How large is the design space for stratospheric aerosol geoengineering?

. Stratospheric aerosol injection (SAI), as a possible supplement to emission reduction, has the potential to reduce some of the risks associated with climate change. Adding aerosols to the lower stratosphere would result in temporary global cooling. However, different choices for the aerosol injection latitude(s) and season(s) have been shown to lead to significant differences in regional surface climate, introducing a design aspect to SAI. Past research has shown that there are at least three independent degrees of freedom (DOF) that can be used to simultaneously manage three different climate goals. Knowing 5 how many more DOFs there are, and thus how many independent climate goals can be simultaneously managed, is essential to understanding fundamental limits of how well SAI might compensate for anthropogenic climate change, and evaluating any underlying trade-offs between different climate goals. Here we quantify the number of meaningfully-independent DOFs of the SAI design space. This number of meaningfully-independent DOFs depends on both the amount of cooling and the climate variables used for quantifying the changes in surface climate. At low levels of global cooling, only a small set of 10 injection choices yield detectably different surface climate responses. For a cooling level of 1-1.5°C, we find that there are likely between 6 and 8 meaningfully-independent DOFs. This narrows down the range of available DOF and also reveals new opportunities for exploring alternate SAI designs with different distributions of climate impacts.

. Understanding the global and spatiotemporal impacts of SAI and even the governance challenges requires that we not treat SAI as one single strategy, but rather understand the range of outcomes across different strategies, the fundamental limits of how well SAI can compensate for GHG-driven climate change, and any underlying trade-offs among SAI strategies.
Choosing where and when to inject aerosols can be thought of as a design problem (Ban-Weiss and Caldeira, 2010;Kravitz et al., 2016;MacMartin and Kravitz, 2019); for a given choice of global cooling, the design space describes the range of 30 all possible such injection strategies. Some strategies produce very different surface climate responses, while others can be relatively similar. These climate responses can be quantified by different metrics, such as surface air temperature, precipitation, and Arctic sea ice. We use the term "degrees of freedom" (DOF) to describe how many independent injection choices there are in the design space. The number of independent injection choices is equivalent to the number of independent climate goals that can be managed by SAI simultaneously. Most studies have only explored a single degree of freedom: injecting SO 2 at one 35 location (often the equator) either at a fixed rate or to meet one climate objective (often global mean temperature (T0)) (e.g., Robock et al., 2008;Rasch et al., 2008;Kravitz et al., 2011Kravitz et al., , 2015. Kravitz et al. (2016Kravitz et al. ( , 2017) demonstrated a strategy in which three DOFs were used to manage three temperature goals: T0, interhemispheric temperature gradient (T1), and equator-to-pole temperature gradient (T2); the same strategy was then used in The Geoengineering Large Ensemble Project (GLENS) (Tilmes et al., 2018). Additional studies have explored variations on these DOFs, such as Visioni et al. (2020c), who injected SO 2 in 40 only one season to meet the same set of climate goals, or Lee et al. (2020a), who used the same set of DOFs as in Tilmes et al. (2018) to meet different sets of climate goals, including T0, the latitude of the Intertropical Convergence Zone (ITCZ), the amount of Arctic September Sea Ice (SSI), and global mean precipitation (P0). Higher-latitude injections in different seasons have been shown to have different efficacies in preserving SSI; e.g., spring-only injection at 60 • N restores twice the amount of SSI compared to annually-constant injection at that latitude (Lee et al., 2021). A key open question from these design studies is 45 how many other strategies are unexplored (e.g., MacMartin and Kravitz, 2019); in other words, how many independent degrees of freedom are there?
In this study, we estimate the number of DOF of the design space for SAI. Knowing how many DOFs there are in the design space quantifies the number of independent climate goals that can be managed simultaneously by a SAI strategy. In order to be managed simultaneously, those independent climate goals cannot conflict. (For example, T0 and P0 are conflicting and cannot 50 be managed simultaneously; see e.g., Bala et al., 2008, Tilmes et al., 2013, and Lee et al., 2020a Knowing the number of DOF also helps understand the full range of possible climate outcomes and what climate outcomes cannot be achieved by SAI strategies. In this study, we focus only on SO 2 injections, and evaluate the range only in one model. However, the results will depend primarily on the constraints imposed by stratospheric circulation and the lifetime of the aerosols in the stratosphere MacMartin et al., 2017;Dai et al., 2018); as such, many of the conclusions can be expected to be applicable 55 regardless of aerosol choice.
The aerosols will primarily stay in the same hemisphere where they are injected and be transported mostly poleward by the stratospheric Brewer-Dobson circulation MacMartin et al., 2017). Thus, injecting in one hemisphere preferentially increases AOD in that hemisphere; injecting further poleward increases the AOD burden further poleward. In-by additional choices of injection latitude. That leads to the question of how many meaningfully-independent patterns of AOD are possible given the constraints imposed by stratospheric circulation.
There are two distinct steps in the analysis herein. The first step is to consider how different the spatiotemporal AOD patterns are for different injection choices. And second, to know whether the AOD patterns from two different injection choices are sufficiently similar to treat them as effectively equivalent, or sufficiently distinct to treat them as two separate DOFs, one 70 needs to relate how similar or dissimilar the patterns of AOD are to how similar or dissimilar the resulting climate responses are. Identifying the number of DOF only needs to consider injections that produce meaningfully different climates; herein we define "meaningfully different" based on the ability to detect the difference in climate after 20 years, given natural variability -and this threshold clearly depends on the choice of climate variables to be considered and the amount of cooling desired.
For example, injecting aerosols at 30 • N only in the summer or only in the fall would yield different patterns of AOD. The 75 difference in resulting climate could be distinguishable against the background climate variability if the desired amount of cooling is high. However, by reducing the level of cooling, the difference would become indistinguishable.
The next section describes the climate model and simulations used. Section 3 assesses the differences in spatiotemporal AOD patterns from 29 different injection choices, sampling different latitudes and seasons of injection, and quantifies the size of design spaces with different numbers of DOF using a metric based on the angle between different patterns of AOD. Section 80 4 identifies a relationship between how similar or dissimilar AOD patterns are and how similar or dissimilar the corresponding climate responses are, using existing simulations that were conducted with various different choices for climate goals and/or different DOFs. Section 5 then quantifies how large the difference in climates needs to be in order to be meaningfully different at different levels of cooling. Finally, we combine these pieces of analysis in Section 6 to show that for a cooling level of 1-1.5 • C, for example, there are between 6 and 8 DOFs. System model which includes atmosphere, ocean, land, and sea ice components . The model has a horizontal resolution of 0.95 • latitude by 1.25 • longitude, with 70 vertical levels that extend from the earth surface to 140 km in altitude, 90 and stratospheric aerosols have been shown to reasonably match observations after the Mt. Pinatubo eruption . With the exception of a few cases noted below, we use existing output from previous simulations for our analysis.
To assess the range of possible spatiotemporal patterns of AOD arising from different injection choices, we sample 29 possible choices in the AOD design space, including injections at low and middle latitudes as described by Visioni et al. (2019), and high latitude injections as described by Lee et al. (2021); the set of 29 possible choices is illustrated in Fig. 1. Visioni et al. 95 (2019) include injections at 5 different latitudes: 30 • N, 15 • N, equator, 15 • S, and 30 • S, as in Tilmes et al. (2017). For each latitude, injections are simulated both annually-constant, and restricted to each season: December-January-February (DJF), March-April-May (MAM), June-July-August (JJA), and September-October-November (SON). In each simulation, 6 Tg yr −1 of SO 2 are injected into the lower stratosphere, approximately 6-7 km above the tropopause at 180 • E (about 25 km for equator and 15 • N/S, 23 km for 30 • N/S, 16 km for 45 • N/S, and 15 km for 60 • N/S). Simulations were conducted for 5 100 years (2040-2044), which is sufficient for estimating the steady-state AOD pattern (Visioni et al., 2019), though of course not for estimating the climate response to this forcing. The high latitude injections included here are not exactly the same as those described in Lee et al. (2021), which have a higher injection rate of 12 Tg yr −1 and only consider injection at 60 • N and further poleward. To be consistent with simulations performed by Visioni et al. (2019), we conducted additional simulations of spring (MAM or SON) injection at 45 • N, 45 • S, 60 • N, and 60 • S for 5 years from 2040 to 2044 with an injection rate 105 of 6 Tg yr −1 , to complete the sample set. Following Lee et al. (2021), the other seasons of injection at high latitude are not expected to be particularly effective. Figure 2 shows the spatiotemporal patterns of AOD in each of these four spring injections at high latitudes (see Visioni et al., 2019 for the remaining cases). This gives us a total of 29 different injection cases, and associated spatiotemporal patterns of AOD. In Section 3, we rank the 29 injection cases based on the uniqueness of their AOD patterns, which are then used to identify the number of meaningfully-independent injection choices. In Section 7, we show that 110 the uniqueness of stratospheric AOD patterns does not depend on the altitude of injection, using simulation data from Tilmes et al. (2017) that include SO 2 injection either 1 km above the tropopause or 6-7 km above the tropopause.
In addition to these shorter simulations that we use to assess the range of possible spatiotemporal AOD patterns from different injection choices, five sets of solar geoengineering simulations from existing studies (Tilmes et al., 2018;Kravitz et al., 2019;Visioni et al., 2020c;Lee et al., 2020a) are used to analyze the connection between the patterns of stratospheric 115 AOD and surface climate responses (see Table 1). These 5 sets were all performed in CESM1(WACCM) with RCP8.5 as the background warming scenario, and used a feedback algorithm  to adjust SO 2 injection rates to maintain one or more climate objectives. Each simulation takes the 20-year average of annual mean temperature from 2010-2029 in the Representative Concentration Pathway 8.5 (RCP 8.5) emissions scenario as the target value for T0. Maintaining T0 constant at 2010-2029 average results in 4 • C of global mean cooling in each of these simulations by 2070-2089. The equatorial case adjusts 120 the single SO 2 injection rate to meet T0. The other simulations adjust SO 2 injection rates at multiple latitudes to simultaneously meet T0 and additional climate objectives (see Table 1); these additional objectives include T1, T2, ITCZ (using the centroid of precipitation between 20 • S and 20 • N as a proxy), and SSI. Table 1 lists the injection seasons and latitudes, the number of ensemble members, and the design objectives of the five sets of simulations.   In this section, we consider 29 different injection choices, sampling from different latitudes and seasons of injection, as well as 3 additional cases that we use to verify that the set of 29 is sufficiently complete. The AOD pattern from a given injection choice (a given latitude and season of injection) is largely determined by the stratospheric circulation and aerosol lifetime, which constrains what spatiotemporal patterns are achievable MacMartin et al., 2017). The AOD resulting from any particular choice of latitude and season can be approximated by a linear combination of other choices. Our goal in 130 this paper is to determine how many distinct injection choices are needed to adequately approximate all of the possible AOD patterns. What constitutes "adequacy" is determined in subsequent sections.
To describe any pattern of AOD, we consider the zonal-mean pattern as a function of both latitude and time of year. In order to treat these two dimensions consistently and yield AOD patterns independent of our sampling resolution in each dimension, we weight the monthly-mean zonal mean AOD at each latitude and month by the corresponding incoming solar 135 energy (petajoules) at the top of the atmosphere (TOA). We then represent the weighted spatiotemporal AOD pattern from each injection choice as a vector a, the length of which is equal to the number of latitudes times the number of months, = lat × month . One way of quantifying how similar or dissimilar two patterns of AOD are is to consider the angle between their vector representations, θ AOD . This implicitly assumes that the patterns of AOD are sufficiently linear for 1-4 • C of cooling, although nonlinearities will become increasingly important at higher forcing levels Visioni et al., dissimilar; the angle between the vector representations of these two AOD patterns is 84 • . By contrast, the AOD patterns of injection at 60 • N and 45 • N are similar; the angle between these two AOD patterns is only 12 • .
With the vector representation explained above, our goal is to select a subset from the set of 29 injection choices such that any possible AOD pattern can be adequately represented by a linear combination of injection choices from this subset.
Determining the dimension of that set necessary to meet this goal is equivalent to determining the number of DOF of SAI. Mathematically, the linear combination that is most similar to the simulated pattern of AOD is the projection of its vector 155 representation onto the space formed by the 29 injection choices. Solving the best approximation of the pattern of AOD can be formed as a constrained linear least-square problem of finding the projection onto the set of 29 injection choices: arg min where d is the vector representation of the AOD pattern of each verification case, which is obtained from CESM1(WACCM) simulation,d is the best approximation of d, Q 29 is the set of vector representations of the 29 injection choices, andx is the vector of best approximating linear coefficients. All linear coefficientsx i are constrained to be non-negative numbers, as injection rates cannot be negative.
By calculating the angle between the vector representation of the simulated AOD pattern and the vector representation of  (WACCM) simulations, the AOD pattern approximated using the set of 29 injection cases, and the difference between these. The angles between the simulated AOD pattern and the AOD pattern approximated using the set of 29 cases are 7.6 • , 5.9 • , and 6.1 • , respectively.
comparison, the approximated AOD patterns are adequately similar to the simulated AOD patterns, as long as the threshold for an "adequate" approximation of the AOD is larger than 7.6 • (which it will be here, as shown in subsequent sections). Allowing 175 this small difference between the approximated and simulated AOD patterns, the set of 29 injection choices thus adequately describes other possible AOD patterns that we have not simulated. In choosing subsets, we enforce hemispheric symmetry such that if an injection choice in one hemisphere is included, then the corresponding choice on the opposite hemisphere is also included (e.g., MAM in the Northern Hemisphere and SON in 190 the Southern Hemisphere). While the seasonal circulation patterns are not exactly symmetric between the hemispheres, they are sufficiently similar that this is a reasonable simplification that reduces the number of sets to search over. For injections at the equator, we similarly either include or don't include opposite seasons (e.g., DJF and JJA, or MAM and SON). With hemispheric symmetry, the only way to have a set with an odd number of DOF is to include annually-constant equatorial injection; we revisit this case in the discussion.

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Mathematically, the steps above can be described as follows. First, for each subset Q j of n injection choices, we identify the maximum angle that can be formed between that subset and any other injection choices in the set of 29 that are not selected: where k = 29 − n, the total number of injection choices that are not selected by the set Q j .
Taking n = 4 as an example, we can identify all possible combinations of 4 injection choices from the set of 29. With Among all possible subsets of n injection choices (Q j ∈ Q), we identify which subset has the smallest maximum angle and denote the "best" subset that minimizes θ max as Q * (n): Q * (n) is the subset of size n that best approximates any achievable pattern of AOD. The angle θ max of the "best" set of n is denoted as θ * (n). For each possible value of n, we find the "best" set and corresponding angle θ * (n), plotted in Fig. 6. Strictly speaking, because we only sampled 29 possible injection choices (out of an infinite theoretical space), θ * (n) is simply our best estimate 215 for the maximum angle between a subspace of n DOFs and any possible AOD pattern of injection choices that does not fall into this subspace.
4 Comparing AOD and Surface Climate Figure 6 shows that there are diminishing marginal returns for how many degrees of freedom are included. However, the analysis in Section 3 does not indicate what degree of approximation is sufficient. The next step in our analysis is to evaluate 220 the relationship between how similar or dissimilar two AOD patterns are, and how similar or dissimilar the corresponding surface climate responses are. This relationship is crucial for determining a threshold for whether two patterns of AOD from To estimate this relationship, we consider the different strategies described in  variables are similar, then changes in other surface climate variables, such as precipitation minus evaporation (P-E), will also 235 be sufficiently similar, and also ignores shifts in the seasonal cycle (Jiang et al., 2019) as these tend to be smaller than the annual-mean changes.
To estimate how large a change in the spatiotemporal pattern of AOD is needed to obtain a detectably different pattern of surface climate response, we consider detectability over a 20-year period. Therefore, we normalize the surface temperature and precipitation changes by the variability in 20-year averages, calculated from the across-ensemble variability from 2010-2029 in 240 RCP8.5 emissions scenario, where 21 ensemble members are available. If the variability were uncorrelated from year to year, this value would simply be a factor of √ 20 smaller than the interannual variability; this would be a reasonable approximation for precipitation but not for temperature. Normalizing by variability also allows temperature and precipitation changes to be compared in consistent units (Ricke et al., 2010).
To analyze the differences in AOD and surface climate for different strategies, we define the AOD space, temperature space 245 and precipitation space. In Section 3, we define the vector representation of AOD patterns as a. a represents an achievable spatiotemporal AOD pattern arising from a possible injection choice. The AOD space A is a -dimensional space, A ⊂ R , that includes all possible values of a. Similarly, the temperature space T and the precipitation space P are both m-dimensional spaces, T ⊂ R m and P ⊂ R m , where m is equal to the number of latitudes times the number of longitudes, m = m lat × m lon .
Any vector T in T represents a possible surface air temperature response to a possible injection choice, and any vector P in 250 P represents a possible precipitation response to a possible injection choice: where T 1 , . . . , T m are temperature responses, and P 1 , . . . , P m are precipitation responses, both in dimensionless units of standard deviations.

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In the AOD space, temperature space and precipitation space defined above, we evaluate the differences between each possible pair of the five SAI strategies described in Table 1, i.e., 10 pairwise comparisons. As in Section 3, the difference between AOD patterns for different pairs of strategies is evaluated by computing the angle between them, θ AOD . For temperature and precipitation, the difference between two strategies is evaluated by computing the temperature distance, d t , or the precipitation distance, d p . The temperature distance is defined as the area-weighted L 2 -norm (root-mean-square) of the difference between 260 the two vector representations of surface air temperature responses, to count all areas on the Earth equally. Similarly, the precipitation distance is defined as the area-weighted L 2 -norm (root-mean-square) of the difference between the two vector representations of precipitation responses. Among the 10 pairwise comparisons, GLENS and EQ have the largest temperature distance, though it's still an order of magnitude smaller than the temperature distance between GLENS and the projected 20- year average (2070-2089) climate response under RCP8.5 ( Fig. 7(a) and (b)). The precipitation distance between GLENS and 265 EQ is also smaller than that between GLENS and RCP8.5 (Fig. 7(c) and (d)). Among all 10 pairwise comparisons, the temperature distances are always larger than the corresponding precipitation distance. That is the changes in temperature, compared to variability, are larger than the changes in precipitation. To conclude, a small angle between the vector representations of AOD patterns indicates the two compared SAI strategies yield similar AOD patterns, and a small value of d t or d p indicates that the two compared SAI strategies have similar surface air temperature responses or precipitation responses. Likewise, a 270 larger AOD angle, temperature distance, or precipitation distance implies less similar AOD patterns, surface air temperature or precipitation.  To estimate the relationships between θ AOD and d t and between θ AOD and d p , we perform linear regressions on the data points obtained from the 10 pairwise comparisons among the 5 different SAI strategies in Table 1 (Fig. 8). We constrain the linear regressions to go through zero because an identical AOD pattern should yield an identical temperature and precipitation 275 response. The linear functions are obtained as: Figure 8. (a) Angle between AOD patterns, θAOD, and the distance between corresponding temperature responses, dt, for each pair of strategies in Table 1; (b) angle between AOD patterns, θAOD, and the distance between corresponding precipitation responses, dp, for each pair of strategies in Table 1. dt and dp are expressed in number of standard deviations (std) of 20-year averages of temperature and precipitation respectively. Orange dots represent the values of θAOD and corresponding dt and blue dots represent the values of θAOD and corresponding dp of all pairwise comparisons between different strategies. The black lines represent the best-fit linear regression functions, constrained to pass through the origin: θAOD =6.1 dt, and θAOD =9.9 dp with the coefficient of determination, R 2 t = 0.62 and R 2 p = 0.65, respectively. The error in estimating each point (calculated from across-ensemble variability) is small (less than 0.2 • in AOD angle, and less than 0.1 std in both temperature distance and precipitation distance) compared to the fitting error, indicating that a linear approximation to the relationship is only an approximation. Points in the upper-right (most dissimilar AOD and dissimilar surface climate) are the pairwise comparisons between the equatorial injection strategy and the other strategies. In (a), the outlier at approximately (1.6,19), which is the comparison between iSpring and iAutumn, shows that the relationship between θAOD and dt isn't exactly linear; similar AOD patterns yield similar climate responses but different AOD patterns do not guarantee different climate responses. θ AOD = 6.1d t (8) θ AOD = 9.9d p As shown in Fig. 8, pairs of strategies with relatively similar AOD patterns have relatively similar temperature and precipitation responses, and conversely, pairs of strategies with very different AOD patterns result in very different temperature and 280 precipitation responses.
Data used here are from SAI designs that were considered in previous studies. Although they are not designed to span either the overall AOD design space or the surface climate design space, these available simulations do provide a useful set of data for analyzing the relationship between how similar or dissimilar the AOD patterns are and how similar or dissimilar the surface climate responses are.
To evaluate how different the surface climate responses are, we first perform Welch's t-test on the five injection strategies, using a single ensemble member of each injection strategy. Welch's t-test assumes that sampled data are independent; we remove the effect of serial autocorrelation from the temperature and precipitation data by estimating the effective sample size assuming both temperature and precipitation follow a first order autoregressive (AR(1)) process (Wilks, 2019). The t-test results for 290 comparing differences in surface air temperature between GLENS and iSpring and between GLENS and EQ are shown in Fig. 9(a)-(b); the t-test results for precipitation comparison are shown in Fig. 10(a)-(b). At a 4°C cooling and a confidence level of 95%, temperature and precipitation responses from GLENS and iSpring are statistically significantly different from each other in 17% and 7% of Earth's area, respectively, as opposed to the comparison between GLENS and EQ, in which 37% and 23% of area on the Earth shows statistically significant difference in temperature and precipitation.

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Here, we define that two strategies are considered to be detectably different if the difference in temperature or precipitation responses between them are detectable at a 95% confidence level over a 20-year period on more than 5% of Earth's area.
With the temperature and precipitation normalized by the standard deviation of 20-year means, the difference between the temperature or precipitation responses at any grid point will be detectably different if the difference between the normalized data is more than 2. To obtain a global aggregate metric, we note that roughly 5% of the Earth's surface area has a temperature 300 difference more than double the overall temperature distance d t considered earlier and 5% of the Earth's surface area has a precipitation difference more than double the precipitation distance d p . (For example, between GLENS and iSpring, only 5.2% of Earth's area have a difference in regional temperature responses that is more than twice the value of d t , and only 4.7% of Earth's area have a difference in regional precipitation responses that is more than twice the value of d p .) Thus when the temperature distance d t between two injection strategies is one standard deviation, then roughly 5% of the Earth's surface area 305 will have detectably different temperature responses at a 95% confidence interval. Similarly, when the precipitation distance d p between two injection strategies is one standard deviation, then roughly 5% of the Earth's surface area will have detectably different precipitation responses at a 95% confidence interval. Thus, we use one standard deviation of the overall root mean square (RMS) normalized temperature distance or precipitation distance as the threshold for determining whether two strategies result in detectably different temperature or precipitation responses. 310 We compare the difference in temperature and precipitation responses between GLENS and iSpring and between GLENS and EQ, and show how the difference changes with levels of cooling using detectability plots, as shown in Fig. 9(c)-(d) and Fig. 10(c)-(d). Figure 9(c) and 10(c) show the detectability of difference in temperature and precipitation between GLENS and iSpring at a cooling level of 4 • C. Figure 9(d) and 10(d) show the comparison between GLENS and EQ at the same cooling level. In each plot, the length of the vector is equal to the area-weighted L 2 -norm of the corresponding temperature 315 or precipitation vector, and the distance between the two vectors is the corresponding temperature distance or precipitation distance. The circle around the tip of each vector represents the temperature or precipitation variability on a 20-year timescale, whose radius is equal to one due to the normalization by the standard deviation of temperature or precipitation variability. In Fig. 9(c), the two temperature variability circles are partially overlapped, while in Fig. 9(d), the two circles are separated from measures the temperature distance, dt, between GLENS and iSpring and between GLENS and EQ, respectively. GLENS and EQ have a larger distance in the temperature response, which indicates the difference between GLENS and EQ is more detectable at the same level of cooling.
each other. The more overlapped these two circles have, the less area has detectably different temperature responses. When 320 the overlapped area is sufficiently large, it indicates that the difference in the temperature responses is small enough such that it is hard to tell whether the difference could be purely due to natural variability. These implications from the observation of temperature responses also apply to precipitation.
From the detectability plots that compare these different SAI strategies, it is clear that the detectability of different injection strategies depends on both the level of cooling and the choice of climate variables. With the underlying assumption of linearity 325 Figure 10. Left column shows the comparison between GLENS and iSpring; the right column shows the comparison between GLENS and EQ. (a) and (b) show the difference in precipitation responses; shaded areas are where no statistically significant difference is observed, based on a t-test with a confidence level of 95%. (c) and (d) are detectability plots that geometrically show the ability of detecting differences in the precipitation responses between GLENS (black) and iSpring (blue) and between GLENS and EQ (red), respectively. The precipitation responses are expressed in number of standard deviations (std) of 20-year averages of precipitation. The length of each vector represents the area-weighted L 2 -norm of precipitation response, and the circle represents precipitation variability in all possible directions, with a radius of 1 std due to the normalization. The dashed line in (c) and (d) measures the precipitation distance, dp, between GLENS and iSpring and between GLENS and EQ, respectively. Similar to temperature response, the difference in precipitation between GLENS and EQ is more detectable than that between GLENS and iSpring at the same level of cooling.
for surface climate responses, we estimate the difference in temperature and precipitation responses between GLENS and EQ at reduced levels of cooling (Fig. 11). For the same pair of strategies, as we reduce the amount of cooling, the temperature distance and the precipitation distance between these two strategies decreases. At 1.8 • C cooling, the resulting temperature responses of these two strategies are 2 standard deviations of temperature variability away from each other ( Fig. 11(b)); at the same level of cooling, the resulting precipitation responses are 1 standard deviation apart (Fig. 11(e)). At 0.9 • C cooling, 330 the resulting temperature responses are exactly 1 standard deviation apart (Fig. 11(c)), and the precipitation responses are 0.5 standard deviation apart (Fig. 11(f)). For any cooling level lower than 0.9 • C, temperature responses of GLENS and EQ Figure 11. Top panels are detectability plots for the difference in temperature responses between GLENS (black) and EQ (red) at three different levels of cooling: (a) 4 • C, (b) 1.8 • C, and (c) 0.9 • C. Bottom panels are detectability plots for the difference in precipitation responses between GLENS (black) and EQ (red) at those three levels of cooling. Dots represent the estimated mean temperature or precipitation responses for the corresponding levels of cooling. Temperature and precipitation responses are expressed in number of standard deviations (std) of 20-year averages of temperature and precipitation respectively. As the cooling level decreases, less area on the Earth has detectably different climate responses and the ability to detect the difference between GLENS and EQ decreases.
will not be detectably different by our metric; that is to say, less than 5% of area on the Earth is expected to have detectably different temperature responses at a 95% confidence level over a 20-year period. On the other hand, to have detectably different precipitation responses between GLENS and EQ, the cooling level needs to be higher than 1.8 • C. The cooling levels of 0.9 • C 335 and 1.8 • C are defined as the cut-off cooling levels for detectable difference in temperature and precipitation, respectively, between GLENS and EQ. Note that for sufficiently small amounts of cooling, the resulting climate from these strategies will also be undetectably different from the climate with neither increased greenhouse gases nor SAI; from Fig. 7, they will be detectably different from the climate with increased GHG except at very small levels of cooling.
inversely proportional to d p : By substituting eq. (10) into eq. (8) and substituting eq. (11) into eq. (9), we obtain two functions that can be used to estimate a threshold value of AOD angle, θ a , which is used to assess the detectability of different injection choices at different levels of 345 cooling: Thus, given a particular level of cooling, we can calculate if two injection strategies are expected to result in detectably different temperature or precipitation responses by comparing their AOD patterns. If the angle between the patterns of AOD 350 is smaller than θ t a or θ p a , these two strategies can be expected to not result in detectably different temperature or precipitation responses. However, if the angle between the patterns of AOD is larger than θ t a or θ p a , these two strategies can be expected to be meaningfully independent in terms of temperature responses or precipitation responses. In Fig. 12, we compare the cutoff AOD angles predicted by eq. (12) and eq. (13). As shown in Fig. 12, the threshold values of AOD angle predicted using temperature responses are always lower than those predicted using precipitation responses.

Estimating the Number of DOF
In the previous sections, we estimate the relationship between the number of DOFs included and the maximum error in approximating AOD, and the relationship between the AOD angles and the level of cooling at which the resulting temperature or precipitation response could be expected to be detectably different. In this section, we combine these two to estimate how many meaningfully-independent DOF there are as a function of the levels of cooling.

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As changes in temperature are more detectable than those in precipitation, the extent to which two AOD patterns are sufficiently similar, and thus the number of DOF in the SAI design space, are determined by the temperature response. Using eq. (12), we calculate the cut-off AOD angle θ t a (listed in Table 2) for cooling levels of 0.5 • C, 1 • C, 1.5 • C, and 2 • C. It is expected that two SAI strategies with AOD differing by any angle smaller than θ t a , will not result in detectably different temperature or precipitation responses. Figure 13 shows the four cut-off AOD angles and the corresponding minimum numbers of 365 DOF required for θ * (n) not exceeding the cut-off value, θ t a (also listed in Table 2).
Any set of injection choices that has a θ max smaller than the cut-off angle θ t a (∆T ) will form a design space that captures all detectably different climate responses. That is, for any possible injection strategy not included in the design space, you can find Figure 12. The cut-off AOD angle θa at different levels of cooling ∆T estimated using temperature responses and precipitation responses. Table 2. The minimum number of DOF, n, of the SAI design space for four targeted levels of cooling. At each level of cooling, θ t a is the maximum angle that can be formed between two AOD patterns that yield undetectably different temperature responses. θ * (n) is the maximum angle that can be formed between any AOD pattern and the design space of n DOF, and must be smaller than θ t a . an injection strategy in the design space such that the angle between their AOD patterns is smaller than θ t a and the difference 370 in the corresponding climate responses is sufficiently small such that they are not meaningfully different.
As shown in Table 2, as the targeted level of cooling increases, the cut-off value of θ t a decreases and the minimum number of DOF increases. For a targeted cooling level of 1 • C, we likely need 6 DOFs. For any possible AOD pattern, the angle between it and the AOD pattern approximated by a design space of 6 DOFs is likely to be smaller than 24 • , and the resulting difference in climate response will be largely undetectable. For a targeted cooling level of 1.5 • C, we likely need 8 DOFs. This finding 375 Figure 13. The minimum number of DOF corresponding to the worst-case error in approximating AOD, θ t a .
significantly reduces the dimension of the design space needed for evaluating the possible climate impacts of different SAI strategies and associated trade-offs. as well as spring injection at 60 • N (as in Lee et al. (2021)) and 60 • S is only slightly worse than this optimal set, within a range of 0.1 • , still sufficient to span the design space for 1 • C cooling.
For a cooling level of 1.5 • C, the best set of 8 is similar to the best set of 6 but with equatorial injections at the other two seasons instead and the addition of summer injections at 15 • N and 15 • S. Note that in the optimization earlier, we constrained 385 our search to hemispherically symmetric pairs of injection choices. Including only annually-constant injection at the equator rather than two seasons yields a set of 7 injection choices that performs almost as well as the optimal set of 8 and is still sufficient for 1.5 • C cooling.

Analysis of injections at different altitudes
The SAI simulations analyzed in the previous sections are all high-altitude injections (6-7 km above the tropopause). Tilmes  while inefficient, would likely result in more significant differences in AOD patterns (Bernstein et al., 2013). Previous studies have shown that different choices of stratospheric aerosol injection latitudes and seasons lead to different surface climate responses. Choosing where and when to inject aerosols to the stratosphere to meet different climate goals can be considered as a design problem. Previous studies have concluded that there are at least 3 degrees of freedom (DOF), that is, at least three independent climate goals can be simultaneously met. These three -basically the global mean aerosol burden, the 405 interhemispheric difference, and the equator-to-pole difference -were motivated by physical intuition regarding stratospheric transport, which will ultimately constrain how many independent DOFs are achievable through different injection choices. A key observation is that the number of DOF effectively depends on the amount of global cooling provided by SAI, because for a small amount of cooling, the difference in the climate response for different strategies may not be detectable. As the amount of cooling increases, the number of meaningfully-independent DOF increases. For a cooling level of 1-1.5 • C, there are likely 410 between 6 and 8 meaningfully-independent DOFs. If only precipitation changes mattered, and not temperature changes, then there would be fewer meaningfully-independent DOFs.
Our estimation of the number of DOF provides useful guidance to bound the number of injection choices that need to be considered when evaluating the range of possible different SAI strategies and the trade-offs among them. If only a small amount of cooling is needed from implementing stratospheric aerosol injections, a small set of selected injection choices 415 would be sufficient to capture the range of possible resulting climate responses and evaluate how different those climate responses could be. As all possible injection choices form an extremely high dimensional design space, only considering the meaningfully-independent injection choices significantly reduces the dimension of the design space.
The number of meaningfully-independent DOF determines the number of independent climate metrics that SAI can manage simultaneously. Thus, for a cooling level of 1-1.5 • C, for example, SAI can manage 6-8 independent climate metrics at the 420 same time. This expands the manageable number of climate metrics relative to what has been considered in previous studies, opening up new opportunities for exploring alternate designs that will have different distributions of impacts.
It is important to note that all of these results are obtained from a single climate model. Other climate models may produce different numerical results. Nonetheless, the number of independent DOF needed to span the range of possible different stratospheric AOD patterns can be reasonably expected to remain consistent as the transport of aerosols are constrained by 425 stratospheric circulation. We make several simplifying approximations in order to make analysis tractable, particularly to estimate the relationship between how similar or dissimilar two patterns of AOD are and how similar or dissimilar the corresponding surface climate is; future research could explore the impact of these approximations. First, we only consider changes in temperature and precipitation, and we only consider changes in annual mean rather than shifts in seasonality, which could matter at high latitudes in particular (Jiang et al., 2019). Second, we only consider whether the difference in climate response 430 would be detectable over a 20-year period. Third, the globally-aggregated metric we use for determining whether two different climate responses are "detectable" is based on whether they are detectably-different at a 95% confidence level over 5% of the surface area. However, changes that are less-confidently detected may still matter, and since social, agricultural and other economic activities are strongly affected by regional climate changes, just because they only happen in a small percentage matter.
A key outcome of this study is that further research should be conducted to explore alternate SAI designs that can manage more than 3 and up to 8 independent climate metrics simultaneously, and to compare the resulting climate responses and associated trade-offs. Research into more than 8 is less policy-relevant, simply because any hypothetical deployment scenario would not reach more than 1.5 • C cooling for many decades, if ever. Ultimately, to evaluate the impacts of stratospheric aerosol 440 geoengineering, regional surface climate needs to be considered, as social and economic activities are significantly affected by regional climate change. Alternate SAI designs may enable a better compensation of the impacts from climate change; alternate designs might also lead to the potential to create more novel climates that optimize some metrics at the expense of others -both of these possibilities are important to understand in order to inform not only future scientific research in SAI but also governance challenges. 2020c) are available at https://doi.org/10.7298/c92j-2p46 , and data for PREC (from Lee et al., 2020a) are available at