Understanding the climate impacts of solar geoengineering is essential for evaluating its benefits and risks. Most previous simulations have prescribed a particular strategy and evaluated its modeled effects. Here we turn this approach around by first choosing example climate objectives and then designing a strategy to meet those objectives in climate models.

There are four essential criteria for designing a strategy: (i) an explicit specification of the objectives, (ii) defining what climate forcing agents to modify so the objectives are met, (iii) a method for managing uncertainties, and (iv) independent verification of the strategy in an evaluation model.

We demonstrate this design perspective through two multi-objective examples.
First, changes in Arctic temperature and the position of tropical
precipitation due to CO

Geoengineering describes a set of technologies designed to offset some of the
effects of anthropogenic climate change by deliberately intervening in the
climate system. There are many proposed methods of

Many of the ongoing efforts in solar geoengineering research involve climate
model simulations designed to ascertain the expected climate effects of
various scenarios of geoengineering

As an example, one of the results from geoengineering that is repeatedly
discussed is that offsetting the global mean radiative forcing from
a CO

Figure

Climate model simulations suggest that many of the proposed methods of
conducting solar geoengineering are likely to have both commonalities and
differences in their climate effects

These three panels show that the “canonical” temperature response
to offsetting global mean temperature increases from CO

Our primary motivation in this study is to introduce a design perspective that can be used to more systematically evaluate some of the potentials and limitations of geoengineering. We do this by exploring two examples of geoengineering strategies designed to meet specific, multifaceted goals. For any strategy, achieving multifaceted goals can be accomplished via following a certain set of criteria:

an explicit definition of specific objectives of geoengineering;

determination of the particular degrees of freedom to be modified to meet the objectives;

a strategy for meeting the objectives in the presence of uncertainty;

verification of the designed strategy in a different evaluation model.

Implicitly included in these four criteria is that it is necessary to determine the feasibility of the objectives. It may not be possible to achieve all objectives due to physical constraints on the climate system. Moreover, the space of possible climates may be further narrowed by technological limitations. As an example, it is not clear how stratospheric transport can be controlled, which may limit the spatial distribution of radiative forcing that is achievable via geoengineering with stratospheric sulfate aerosols. Our analyses inherently include the assumption that the radiative forcing is achievable.

In a system in which the relationships between adjustable climate parameters
and the desired pattern of radiative forcing are well characterized, one
could optimize the relative contributions of the parameters such that the
desired climate objectives are approximately met. This was the approach taken
by

One method of managing uncertainties is to use

Although these past studies were instrumental in developing applications of explicit feedback for geoengineering, their applicability is limited in that they do not address the potential for multifaceted geoengineering goals. Offsetting multiple independent features of climate change requires modifying multiple simultaneous degrees of freedom. Ensuring that those climate objectives are met in the presence of uncertainty requires explicit feedback. The present study is the first to combine these two aspects, illustrating some of the potentials and limitations associated with designing geoengineering strategies.

Addressing criterion 4 requires a two-stage process, as was illustrated by

We illustrate the design approach through two examples: one
regionally focused and the other globally focused, described in
Sect.

Here we illustrate the nature of geoengineering as a design problem through
two examples, which we will call 2

The 2

The degrees of freedom that were modified in the two cases
considered here, referred to as 2

More concretely, the two inputs in the 2

The 3

As metrics for these, we define

All of these simulations are conducted using the method of explicit feedback,
as described by

While the previous section introduced the idea of choosing multiple spatial
degrees of freedom to balance multiple criteria, this section is concerned
with how to choose the amplitude of each of these degrees of freedom

Feedback design (design of the explicit feedback algorithm) requires some information about the system response to an input. This information is provided by the design model, and feedback is then used to bridge the gap between the modeled response and the real-world response if this design were implemented.

Specifying exactly what information is needed to design the feedback
algorithms is not immediately obvious. We begin this section with
a discussion of dynamic modeling for feedback design
(Sect.

The feedback algorithm defines the rule by which the “input” (e.g., solar
reduction) is adjusted in response to observations of the “output” (e.g.,
difference between measured and desired global mean temperature). The design
of this algorithm starts with a dynamic model of the input–output behavior of
the system. This dynamic model does not describe how the entire climate state
responds to a perturbation in the input signal but specifically the response
of the output signal. We use the term

A general linear dynamic input–output relationship can be described by
a convolution equation in the time domain. However, many of the expressions
we wish to evaluate are greatly simplified when expressed in the frequency
domain, because convolution is replaced by multiplication, and coupled
differential equations in the time domain become algebraic relationships in
the frequency domain. A time-domain equation

In illustrating feedback design guidelines in the next subsection, it is
convenient to consider a first-order linear (i.e., first-order
autoregressive) description of the input–output relationship, including
a time delay

Taking the Laplace transform of Eq. (

At any frequency

Bode plot showing the frequency response of the transfer function

A semi-infinite diffusion model has been shown by

Bode plot (as in Fig.

We now consider the design of the feedback algorithm, using the model in
Eq. (

The full system is now described in the frequency domain by

There are three critical observations. (1) At very low frequencies
(

The frequency where the magnitude

With proportional–integral control, the control gains

We now outline a process for determining choices for

As in Fig.

First consider a pure integral control (

Adding the proportional gain

We now provide a more detailed recipe for determining control gains for
a particular application. Let

We choose

Note that Eqs. (

However, the model form does influence characteristics such as amplification
of natural variability at frequencies away from

Magnitude of the sensitivity function

Trade-offs between convergence timescale and amplification of natural
variability are choices in designing a feedback algorithm. Higher bandwidth
leads to faster convergence and tighter management of the specified climate
objectives. However, at higher frequencies, the system response has greater
phase lag (see Fig.

Time-domain response of the first-order linear (ARX(1);
Eq.

The discussion thus far has focused on a single-input, single-output (SISO) feedback algorithm case. However, both of our design examples
(Sect.

If

As an illustrative example of how to design a multivariate feedback
algorithm, we consider the following 3

As was mentioned in Sect.

A step input perturbation is quite common in climate science, e.g., the
abrupt4xCO2 simulation in CMIP5

An alternative is to use single-frequency sinusoidal input signals, as was
done by

Another alternative is to input a band-limited signal, which is useful for characterizing system behavior over a small range of frequencies; this can be helpful if the different input–output relationships have different timescales of response. This method has an advantage over step-response simulations, in that the input signal is not heavily weighted toward some frequencies at the expense of others. This has a disadvantage as compared to sinusoidal inputs, in that the input signal is more distributed, resulting in lower signal-to-noise ratios. If the loop crossover frequency falls within the quasi-static response of the system, then a sinusoidal input and a band-limited input will yield similar information.

Step responses for the 2

Natural climate variability limits the accuracy of estimating the transfer
function in simulations. Errors can be estimated from the frequency-dependent
signal-to-noise ratio (SNR) denoted

Our characterization of the 2

For the step response simulations, beginning from a stable preindustrial
control run, insolation over the Arctic or Antarctic was abruptly reduced by 2,
4, 8, and 12 %; the results from these simulations are summarized in
Fig.

Linear regression over the precipitation centroid (

These simulations can already inform the influence matrix for this particular
case. Reductions in Arctic insolation reduce Arctic temperature and shift
tropical precipitation southward. Reductions in Antarctic insolation shift
tropical precipitation northward but do not discernibly affect Arctic
temperatures. Therefore, using notation to suppress any potential time or
frequency dependence, we can write the influence matrix as

The step-response results can be fit to the functional form in
Eq. (

To circumvent these shortcomings, it is useful to complement the step
response information with sinusoidal input signals. Beginning from
a preindustrial control run, insolation over the Arctic or Antarctic was
reduced according to the function

Figure

From visual inspection of Figs.

Results from the sinusoidal perturbations in the 2

Results for the sinusoidal perturbations in the 3

Based on calculations of SNR (Eq.

From Eq. (

Following the procedure described in the previous sections, we first choose
SISO feedback gains to adjust high-latitude Northern Hemisphere forcing in
response to deviation in Arctic temperature from the desired value. Choosing

As described in Sect.

Thus, in summary, we have

We now consider system identification and feedback design for the three-input,
three-output design example described in Sect.

Similarly to Eq. (

We characterize the system response solely through sinusoidal input signals,
as shown in Fig.

The best estimate of the magnitude and phase of the input–output response at

The phase estimates include a half year of time delay due to annual
averaging. Climate variability clearly introduces uncertainty in these
estimates, particularly for the small elements
(Fig.

Performing the same error calculations as in Sect.

The errors in magnitude (1

We first choose feedback gains to adjust globally uniform solar reduction to
maintain global-mean temperature, corresponding to the (1, 1) entry of the
system dynamics matrix. Again, note that there is a 1-year time delay
introduced by averaging over the previous year before making a decision and
holding that decision fixed for an entire year; at a frequency of
0.2 rad yr

If the system were diagonal, the additional degrees of freedom could be
similarly adjusted with just a rescaling of both

Results for the 2

We now proceed with an evaluation of the effectiveness of our designed feedback algorithms.

Figure

Because of the net northward shift with only Arctic insolation reductions,
bringing the precipitation centroid southward actually requires an increase
in Antarctic insolation in this model. (The feedback algorithm was not given
any information regarding feasibility of the applied radiative forcing; there
is no known method of modifying shortwave radiation between 60
and 90

Root-mean-square (RMS) differences in Arctic
temperature (

Figure

Maps of temperature change (left column panels;

Climatology of percent change (with respect to piControl) in total
precipitation (shading; %) and shift in the precipitation centroid (

Same as Fig.

Figure

Given sufficient simulation time, the results above with the design model
could have been achieved without feedback simply by estimating the model
sensitivities to forcing from CO

The required reduction in Arctic insolation to achieve the Arctic temperature
goal is approximately 7 %, or about half of the required value for CESM.
Unlike the design model, achieving the goal for

Same as Fig.

Like the results for CESM (Fig.

Figure

Overall, we have demonstrated the ability to successfully design a 2

Figure

Results for the 3

Root-mean-square (RMS) differences in

Reductions in

In Fig.

Figure

Percent reduction in insolation for the 3

Maps of temperature change from the preindustrial control simulation
(

Same as Fig.

Figure

Early in the 2

For the full 3

Because the objectives of the 3

Residual changes in global mean precipitation (

All of the GISS results (Fig.

Same as Fig.

Same as Fig.

Same as Fig.

Geoengineering is not a binary decision of “on” or “off”. Rather, if it is ever deployed, multiple separate degrees of freedom could be adjusted to simultaneously meet multiple objectives. Climate models can be used to predict the response of multiple “output” variables in response to multiple “input” variables, but the actual climate response will not be identical. For this reason, the radiative forcing introduced by geoengineering would need to be adjusted in response to the observed climate outcomes; this feedback process compensates for uncertainty between models and reality. Here we have demonstrated this design process, and in particular the ability to simultaneously adjust multiple patterns of radiative forcing in response to multiple observed climate variables. Using a two-model approach with separate design and evaluation models is essential for demonstrating that the feedback process results in a strategy that is not overly dependent on the specific details of an individual model but is instead robust across models.

We reiterate two key points. First, attempts to generically characterize the
climate effects of solar geoengineering are ill-posed, because these effects
depend both upon the specific technology used and the objectives. There is
a broad range of potentially achievable climates, each with its associated
impacts on society (such as effects on water scarcity or agriculture).
Second, by demonstrating a multivariable feedback strategy to adjust multiple
distinct spatial patterns of radiative forcing, and demonstrating that
a strategy designed in one model can meet defined objectives in a separate
evaluation model, this work reinforces previous research, suggesting that an
accurate climate model is not necessarily required to implement solar
geoengineering

As we stated in Sect.

There are two obvious directions for future research.

First, what are the limits to such a strategy? We have intentionally chosen a small number of objectives and chosen corresponding input variables where the physical relationship between inputs and outputs is well understood, so the input–output response is likely to be similar between different models, as well as between models and reality. Increasing the number of adjusted patterns of radiative forcing and the number of different climate objectives is likely at some point to be limited by uncertainty. Put more bluntly, one cannot necessarily control 100 different climate fields in 100 regions just because a model says it is possible. While feedback provides robustness, some knowledge is required about the input–output dynamics; if not even the sign of the relationship is known, for example, then it is challenging to design an algorithm that converges. This is where the role of clear physical mechanisms becomes crucial: in the absence of mechanisms, it is not known whether any discovered input–output relationships are robust on the timescales of interest, or if a mechanism is known to have highly nonlinear behavior, linear feedback may not be effective even with large expenditure of effort on feedback design. Furthermore, even if some complicated strategy converged to a slightly better solution than a simpler one, natural variability may limit the ability to detect that difference on societally relevant timescales, let alone attribute those changes to geoengineering. Open questions that require further research include understanding the boundaries of what is achievable and what robust conclusions can be obtained about any particular strategy.

Second, we have demonstrated the ability to simultaneously manage multiple climate criteria using the common approach of changing solar irradiance, here as a function of latitude. Accomplishing the objectives with physically achievable mechanisms, such as with stratospheric aerosols or marine cloud brightening, introduces additional complications, even beyond the example shown in Fig. 12, where meeting the objectives required an increase in Antarctic insolation (i.e., it may not be possible to achieve all objectives due to physical constraints). For example, in the case of stratospheric aerosols, one could choose both the latitude and altitude of injection. However, (i) this does not give arbitrary ability to influence the resulting latitudinal dependence of aerosol optical depth or radiative forcing, (ii) the resulting radiative forcing patterns cannot be adjusted instantaneously, and (iii) the relationship between injection parameters and spatial patterns of radiative forcing introduces additional uncertainty, in no small part due to model-dependent results and insufficient validation of models as compared to reality. Using our methodology with stratospheric aerosols requires two distinct steps. One is to characterize the relationship between injection parameters (e.g., altitude, latitude, season) and distributions of aerosol radiative forcing. The second is determining the relationship between that radiative forcing and climate effects. Each of these steps has substantial uncertainties, and overcoming these uncertainties to meet climate objectives by using stratospheric aerosols (again, assuming those objectives are even achievable, independent of the ability of feedback to meet those objectives) would require a separate feedback process for each step. Marine cloud brightening would introduce further challenges and opportunities from the spatial heterogeneity of radiative forcing in both latitude and longitude, as well as the potentially rapid temporal response. This is intimately tied to the abovementioned area of research: feedback is essential for managing some of these uncertainties, but there are limits to what feedback can achieve.

We thank the reviewers and the editor for thorough comments that greatly improved the manuscript. The Pacific Northwest National Laboratory is operated for the US Department of Energy by Battelle Memorial Institute under contract DE-AC05-76RL01830. CESM simulations were performed using PNNL institutional computing resources. GISS ModelE2 simulations were supported by the NASA High-End Computing (HEC) Program through the NASA Center for Climate Simulation (NCCS) at Goddard Space Flight Center. Edited by: H. Held