate from biogenic sources. The first and secondorder sensitivities of AMTNO_{3} to NO_{x} or TERP can help researchers and environmental decision makers estimate the nonlinear effects of emissions changes on concentrations of secondary pollutants. The crosssensitivity of AMTNO_{3} with respect to both NO_{x} and TERP emissions, ${s}_{{\mathrm{NO}}_{x},\mathrm{TERP}}^{\left(\mathrm{2}\right){\mathrm{AMTNO}}_{\mathrm{3}}}$, is a valuable tool for answering complex research questions. For instance, researchers can use ${s}_{{\mathrm{NO}}_{x},\mathrm{TERP}}^{\left(\mathrm{2}\right){\mathrm{AMTNO}}_{\mathrm{3}}}$ to predict how an increase in monoterpene emissions would affect the firstorder sensitivities of AMTNO_{3} to NO_{x}. Since biogenic emissions of monoterpenes are temperature dependent, understanding how anthropogenic emissions of NO_{x} will affect AMTNO_{3} formation with changing terpene emissions in future scenarios is crucial for designing effective air pollution control strategies. Computing the crosssensitivity is especially challenging with traditional methods since determining the proper perturbation for two species using FDM is even harder than calculating secondorder sensitivities with FDM. The distinct values of ${s}_{{\mathrm{NO}}_{x}}^{\left(\mathrm{2}\right){\mathrm{AMTNO}}_{\mathrm{3}}}$, ${s}_{\mathrm{TERP}}^{\left(\mathrm{2}\right){\mathrm{AMTNO}}_{\mathrm{3}}}$, and ${s}_{{\mathrm{NO}}_{x},\mathrm{TERP}}^{\left(\mathrm{2}\right){\mathrm{AMTNO}}_{\mathrm{3}}}$ demonstrate the value of the HYD method (Fig. 7). The spatial distributions of ${s}_{{\mathrm{NO}}_{x}}^{{\mathrm{AMTNO}}_{\mathrm{3}}}$ and ${s}_{\mathrm{TERP}}^{{\mathrm{AMTNO}}_{\mathrm{3}}}$ are included in Fig. S1 in the Supplement. Overall, the secondorder sensitivities are negative over land in the Southeast US. ${s}_{{\mathrm{NO}}_{x}}^{\left(\mathrm{2}\right){\mathrm{AMTNO}}_{\mathrm{3}}}$ is generally smaller than ${s}_{\mathrm{TERP}}^{\left(\mathrm{2}\right){\mathrm{AMTNO}}_{\mathrm{3}}}$, indicating that the relationship between AMTNO_{3} and TERP emissions is more nonlinear than that between AMTNO_{3} and NO_{x}. The crosssensitivities ${s}_{{\mathrm{NO}}_{x},\mathrm{TERP}}^{\left(\mathrm{2}\right){\mathrm{AMTNO}}_{\mathrm{3}}}$ are mostly positive over the Southeast US. Based on the crosssensitivity results, we can conclude that an increase in TERP emission will make the firstorder sensitivity of AMTNO_{3} to NO_{x} larger. A warmer climate in the future would likely increase the impact of anthropogenic NO_{x} on the AMTNO_{3} concentration in the atmosphere. This kind of information is invaluable for researchers and environmental decision makers aiming to evaluate complex secondary organic aerosol formation with multiple anthropogenic and biogenic precursors.
3.3 Computational cost of CMAQhyd
The practical application of any sensitivity analysis in CTMs depends heavily upon its computational cost. Previous works using operator overloading approaches resulted in high computational costs due to additional mathematical operations, making this approach computationally unfavourable compared to other existing methods. For instance, the implementation of CVM in GEOSCHEM results in a 4.5fold increase in computational overhead when compared to the standard model (Constantin and Barrett, 2014). To evaluate the computational efficiency of CMAQhyd, we compared it with a standard CMAQ model using different computational resources in a supercomputing cluster. The total wall times needed when different numbers of computational nodes are used for an identical run of the original CMAQ model and CMAQhyd are displayed in Fig. 8. The CMAQhyd and standard CMAQ runs were performed with one, two, four, and eight nodes of the supercomputing cluster. The configuration of computing resources is detailed in Sect. S4 of the Supplement. Profiling of the model was completed at the level of the scientific modules, special subroutines, or other important components of CMAQ. The scientific processes are Chem (gasphase chemistry), Aero (aerosol dynamics and thermodynamics), Vdiff (vertical diffusion), Hadv (horizontal advection), Phot (photolysis), Cldproc (cloud processes), Hdiff (horizontal diffusion), and Zadv (vertical advection). MPI_Barrier is a special subroutine used for synchronizing processes among parallel processors after vertical diffusion and before the other three transport processes. Other processes necessary for CMAQ, including the I/O processes, are included in the “Other” category. Details about the highperformance computing cluster used can be found in the Supplement.
With the same computing resources, the total computation time of CMAQhyd is approximately 2.5 (2.44–2.56) times longer. Despite the additional computational burden, CMAQhyd remains computationally competitive with the traditional FDM when calculating derivatives. One run of CMAQhyd generates the same amount of first and secondorder sensitivity information as at least three runs of standard CMAQ. The relatively low computational cost of CMAQhyd compared to the previous operator overloading approach may be due to the selective modification of the source code. In contrast to GEOSCHEM CVM (Constantin and Barrett, 2014), only the parts of the model that involve calculating the main species concentration array use hyperdual calculations.
The computational times of scientific modules in CMAQhyd generally scale well with increases in computational resources, similar to standard CMAQ. Chem, Aero, and Vdiff are the most computationally expensive modules in both CMAQ and CMAQhyd. The relative computational cost of Aero is higher in CMAQhyd than in standard CMAQ. The ratio of the computational time of Chem to that of Aero is 1.53 (1.49–1.56) for the CMAQhyd runs and 3.98 (3.85–4.19) for the standard CMAQ runs (Fig. 8). Future work can potentially reduce the computational cost by ignoring sensitivity propagations during the iterative rootfinding process in select subroutines, since only the output concentrations from these subroutines are used in the later part of the model. This is also a significant advantage of any operatoroverloadingbased approach (Fike and Alonso, 2011). The computational time of each module is detailed in Table S3 in the Supplement, and the full relative percentage of the computational time of each module in eight runs is shown in Fig. S2 in the Supplement.
The MPI_Barrier function also scales well with an increasing number of processors. To a certain point, subdividing the domain further reduces the variability of the time required for scientific processes to be completed across different nodes, resulting in a reduction in the amount of time the program spends waiting for all processes to be synchronized. One important thing to note here is that the scaling of MPI_Barrier is dependent on the ratio of the number of nodes to the number of grid cells. A demonstration of the subdivision of the modelling domain by processors is shown in Fig. S3 in the Supplement.
The I/O process for newly added first and secondorder sensitivity output files increases the computational cost; however, the I/O of species concentration files has a much lower computational cost than other modules in CMAQ for this specific scenario. The I/O processes of CMAQhyd and CMAQ take 193 (181–206) s and 52 (47–56) s, respectively. The I/O process of CMAQhyd takes approximately 3.7 times longer on average than that of standard CMAQ. The overall memory overhead of CMAQhyd is approximately 25 GB for this simulation. A parallel input/output (I/O) approach may be applied to reduce the possibility of potential memory overflow in processor 0 (Wong et al., 2015).
We have demonstrated the implementation of the hyperdualstep method in CMAQ version 5.3.2 to formulate CMAQhyd. This novel model retains the majority of the CMAQ code with slight modifications in the declarations of selected variables and the addition of sensitivity computation modules. The novel model can be applied to compute exact firstorder sensitivities, secondorder sensitivities, and crosssensitivities of pollutant concentrations to precursor emissions efficiently and accurately with a single model run. Compared with traditional sensitivity analysis methods, CMAQhyd is computationally competitive with conventional methods and easier to maintain than other existing advanced methods (DDM and adjoint). The development process of CMAQhyd is also more straightforward than that of other advanced methods, since all that is needed is to change the type of newly declared variables to hyperdual.
We developed and validated the hyperdualstep module “HDMod”, which is limited to analytically verifiable mathematical operations of hyperdual numbers. This module can also be applied to other numerical models where first and secondorder sensitivities are of interest. We further evaluated the development of CMAQhyd against the FDM and the FDMHYD hybrid method to ensure the correctness of the implementation. During the evaluation process, CMAQhyd demonstrated the ability to compute sensitivities free from truncation and subtractive cancellation errors, unlike those calculated by the FDM. HDMod can potentially be applied to other numerical models written in Fortran to produce first and secondorder sensitivities.
The computation of secondorder sensitivities is crucial for researchers and environmental decision makers who need to decide the priority and extent of controls on specific types of emissions to reduce atmospheric pollutant concentrations. For instance, the secondorder sensitivity of PM_{2.5} concentration to monoterpenes and αpinene provided additional information about the relationships of emissions to concentrations in CMAQ. With the additional secondorder sensitivity information, the curvature of the concentration responses to emissions changes improves the estimate of how a specific pollutant concentration would respond to changes in emissions. The simplicity of computing crosssensitivities with CMAQhyd is another advantage of this augmented model. Crosssensitivities are especially useful in nonlinear processes with two precursors. Knowledge of the synergistic effect of anthropogenic and biogenic emissions on aerosol concentrations (e.g. NO_{x} and monoterpene on AMTNO_{3}) is essential for researchers to predict the dynamics between two potential pollutants and for environmental decision makers to propose policy implementations under different climate scenarios in the future.
Although CMAQhyd remains computationally competitive with the traditional finitedifference method, it is still computationally intensive and has a memory overhead. We plan to implement optimizations for iterative processes in CMAQ and to apply the parallel I/O approach to reduce the memory overhead on the compute node where all the information is gathered. The implementation of checkpointing of sensitivities after specific subroutines is also a potential advantage of CMAQhyd and will provide valuable information on how each module or even each line of the model affects the sensitivities, akin to a process analysis approach. This checkpointing feature cannot be easily implemented with other methods such as FDM, DDM, and the adjoint method.
In conclusion, we have developed and evaluated CMAQhyd, a novel augmented model to compute firstorder sensitivities, secondorder sensitivities, and crosssensitivities free from subtractive cancellation and truncation errors in CMAQ. Our successful implementation also provides an example of the hyperdualstep method that may be applicable for other CTMs where sensitivities are helpful.
CMAQv5.3.2 is available at https://github.com/USEPA/CMAQ/tree/5.3.2 (last access: 1 June 2021) and is archived at https://doi.org/10.5281/zenodo.4081737 (US EPA Office of Research and Development, 2020). The CMAQhyd model is archived at https://doi.org/10.5281/zenodo.10119026 (Liu et al., 2023). Both the CMAQv5.3.2 and the CMAQhyd models are under MIT licences. The input data for the simulation experiments are available at https://doi.org/10.15139/S3/IQVABD (US EPA, 2019).
The supplement related to this article is available online at: https://doi.org/10.5194/gmd175672024supplement.
JL developed the model code and performed the simulations with help from SLC. EC helped develop and test the “HDMod” library. JL prepared the manuscript with guidance from SLC.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.
The authors would also like to thank Ryan P. Russell for kindly providing the testing framework for multicomplex numbers, which inspired the development of the testing framework for hyperdual numbers.
The work is supported by National Science Foundation CAREER Award grant no. 1944669 to Shannon L. Capps.
This paper was edited by Yilong Wang and reviewed by Jixiang Li, Jaroslav Resler, and one anonymous referee.
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