1 Introduction

Forecasting and incorporating future climate conditions is a key to improve community resilience and promote engineering adaptation against climate change (ASCE-CACC 2018; Moss et al. 2019; Wright et al. 2019). Future climate information can be obtained from global climate models (GCMs; USGCRP (2017)), although their projections exhibit large uncertainty (Knutti and Sedláček 2013; Nissan et al. 2019) and it is challenging to incorporate them in engineering practice (Douglas et al. 2017). Substantial progresses and efforts have been made to improve and employ climate models (Eyring et al. 2016), including the recently released results from the Coupled Model Intercomparison Project phase 6 (CMIP6; Eyring et al. (2016)) with Shared Socioeconomic Pathways (SSPs; O’Neill et al. (2017)). However, practical applications of climate model projections (especially in engineering) are generally challenging (Douglas et al. 2017; Moss et al. 2019; Wright et al. 2021), and are subject to limitations related to temporal and spatial precision (Nissan et al. 2019) and large uncertainty (Steinschneider et al. 2015a; Cook et al. 2020; Lopez-Cantu et al. 2020; Helmrich and Chester 2022; Lai et al. 2022).

Flexible engineering adaptation strategies have been proposed and assessed in recent studies (Pozzi et al. 2017; Hui et al. 2018; Fletcher et al. 2019; Cohen and Herman 2021) to provide an alternative approach to traditional, rigid methods. Compared to traditional approaches of making fixed, long-term decisions on adaptation efforts (as many types of infrastructure have design life of over 100 years (ASCE-CACC 2015)), the alternative flexible strategies aim to introduce flexibility to engineering decision-making (Chester and Allenby 2019), e.g., by providing future expansion and modification options. These flexible strategies allow engineers to delay decisions or revisit adaptation sequentially, leverage the anticipatory reduction of climate uncertainty in the future, and increase benefit-to-cost ratios of infrastructure investments. Flexible strategies have been proposed and utilized in engineering with respect to other forms of uncertainty such as transportation demand (Fawcett et al. 2015) and economic development (Guma et al. 2009). With respect to climate uncertainty, flexible strategies have been studied in water resource management as optimal control problems (Herman et al. 2020) and in economic, adaptive decision-making such as using real-option analyses (Guthrie 2019; Ginbo et al. 2021; Kim et al. 2022). Similar concepts have been named as “adaptive” or “dynamic” (Hui et al. 2018; Herman et al. 2020), here we refer to this strategy as “flexible” adaptation (compared to traditional, rigid approaches).

Probabilistic estimation of climate variables – quantifying the value of information (Memarzadeh and Pozzi 2016) from more observations – are needed to facilitate flexible adaptation. Described as “learning scenarios” in Völz and Hinkel (2023), such probabilistic estimations – compared to the existing, “static” projections such as the ones provided in the Intergovernmental Panel on Climate Change (IPCC) Assessment Reports (ARs) – can be made at future moments in time (e.g., in 2050) with additional observations to assess the corresponding reduction of uncertainty. While other factors such as scientific advances can also contribute to reduction of uncertainty, this work focuses on quantifying the reduction of uncertainty from observing climate change.

The main objective of this work is to develop a probabilistic modeling framework – based on climate science and reasonably simplified to improve efficiency – to investigate the expected reduction of climate uncertainty based on future temperature observations and facilitate flexible adaptation strategies. As discussed in Völz and Hinkel (2023), many existing methods used to generate climate learning scenarios for flexible adaptation such as real-option analyses are simplified approaches and are inadequate of representing climate science; one promising option is to combine Bayesian approaches with climate models or with statistical approximation models. Time series models – which have been applied or developed for climate studies (Mudelsee 2010) such as the ARIMA-based (autoregressive integrated moving average model) approaches in Lai and Dzombak (2020, 2021) – are combined in this work with a Bayesian method (which, similarly, has been used in the studies like Tebaldi and Sansó (2009), Steinschneider et al. (2015b), Hui et al. (2018), and Fletcher et al. (2019)) to provide such a probabilistic framework.

This framework models physical parameters to describe global mean temperature anomaly considering multiple sources of uncertainty. The simplified energy-balance equations, describing global temperature response to radiative forcing (Gregory et al. 2004; Lewis and Curry 2015; Meehl et al. 2020), serve as a basis for a parametric form of the state-space model (SSM) (Cummins et al. 2020). Similar physical equations have been used in the integrated assessment models (Calel and Stainforth 2017) and for estimating parameters such as climate sensitivity of different GCMs (Cummins et al. 2020). Importantly, these energy-balance equations, as described in the subsequent section, provide the foundation to model the uncertainty from several key sources such as climate sensitivity, ocean heat uptake (Webster et al. 2008), and aerosol forcing (Myhre et al. 2013; Rotstayn et al. 2015). Inclusion of physical parameters in the modeling framework facilitates the assessment of these sources of uncertainty by using Bayesian inference with informative prior. It should also be noted that the uncertainty from socioeconomic development and mitigation efforts (e.g., among SSP scenarios) is incorporated in the framework as individual radiative forcing time series; this work models these SSP scenarios separately and does not address the reduction of uncertainty among SSP scenarios, however.

A SSM – integrated with physical parameters and calibrated via Bayesian inference – is consequently developed in this work to analyze time series of global mean temperature anomaly. Changes of global mean temperature are an important climate change indicator which has been used in policy documents such as the Paris Agreement and linked with different levels of regional impacts (Arnell et al. 2019; He et al. 2022). The methodology and results of this work focus on global mean temperature anomaly, although this SSM framework can be further modified to model changes in regional variables, e.g., based on pattern scaling (Tebaldi and Arblaster 2014) and linear relationships to model local response and variability (Beusch et al. 2020).

In addition to the use of historical observations of global mean temperature anomaly, the simulations from GCMs are used in this work to: (a) calibrate the SSM and (b) serve as synthetic observations (or “pseudo-observations” (Eyring et al. 2019)) to investigate relative accuracy and reduction of uncertainty in the future (in 2050 and 2080). The overall approach is described in Section 2. The results from the probabilistic inference of physical parameters of GCMs using the SSM are discussed in Section 3. The results of parameters’ posterior distribution and updated climate projections (using both pseudo- and historical observations) are discussed in Section 4. Summary, conclusions, and recommendations are provided in Section 5.

2 Methodology

The methodology of this work consists of three main components: the physical equations and parameters for describing global mean temperature anomaly, the selected parametric form of the SSM, and the application of Bayesian inference. These three components are described in the following sub-sections.

2.1 Physical modeling of global mean temperature anomaly

The time series of global mean temperature anomaly \(T\) responding to radiative forcing \(F\) can be described using a simplified, two-layer energy-balance model. The heat flux to the climate system, which is largely absorbed by ocean, can be expressed as \(F-\lambda T\) (Gregory et al. 2002), where \(\lambda\) is a climate feedback parameter representing the sensitivity of temperature response (Gregory and Andrews 2016). The first layer of the two-layer model represents air and surface ocean, whereas the second layer represents deep ocean which, due to its high heat capacity, responds to heat exchange more slowly than the surface layer (Calel and Stainforth 2017). A one-layer model (without an explicit separation of a deep ocean layer) and multiple-layer models can also be applied (such as in Cummins et al. (2020)), with less and more accurate modeling of energy exchange and at a lower and higher computational cost, respectively. In this work, the two-layer model is adopted because it is more accurate than the one-layer model (further details for the one-layer model can be found in the Supplemental Materials), includes a reasonably small number of model parameters, and can be efficiently calibrated via Bayesian inference.

This two-layer model is associated with two temperature anomaly series, i.e., the anomalies of the surface layer and of the deep ocean layer. The global mean surface air temperature anomaly is used for the surface layer, assuming that the heat exchange between the air and surface ocean occurs spontaneously and uniformly. Heat exchange between the surface layer and deep ocean layer is incorporated to the two-layer model. The temperature anomalies of these two layers at time \(t\) are therefore modeled as (Calel and Stainforth 2017):

$$\begin{array}{c}{C}_{1}\frac{{\mathrm{d}}T}{{\mathrm{d}}t}=F-\lambda T-\beta (T-{T}_{\mathrm{LO}})\\ {C}_{2}\frac{{\mathrm{d}}{T}_{\mathrm{LO}}}{{\mathrm{d}}t}=\beta (T-{T}_{\mathrm{LO}})\end{array}$$
(1)

where \(T\) and \({T}_{\mathrm{LO}}\) (in ºC or K) are the temperature anomalies at the surface layer and deep ocean (i.e., Lower Ocean) layer, respectively, \({C}_{1}\) and \({C}_{2}\) (in Wm−2 K−1 yr) are the heat capacities of the surface and deep ocean layers, respectively, \(\beta\) and \(\lambda\) (in Wm−2 K−1) are the heat exchange coefficient and the climate feedback coefficient, respectively, and \(F\) (in Wm−2) is the radiative forcing.

To integrate these equations, a simple finite-difference numerical scheme of approximating the solution is used (as also adopted by some integrated assessment models (Calel and Stainforth 2017)):

$$\begin{array}{c}{C}_{1}\frac{T\left(t+\Delta t\right)-T(t)}{\Delta t}\simeq F(t)-\lambda T(t)-\beta [T\left(t\right)-{T}_{\mathrm{LO}}\left(t\right)]\\ {C}_{2}\frac{{T}_{\mathrm{LO}}\left(t+\Delta t\right)-{T}_{\mathrm{LO}}(t)}{\Delta t}\simeq \beta [T\left(t\right)-{T}_{\mathrm{LO}}\left(t\right)]\end{array}$$
(2)

hence, the evolution of surface and the deep ocean temperature anomalies can be expressed as:

$$\begin{array}{c}T\left(t+\Delta t\right)\simeq \frac{{C}_{1}-\lambda\Delta t-\beta\Delta t}{{C}_{1}}T\left(t\right)+\frac{\beta\Delta t}{{C}_{1}}{T}_{\mathrm{LO}}\left(t\right)+\frac{\Delta t}{{C}_{1}}F(t)\\ {T}_{\mathrm{LO}}\left(t+\Delta t\right)\simeq \frac{\beta\Delta t}{{C}_{2}}T\left(t\right)+\frac{{C}_{2}-\beta\Delta t}{{C}_{2}}{T}_{\mathrm{LO}}\left(t\right)\end{array}$$
(3)

where \(\Delta t\) is the time step size.

The variability among the GCM simulations of future global mean temperature anomaly can be largely attributed to the different values of the parameters in Eq. (1) or Eq. (3), including \(\lambda\), \({C}_{1}\), \({C}_{2}\) and function \(F(t)\) (Geoffroy et al. 2013).

Parameter \(\lambda\) indicates the sensitivity of temperature response to the imposed radiative forcing and can be calculated using the Equilibrium Climate Sensitivity (ECS) as \(\lambda ={F}_{2\times {{\mathrm{CO}}}_{2}}/{\mathrm{ECS}}\), where \({F}_{2\times {{\mathrm{CO}}}_{2}}\) is the forcing value with a doubled CO2 concentration from the pre-industrial level and is around 3.7 W/m2 (Lewis and Curry 2015). As shown in Eq. (1), the temperature anomaly at a steady state is equal to \(F/\lambda\) and \(1/\lambda\) therefore represents the temperature change at equilibrium per unit radiative forcing. ECS is defined as the temperature change at equilibrium with a doubling of CO2 concentration (Meehl et al. 2020) and with a radiative forcing \({F}_{2\times {{\mathrm{CO}}}_{2}}\).

The Transient Climate Response (TCR) is another commonly used climate sensitivity parameter: the TCR represents the temperature change when the CO2 concentration is doubled during a period of 70 years with a 1% increase of concentration each year (Richardson et al. 2016). As the system does not reach equilibrium at the end of the 70-year period, the TCR is lower than the ECS, and it is affected by the rate of heat exchange, which in this model is related to parameters \({C}_{1}\), \({C}_{2}\), and \(\beta\).

Radiative forcing \(F\) is one major source of uncertainty in climate projections because of the uncertainty related to greenhouse gas (GHG) emissions and aerosol forcing (Andreae et al. 2005; Hawkins and Sutton 2009). The different climate change scenarios (such as the SSP2-4.5 and SSP5-8.5) represent the different trajectories of future GHG emissions (Hawkins and Sutton 2009). Aerosol forcing also greatly contributes to climate change uncertainty (Andreae et al. 2005), as an underestimated present-day aerosol cooling effect could lead to the underestimation of climate sensitivity, and consequently the underestimation of temperature increases when future aerosol cooling effect decreases (Myhre et al. 2013). Given that the objective of this work is to assess climate projection uncertainty including forcing uncertainty, an average forcing series for each SSP scenario with an additional consideration of its uncertainty is used instead of a fixed forcing series (such as the forcing with a quadrupling of CO2 concentration used in Cummins et al. (2020)). The average forcing series of different SSP scenarios estimated in and provided by IPCC AR6 (Smith et al. 2021) are used in this work.

Two linear scaling coefficients (\({\upgamma }_{1}\) and \({\upgamma }_{2}\)) are used to model the uncertainty of the GHG and aerosol forcings:

$$F\left(t\right)={\gamma }_{1}{F}_{(t)}^{\mathrm{GHG}}+{\gamma }_{2}{F}_{(t)}^{\mathrm{aerosol}}+{F}_{(t)}^{\mathrm{other}}$$
(4)

where \({F}_{(t)}^{\mathrm{GHG}}\) models the contributions from GHG forcing, \({F}_{(t)}^{\mathrm{aerosol}}\) those of anthropogenic aerosol forcing, and \({F}_{(t)}^{\mathrm{other}}\) those of other sources, including natural forcing and the forcing from land use change.

2.2 The physical-parameter-based SSM

The two-layer model presented in Eq. (3) and (4) is subsequently developed into a parametric SSM. A SSM consists of a state transition function and a measurement function (Shumway and Stoffer 2017). The state transition function describes the changes of latent variable vector \({\varvec{x}}\left(t\right)\), including the surface and the deep ocean temperature anomalies, i.e., \(T\left(t\right)\) and \({T}_{\mathrm{LO}}\left(t\right)\), at time \(t\). The measurement function describes the relation between the latent and the measured temperature, including noise and errors to explain the discrepancies between latent temperature and GCM simulations or historical observations of global mean temperature anomalies.

Based on Eq. (3) and (4), the transition function is:

$$\begin{array}{c}{\varvec{x}}\left(t+\Delta t\right)={\varvec{A}}{\varvec{x}}\left(t\right)+{\varvec{B}}{\varvec{F}}\left(t\right)+{\varvec{\omega}}(t)\\ {\varvec{x}}\left(t\right)=\left[\begin{array}{c}T\left(t\right)\\ {T}_{\mathrm{LO}}\left(t\right)\end{array}\right]; {\varvec{A}}=\left[\begin{array}{cc}1-\frac{{F}_{2\times {{\mathrm{CO}}}_{2}}}{{C}_{1}{\mathrm{ECS}}}\Delta t-\frac{\beta }{{C}_{1}}\Delta t& \frac{\beta }{{C}_{1}}\Delta t\\ \frac{\beta }{{C}_{2}}\Delta t& 1-\frac{\beta }{{C}_{2}}\Delta t\end{array}\right]\\ {\varvec{B}}=\frac{\Delta t}{{C}_{1}}\left[\begin{array}{ccc}{\upgamma }_{1}& {\upgamma }_{2}& 1\\ 0& 0& 0\end{array}\right]; {\varvec{F}}\left(t\right)=\left[\begin{array}{c}{F}_{\left(t\right)}^{\mathrm{GHG}}\\ {F}_{\left(t\right)}^{\mathrm{aerosol}}\\ {F}_{\left(t\right)}^{\mathrm{other}}\end{array}\right];{\varvec{\omega}}\left(t\right)=\left[\begin{array}{c}{\omega }_{1}\left(t\right)\\ {\omega }_{2}\left(t\right)\end{array}\right]\end{array}$$
(5)

where vector \({\varvec{\omega}}\left(t\right)\) is a noise term including independent, zero-mean normal noises \({\omega }_{1}\left(t\right)\) and \({\omega }_{2}\left(t\right)\). For the analyses carried out in this work, \(\Delta t\) is fixed to one year.

Natural climate variability can affect temperature series by introducing additive noise to the long-term climate change trend (Lai and Dzombak 2019). The GCM simulations and historical observations are modeled as measurement variable vector \({\varvec{y}}\left(t\right)\) with additional noise as:

$$\begin{array}{c}{\varvec{y}}\left(t\right)={\varvec{D}}{\varvec{x}}\left(t\right)+{\varvec{\nu}}(t)\\ \begin{array}{cc}{\varvec{D}}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right];{\varvec{\nu}}\left(t\right)=\left[\begin{array}{c}{\nu }_{1}\left(t\right)\\ {\nu }_{2}\left(t\right)\end{array}\right]& (\mathrm{if\;deep\;ocean\;temperature\;measurements\;are\;processed})\end{array}\\ \begin{array}{cc}{\varvec{D}}=\left[\begin{array}{cc}1& 0\end{array}\right];{\varvec{\nu}}\left(t\right)={\nu }_{1}\left(t\right)& (\mathrm{if\;deep\;ocean\;temperature\;measurements\;are\;not\;used})\end{array}\end{array}$$
(6)

where \({\nu }_{1}\left(t\right)\) and \({\nu }_{2}\left(t\right)\) are independent, zero-mean normal noise terms affecting the surface and the deep ocean temperature anomalies, respectively.

As indicated in Eq. (6), the SSM can be applied with or without processing deep ocean temperature measurements. The analyses without using ocean temperature measurements are applied and presented subsequently, while some results from using GCM ocean temperature simulations are presented in the Supplemental Materials. One reason for excluding deep ocean temperature measurements is related to the relatively limited amount of historical observations of deep ocean temperature (Abraham et al. 2013). Additionally, the ocean temperature simulations from GCMs are commonly provided in separate ocean layers and additional assumptions and procedures are needed to process the multiple layers of ocean temperature simulations for the two-layer model, which can lead to challenges and high sensitivity of the parameter posterior distributions (as described in the Supplemental Materials).

It is also worth noting that, although not further assessed in this work, Eq. (6) provides a flexible parametric form to model regional variables. For example, matrix \({\varvec{D}}\) can be modified to include linear scaling factors for modeling regional responses and variables; \({\varvec{\nu}}(t)\) can also be modified to include additional regional variability and noise (e.g., following a framework similar to the one used in Beusch et al. (2020)). Such alternative modeling approaches are also described in the Supplemental Materials.

Combining Eq. (5) and (6), a parametric SSM form is developed with ten parameters \({\theta }_{1}\) to \({\theta }_{10}\), as presented in Table 1. Vector \({\varvec{\Theta}}\) lists all ten parameters, representing a linear transformation of the natural logarithm of the physical parameters: \({\varvec{\Theta}}={\varvec{H}}{\mathrm{log}}({[\begin{array}{ccc}{\mathrm{ECS}}& {C}_{1}& \begin{array}{ccc}{C}_{2}& \beta & \begin{array}{ccc}{\upgamma }_{1}& {\upgamma }_{2}& \begin{array}{ccc}{q}_{1}& {q}_{2}& \begin{array}{cc}{r}_{1}& {r}_{2}\end{array}\end{array}\end{array}\end{array}\end{array}]}^{\mathrm{T}})\), where 10-by-10 matrix \({\varvec{H}}\) can be deduced from the definitions in Table 1 (the matrix and its inverse is explicitly reported in the Supplemental Materials) and \({q}_{1}\), \({q}_{2}\), \({r}_{1}\), and \({r}_{2}\) are the standard deviations of zero-mean white noise \({\omega }_{1}(t)\), \({\omega }_{2}(t)\), \({\nu }_{1}(t)\) and \({\nu }_{2}(t)\), respectively.

Table 1 Summary of the ten SSM parameters and their representation of the physical parameters in Eq. (5) and (6)
Full size table

2.3 A two-step Bayesian inference procedure based on the SSM

Following the Bayesian method, posterior distributions of parameters are obtained integrating prior distributions and likelihood functions. Bayes’ formula is applied twice in this work, during the processing of GCM simulations and subsequently the processing of pseudo- or historical observations. When processing GCM simulations, broad prior distributions informed by the literature are used (this literature prior is discussed in Section 3). The posterior distributions obtained from this first step of processing GCM simulations are then integrated into a new distribution (i.e., the GCM-informed prior distribution) for processing observations.

This additional step of processing GCM simulations aims at leveraging long-term simulations from GCMs (up to 2099 used in this work) to inform the SSM parameters before processing observations. Compared to the distributions based on values taken from the literature, the parameters informed by the GCM simulations can better characterize some features of parameter uncertainty, e.g., highlighting the correlation among parameters.

2.3.1 Processing GCM simulations

In general, an approximate posterior distribution of the SSM parameters is obtained in this work by applying the Laplace approximation to the product of the model likelihood functions computed by the Kalman Filter and prior distributions (Shumway and Stoffer 2017). Additional technical details for this section including the Kalman Filter are also provided in the Supplemental Materials.

The posterior distribution is obtained from Bayes’ formula when processing the GCM-simulated temperature series \({\varvec{y}}\):

$$\begin{array}{c}p\left({\varvec{\Theta}}|{\varvec{y}}\right)=\frac{p\left({\varvec{y}}|{\varvec{\Theta}}\right)p\left({\varvec{\Theta}}\right)}{\int p\left({\varvec{y}}|{\varvec{\Theta}}\right)p\left({\varvec{\Theta}}\right){\mathrm{d}}{\varvec{\Theta}}}\\ {\varvec{y}}|{\varvec{\Theta}}\sim \mathrm{KF(}{\varvec{\Theta}}\mathrm{)}\\ p\left({\varvec{\Theta}}\right)\propto \mathcal{N}\left({\varvec{\Theta}},{{\varvec{\mu}}}_{0},{{\varvec{\Sigma}}}_{0}\right){f}_{\mathrm{TCR}}\left({\varvec{\Theta}}\right); {f}_{\mathrm{TCR}}\left({\varvec{\Theta}}\right)\propto \mathcal{N}\left(\mathrm{TCR(}{\varvec{\Theta}}\mathrm{)}, {\mu }_{\mathrm{TCR}},{\upsigma }_{\mathrm{TCR}}^{2}\right)\end{array}$$
(7)

where \(p\left({\varvec{\Theta}}|{\varvec{y}}\right)\) is the posterior distribution of parameters; \(p\left({\varvec{y}}|{\varvec{\Theta}}\right)\) is the likelihood function of time series computed from the Kalman filter; \(\mathrm{KF(}{\varvec{\Theta}}\mathrm{)}\) indicates the processes of using the Kalman Filter with the parameter vector \({\varvec{\Theta}}\); \(p\left({\varvec{\Theta}}\right)\) is the parameter prior, proportional to the product of a function \({f}_{\mathrm{TCR}}\) and a normal distribution with the mean vector \({{\varvec{\mu}}}_{0}\) and covariance matrix \({{\varvec{\Sigma}}}_{0}\); and symbol \(\mathcal{N}\) indicates the normal distribution. Moments \({{\varvec{\mu}}}_{0}\) and \({{\varvec{\Sigma}}}_{0}\) are selected based on the values reported in the literature. \(\mathrm{TCR(}{\varvec{\Theta}}\mathrm{)}\) is calculated deterministically from Eq. (5) (i.e., assuming the noise variance is zero) under an annual 1% increase of CO2 forcing for a 70-year period. The distribution of \(\mathrm{TCR(}{\varvec{\Theta}}\mathrm{)}\) is based on a likelihood function related to a nominal value of TCR as \({\mu }_{\mathrm{TCR}}=1.8{\mathrm{K}}\) and a standard deviation \({\upsigma }_{\mathrm{TCR}}=0.5{\mathrm{K}}\) (selected based on the reported range of TCR), and it is included to align parameters \({\varvec{\Theta}}\) with a reasonable value of TCR.

The Kalman Filter allows for inferring the hidden state and also for computing the likelihood related to entire time series (Rings et al. 2012). In this work the term “prediction” specifically refers to the prediction procedure of the Kalman Filter and “projection” refers the future temperature projection obtained from the original GCM simulations and the SSM (these terms can have different meaning in the literature (Merryfield et al. 2020)).

To apply the Laplace’s approximation, the Maximum A Posteriori (MAP) is identified by a numerical optimization algorithm applied to the unnormalized posterior (i.e., \(p\left({\varvec{y}}|{\varvec{\Theta}}\right)p\left({\varvec{\Theta}}\right)\) in Eq. (7)). In this work, the “L-BFGS-B” method (Byrd et al. 1995) is used. The Hessian matrix is subsequently estimated at the MAP point, also by the L-BFGS-B algorithm to approximate the posterior with a Gaussian distribution (Nocedal and Wright 2006; Barber 2011).

Additionally, the posterior distribution related to a specific GCM is obtained by processing multiple ensemble members (i.e., different simulation runs) of this GCM (if available). Let \({\mathbf{Y}}_{i}=\{{{\varvec{y}}}_{i,1}, {{\varvec{y}}}_{i,2},{{\varvec{y}}}_{i,3},\dots \}\) list the ensemble members (\({{\varvec{y}}}_{i,1}\) as ensemble member 1, for example) of GCM \(i\), characterized by parameter value \({{\varvec{\Theta}}}_{i}\). The different ensemble members of a GCM are assumed to be independent of each other conditionally to \({{\varvec{\Theta}}}_{i}\) (hence ensemble members of GCM \(i\) share the same parameter value \({{\varvec{\Theta}}}_{i}\) but not the hidden variables \(\mathbf{x}(t)\)). From the Laplace’s approximation applied to Eq. (7), the posterior distribution is:

$${\boldsymbol{\Theta }}_{i}|{{\varvec{Y}}}_{i} \stackrel{approx}{\sim } \mathcal{N}\left({\widehat{{\varvec{\mu}}}}_{i},{\widehat{\varvec{\Sigma }}}_{i}\right)$$
(8)

where \({\widehat{{\varvec{\mu}}}}_{i}\) and \({\widehat{{\varvec{\Sigma}}}}_{i}\) are the posterior mean vector and covariance matrix estimated by the Laplace’s approximation, respectively.

2.3.2 Processing pseudo- or historical observations

The GCMs are considered as different realizations of the Earth climate system and are represented by the posterior distributions of the parameters obtained in the previous step; the prior distribution used subsequently is informed by these intermediate, posterior distributions. Specifically, the posterior distributions identified for individual GCMs in the previous step are integrated into a single, unified prior distribution (i.e., the GCM-informed prior distribution) to process pseudo- or historical observations. Note that, simulated time series from a GCM can also be used as pseudo-observations in this work, if a GCM is used to provide pseudo-observations, this GCM is then excluded from being used to obtain the GCM-informed prior distribution.

Let \({\varvec{Y}}=\{{\mathbf{Y}}_{1},\boldsymbol{ }{\mathbf{Y}}_{2},\dots ,{\mathbf{Y}}_{m}\}\) list all simulations from m GCMs. By leveraging Eq. (8), an integrated posterior distribution conditional to \({\varvec{Y}}\) can be expressed as a mixture of Gaussians:

$$\begin{array}{c}p\left({\varvec{\Theta}}|{\varvec{Y}}\right)\simeq \sum_{i=1}^{m}{f}_{i}\left({\varvec{\Theta}}\right){P}_{i}\\ {f}_{i}\left({\varvec{\Theta}}\right)=\mathcal{N}\left({\varvec{\Theta}},{\widehat{{\varvec{\mu}}}}_{i},{\widehat{{\varvec{\Sigma}}}}_{i}\right)\end{array}$$
(9)

where \({f}_{i}\) is the posterior distribution related to the simulations of GCM \(i\), and \({P}_{i}\) is the prior probability for this GCM, and a non-informative, uniform distribution \({P}_{i}=1/m\) is adopted in this work. The integrated distribution can be approximated via a single normal distribution, by matching moments:

$$\begin{array}{c}{\varvec{\Theta}}|\mathbf{Y}\stackrel{{\mathrm{approx}}}{\sim } \mathcal{N}\left(\widehat{{\varvec{\mu}}},\widehat{{\varvec{\Sigma}}}\right)\\ \widehat{{\varvec{\mu}}}=\sum_{i=1}^{m}{P}_{i}{\widehat{{\varvec{\mu}}}}_{i}; \widehat{{\varvec{\Sigma}}}=\sum_{i=1}^{m}{P}_{i}\left[{\widehat{{\varvec{\Sigma}}}}_{i}+{\left({\widehat{{\varvec{\mu}}}}_{i}-\widehat{{\varvec{\mu}}}\right)\left({\widehat{{\varvec{\mu}}}}_{i}-\widehat{{\varvec{\mu}}}\right)}^{\mathrm{T}}\right]\end{array}$$
(10)

where \(\widehat{{\varvec{\mu}}}\) and \(\widehat{{\varvec{\Sigma}}}\) are the estimated moments.

Instead of integrating into a single unified prior distribution, the Supplemental Materials also provide the description of an alternative approach to assign and estimate posterior probabilities of individual GCMs by using posterior distribution identified previously for each GCM (although this alternative approach is sensitive to the particular set of GCMs included in the analyses).

To process pseudo- or historical observations \({{\varvec{y}}}_{h}\), the Laplace’s approximation is implemented for a second time, with the same “L-BFGS-B” method to identify the MAP. Different from Eq. (7), the likelihood function related to TCR, which has been already used for informing prior distributions in Eq. (7), is not integrated in this step. The posterior distribution \(p\left({\varvec{\Theta}}|{\varvec{Y}},{{\varvec{y}}}_{h}\right)\) is obtained by the Bayes’ formula:

$$\begin{array}{c}p\left({\varvec{\Theta}}|{\varvec{Y}},{{\varvec{y}}}_{h}\right)\propto p\left({{\varvec{y}}}_{h}|{\varvec{\Theta}}\right)p\left({\varvec{\Theta}}|{\varvec{Y}}\right)\\ {{\varvec{y}}}_{h}|{\varvec{\Theta}}\sim {\mathrm{KF}}({\varvec{\Theta}})\end{array}$$
(11)

where \(p\left({{\varvec{y}}}_{h}|{\varvec{\Theta}}\right)\) is the likelihood function related to the observations, computed by the Kalman Filter.

The projections of future temperature anomaly series y∗ are obtained as:

$$\begin{array}{c}p\left({{\varvec{y}}}^{*}|{{\varvec{y}}}_{h},{\varvec{Y}}\right)=\int p\left({{\varvec{y}}}^{*}|{\varvec{\Theta}}, {{\varvec{y}}}_{h}\right)p\left({\varvec{\Theta}}|{\varvec{Y}},{{\varvec{y}}}_{h}\right){\mathrm{d}}{\varvec{\Theta}} \\ {{\varvec{y}}}^{*}|{\varvec{\Theta}}, {{\varvec{y}}}_{h}\sim {\mathrm{KF}}({{\varvec{y}}}_{h}, {\varvec{\Theta}})\end{array}$$
(12)

where \(p\left({{\varvec{y}}}^{\boldsymbol{*}}|{{\varvec{y}}}_{h},{\varvec{Y}}\right)\) is the updated, probabilistic future temperature anomaly projections given with observations, and \({\mathrm{KF}}({{\varvec{y}}}_{h},{\varvec{\Theta}})\) indicates the Kalman Filter with parameters \({\varvec{\Theta}}\) and observations \({{\varvec{y}}}_{h}\) up to the present time.

Projections in Eq. (12) are based on \({{\varvec{y}}}^{*}\) (i.e., the future series of “measurement” variables in the Kalman filter) instead of the latent variable \({{\varvec{x}}}^{*}\), as the former incorporates the additional noise related to natural variability (although distribution \(p\left({{\varvec{x}}}^{*}|{{\varvec{y}}}_{h},{\varvec{Y}}\right)\) can also be easily obtained from the Kalman Filter).

To account for parameter uncertainty, a sampling and simulation procedure is applied following Eq. (12). Specifically, samples of parameters \({\varvec{\Theta}}\) is generated from the posterior distributions \(p\left({\varvec{\Theta}}|{\varvec{Y}},{{\varvec{y}}}_{h}\right)\), and the Kalman Filter is then used to fit observations \({{\varvec{y}}}_{h}\) and simulate future series \({{\varvec{y}}}^{*}\). The generated different \({{\varvec{y}}}^{*}\) series are intended as possible realizations of future temperature anomaly series and indicate the trajectories of future changes incorporating the estimation of the moments of projections.

3 Results of probabilistic estimation of physical parameters of GCMs

3.1 The literature prior

As discussed previously, an identical prior distribution informed by the reported physical parameter values from the literature is used to process GCM simulations. Specifically, as reported from the recent IPCC AR6 (IPCC 2021), ECS has a range of 2 to 5K (90% confidence level), whereas TCR is reported with a range of 1.2 to 2.4K. These ranges are used to determine the distribution of ECS and TCR. Geoffroy et al. (2013) analyzed different GCMs of CMIP5 based on a similar two-layer model and suggests that the average \({C}_{1}\) among GCMs is around 7.3 Wm−2K−1yr corresponding approximately to 80m of ocean depth, the average \({C}_{2}\) is around 106 Wm−2K−1yr corresponding approximately to 1100m of ocean depth, and the average \(\beta\) value is around 0.73 Wm−2K−1. The forcing time series estimated from the IPCC AR6 (Smith et al. 2021) are used in this work, and the distributions of the two scaling coefficients \({\upgamma }_{1}\) and \({\upgamma }_{2}\) are selected to represent the uncertainty of the GHG and aerosol forcings reported. For example, the IPCC-AR6-estimates of GHG forcing and aerosol forcing at 2019 are from 3.4 to 4.4 Wm−2 and from -1.94 to -0.06 Wm−2 for a 90% interval, respectively (Smith et al. 2021). The selected prior distribution of the first six physical parameters (along with the TCR generated from this prior distribution) are presented in Fig. 1. For the four parameters related to noise, the empirical standard deviations of GCM temperature simulations are adopted. Specifically, empirical standard deviation values are derived from simulations of each GCM and log-normal distributions are then calibrated from the derived values of different GCMs. Further discussion is provided in the Supplemental Materials, together with the mean and variance selected for these parameters.

Fig. 1
figure 1

The literature prior distribution selected for the six physical parameters and for the TCR generated from this prior distribution. The GHG and aerosol forcings are estimated using the probability densities of the two scaling coefficients γ1 and γ2 and using the forcing values at year 2019 and (or) 2099. The TCR density is estimated using the simulations from the Kalman Filter based on the prior probabilities of other parameters (and assuming zero noise variance). The three vertical dotted lines present the medians and 95% confidence intervals

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Compared to ECS and TCR which have been calculated and evaluated in a number of existing studies (e.g., IPCC AR6 (IPCC 2021) and Meehl et al. (2020)), the parameters such as \({C}_{1}\), \({C}_{2}\), and \(\beta\) in this work are derived specifically based on the two-layer model, therefore a wide prior distribution is selected for these parameters and may be slightly different from the estimated values of similar parameters in the other studies. For example, the selected prior distribution of \({C}_{1}\) has a greater mean and larger uncertainty than the estimated values in Geoffroy et al. (2013).

3.2 Parameters inferred from GCM simulations

Processing GCM simulations from 1850 to 2099 under the SSP5-8.5 scenario, the posterior distributions of parameters are obtained for different GCMs of CMIP6, and the estimated ECS and TCR are compared with the reported values in Fig. 2. SSP5-8.5 scenario is used because temperature increase is greater under this scenario (i.e., with larger signal-to-noise ratios).

Fig. 2
figure 2

Estimated ECS and TCR (from processing GCM simulations via SSM) vs the corresponding reported values. The estimation is conducted based on the SSP5-8.5 scenario, whereas the reported values are obtained from Meehl et al. (2020). GCMs with available reported ECS or TCR values in Meehl et al. (2020) are presented. The point estimates based on MAP are indicated by black dots, whereas the 95% confidence intervals by grey error bars

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The estimates of ECS and TCR are generally in good agreement with the reported values, with the results of ECS more accurate than the results of TCR, according to Fig. 2. The reported values are inside the 95% posterior interval in 21 out of 29 cases for ECS and 17 out of 29 cases for TCR. The greater errors of estimating ECS and TCR for some GCMs are likely caused by the limitation of the two-layer model (e.g., the limited number of layers, the one-year time step, and its simplified modeling of temperature response).

The parameter inference of a GCM (CNRM-CM6-1) is further used as an example to assess the difference between the GCM simulations and the new simulations generated by the SSM (calibrated using the simulations from this GCM). The SSM simulations are obtained by generating random parameter values from the posterior distribution and generating random noise. The comparison is presented in Fig. 3.

Fig. 3
figure 3

Comparisons of the original CNRM-CM6-1 simulations and the results of the surface-layer temperature anomalies from the SSM under (a) SSP2-4.5 and (b) SSP5-8.5. The first column presents five original CNRM-CM6-1 ensemble members and the forcing time series used in this work; the middle column presents the latent temperature \({\varvec{x}}\left(t\right)\) series obtained from the SSM for each ensemble member using the Kalman Filter and using the point estimates of parameters at MAP; the third column presents the five new simulations generated from the SSM using the resampled parameter values from the parameter posterior distribution. The original SSP5-8.5 simulations are used to estimate the SSM parameter posterior distribution, which are subsequently used to generate new simulations using the SSP2-4.5 and SSP5-8.5 forcing series in the third column. The temperature anomalies are calculated relative to 1850–1879 period

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Figure 3 suggests that the SSM can be used to project future temperature with the inferred physical parameters. The latent series \({\varvec{x}}\left(t\right)\) for the temperature anomaly of surface layer, as presented in the middle column of Fig. 3, exhibit less noise compared to the original GCM simulations in the first column, supporting the assumption that the \({\varvec{x}}\left(t\right)\) values represent an underlying smoothed temperature response to the forcings applied. The new simulated series from the SSM in the third column exhibit long-term temperature trend comparable to the original simulations presented in the first column (although the end-of-the-century temperature level under SSP2-4.5 may be slightly higher than the SSM-simulated series). Consequently, the results of Fig. 3 suggest that GCMs can be reasonably described or represented by the SSM by adopting the posterior distribution of the parameters inferred for these GCMs.

4 Results from processing pseudo- and historical observations

4.1 Posterior distributions of ECS and TCR

The posterior distributions of ECS and TCR after processing GCM simulations as pseudo-observations, are presented and discussed in this section. As TCR is not included in the parameters of the SSM, its posterior distribution is derived from the Kalman Filter simulations. Using one realization series from CNRM-CM6-1 and GFDL-ESM4 as pseudo-observations (selected because CNRM-CM6-1 has a relatively larger reported ECS value, whereas GFDL-ESM4 has a smaller ECS value), the posterior distributions of ECS and TCR – from processing different amounts of pseudo-observations – are presented in Fig. 4.

Fig. 4
figure 4

Posterior probabilities of ECS and TCR using one realization from CNRM-CM6-1 and GFDL-ESM4 as pseudo-observations (under SSP5-8.5): (a) probability density functions when pseudo-observations are available up to 2020 and 2080 and (b) posterior means and 95% confidence intervals given observations available up to a specific time. Vertical dotted lines in part (a) indicate the reported ECS and TCR values from Meehl et al. (2020)

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The results of Fig. 4 suggest that the SSM method can provide reasonable estimates of the physical parameters and additional pseudo-observations can reduce the uncertainty, although the results are dependent on the particular pseudo-observation series assessed. Similar to the results in Fig. 2, ECS results align more closely with reported values than the TCR results. Additionally, the uncertainty of ECS and TCR estimated for the GFDL-ESM4 pseudo-observation series largely reduces with the increase of observations. The results of CNRM-CM6-1 in Fig. 4 show a less observable reduction of uncertainty and an uptick of ECS starting around 2050. Such results suggest the relatively high sensitivity of ECS and TCR estimation with respect to the specific pseudo-observation series used. These features in the estimation of ECS and TCR (e.g., the uptick of ECS for CNRM-CM6-1 starting from 2050) are further investigated and are generally related to the average forcing time series used in this work. Specifically, the forcings and simulated temperature response in GCM can exhibit temporal changes different from the ones assumed and used in this work, causing the ECS and TCR posterior probabilities not temporally consistent and also dependent on pseudo-observation series (more detailed discussions are offered in the Supplemental Materials). The forcing uncertainty is considered and modeled in this work with \({\upgamma }_{1}\) and \({\upgamma }_{2}\), which are time-independent and can be a limitation of this SSM-based approach. Additional uncertainty on the forcing series can be incorporated to the SSM, although this addition is not further investigated.

4.2 Future temperature projections using pseudo-observations

Future temperature projections with the processing of increased amounts of pseudo-observations is examined in this section. Two pseudo-observation series from the two GCMs (CNRM-CM6-1 and GFDL-ESM4) are processed to project future temperature in Fig. 5 as examples, assuming that the pseudo-observations are available from 1850 to 2020, 2050, and 2080.

Fig. 5
figure 5

The projections of global mean temperature anomaly using one realization from (a) CNRM-CM6-1 and (b) GFDL-ESM4 as pseudo-observations, which are assumed to be available up to 2020, 2050, and 2080

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Figure 5 shows how the uncertainty of future projections is reduced when more observation is processed. Compared to the processing of observations up to 2020, the projections with observations up to 2050 and 2080 exhibit gradually smaller uncertainty in each case of Fig. 5. This reduction of uncertainty is the result of reduced posterior uncertainty such as presented previously in Fig. 4.

Further quantitative analyses are carried out to evaluate the reduction of uncertainty and projection accuracy. The simulations from each of 36 (for the SSP2-4.5) or 37 (for the SSP5-8.5) GCMs are used as pseudo-observations, of which up to 2020, 2050, and 2080 are separately used to obtain and compare future projections. The results among the 36/37 pseudo-observation series are summarized in Fig. 6. In addition to assessing projection accuracy with absolute errors, the continuous ranked probability score (CRPS; Gneiting and Raftery (2007)) – the higher the score, the lower the forecast performance – is also used. Uncertainty ranges of the 95% prediction intervals (i.e., calculated as the upper bound minus lower bound) are also summarized in Fig. 6. For example, using pseudo-observations up to 2020 under SSP5-8.5, the end-of-the-century 95% prediction intervals have a 2.7 °C range (averaged among 37 pseudo-observation series) and 3.2 °C and 2.1 °C as the upper and lower quartiles among the 37 pseudo-observation series.

Fig. 6
figure 6

Time series of absolute errors, CRPS, and uncertainty ranges (of the 95% prediction intervals) for the SSM projections from using the pseudo-observations up to 2020, 2050, and 2080 under the (a) SSP2-4.5 and (b) SSP5-8.5. The x-axes indicate the projected future years starting from 2021, 2051, or 2081. The bold lines and shaded area indicate the median and empirical lower and upper quartiles (quartiles are presented because of sample sizes, i.e., 36 for SSP2-4.5 and 37 for SSP5-8.5 pseudo-observation time series used)

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The results of Fig. 6 suggest that for a specific future year, more observations (30 years of additional observations in this case) can largely reduce projection errors and uncertainty, whereas for the projections with a specific lead time (such as same 20 years ahead), the uncertainty can be reduced moderately and projection errors are similar. For example, for the end-of-the-century projections, the absolute errors on average are around 0.5 °C under the SSP2-4.5 when observations are available up to 2020 and are then reduced to 0.4 and less than 0.2 °C when the amount of observations increased to the end of 2050 and 2080, respectively. More noticeably, the uncertainty ranges substantially decrease for the end-of-the-century projections in parts (a3) and (b3): for example, increasing the availability of pseudo-observations from 2020 to 2050 results in the uncertainty ranges decreasing from 1.9 °C on average to 1.0 °C (SSP2-4.5) or from 2.7 °C to 1.2 °C (SSP5-8.5); additional 30 years of observations (up to 2080) lead to the further reduced 95% prediction interval ranges of 0.6 to 0.7 °C for the two SSP scenarios.

4.3 Results from processing historical observation records

Historical observations of global mean temperature anomaly (obtained from the NOAA Climate at A Glance as listed in Data Availability Statement; anomaly related to the 1850–1900 period is calculated and used in this case) are further assessed in this section and the results are presented in Fig. 7.

Fig. 7
figure 7

The results from the SSM with the processing of historical observations of global mean temperature anomaly (from 1850–1900 average): (a) the comparisons between original GCM simulations (including different ensemble members of same GCMs) and the SSM projections made at 2022 (using the GCM-informed prior); (b) the posterior 95% confidence intervals of ECS and TCR given with different amounts of observations; and (c) the posterior joint distributions of ECS and aerosol forcing estimated using observations up to 2022

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As presented in Fig. 7 part (a), the future temperature projections from the SSM already exhibit reduced uncertainty compared to the original GCM projections with the processing of historical observations. For example, the end-of-the-century, empirical 95% intervals among the realizations of GCMs are around 2.2–4.6 °C (SSP2-4.5) and 3.9–7.9 °C (SSP5-8.5), whereas the 95% prediction intervals from the SSM are 2.0–3.8 °C (SSP2-4.5) and 3.2–5.7 °C (SSP5-8.5) for the end-of-the-century projections. This reduction of projection uncertainty is consistent with the results in the previous studies such as Ribes et al. (2021), which suggests that the end-of-the-century, 90% prediction intervals are 2.3–3.7 °C (SSP2-4.5) and 3.8–6.0 °C (SSP5-8.5). The 95% uncertainty ranges of 1.8 °C for SSP2-4.5 and 2.5 °C for SSP5-8.5 in Fig. 7 also align with the results from using pseudo-observations in Fig. 6.

After processing historical observations, the SSM temperature projections in Fig. 7 are notably lower than the GCM original projections. This is likely related to the “hot model” problem (Hausfather et al. 2022) of some GCMs in CMIP6. As presented previously in Fig. 2, many GCMs in CMIP6 have ECS values greater than 4.7K. In comparison, no GCM has a ECS value greater than 4.7K in the previous CMIP5 phase (Meehl et al. 2020) and these large ECS values do not align well with other evidences (Hausfather et al. 2022). Consequently, these GCMs in CMIP6 with large climate sensitivity simulated larger temperature increases and result in higher upper bound for the original GCM projections in Fig. 7, compared to the SSM projections which are based on the processing of historical observations.

Posterior probabilities of ECS, TCR, and aerosol forcings are subsequently examined in Fig. 7. Part (b1) of Fig. 7 suggests some reduction of uncertainty especially when observations after 2000 are used. Because of the GCM-informed prior is based on the GCMs of CMIP6 (which include many GCMs with large ECS values as previously described), the ECS posterior probability using the GCM-informed prior in part (b1) of Fig. 7 (1.6–6.2K; 95% level) has a greater range than the 2-5K (although a 90% level is reported) given by IPCC AR6 (IPCC 2021). To further investigate such results, the literature prior (i.e., based on the reported parameter values in the literature, as mentioned previously) is also used to process historical observations to assess the ECS and TCR; the results are presented in part (b2) and (c2) of Fig. 7. Using the literature prior, the ECS posterior probability (2–4.5K as summarized by the 95% interval) has a smaller uncertainty range, although the mean ECS value updated at 2022 is comparable to the mean ECS in part (b1). Additionally, the joint posterior distributions of ECS and aerosol forcing are presented in part (c) of Fig. 7, which suggests a correlation between ECS and aerosol forcing. Such a correlation is consistent with the expectation that, if the present-day negative aerosol forcing is stronger (i.e., the net forcings combining aerosol and GHG forcings are smaller), then the Earth climate system should have greater climate sensitivity (i.e., the ECS in this case) given the observed temperature change (Andreae et al. 2005).

5 Summary, conclusions, and recommendations

To facilitate flexible decision-making in climate adaptation, a physical-parameter-based SSM (using a two-layer, energy-balance model) with Bayesian inference is developed and assessed in this work, serving as a modeling framework – consistent with climate science and computationally efficient – to investigate reduction of projection uncertainty of global mean temperature anomaly from additional observations. The method involves two steps: (1) leveraging long-term simulations (up to the year 2099 in this work) from GCMs to obtain the distributions of SSM parameters for each GCM, which are then integrated as a unified, GCM-informed prior distribution; (2) processing pseudo- or historical observations with the GCM-informed prior distribution, obtaining parameter posterior distributions conditional on observations, and projecting future temperature change.

The integration of the two-layer, energy-balance model and Bayesian inference allows the SSM to consider and model different sources of climate change uncertainty (including natural variability, climate sensitivity, ocean heat uptake, and forcing uncertainty from aerosol) and relate these sources of uncertainty to temperature projection. As described in the previous studies (Andreae et al. 2005; Webster et al. 2008), a given temperature change can be related to different combinations of variables related to climate sensitivity, ocean heat uptake, and aerosol forcing; these various sources of uncertainty make the task of reducing projection uncertainty difficult. This issue is addressed in this work by integrating the energy-balance equations to the SSM and explicitly including parameters related to different sources of uncertainty.

Using GCM simulations as pseudo-observations, the SSM method is assessed with respect to the posterior probabilities of physical parameters, the projections of global mean temperature anomaly, and the reduction of projection uncertainty with additional observations. The SSM provides reasonable estimates of physical parameters: for example the estimated ECS and TCR values for each GCM are generally consistent with their reported values. Reduction of parameter uncertainty (such as ECS and TCR) can be observed when further observations are processed, leading to the decreased uncertainty. Analyzing the projections sequentially made with the processing of pseudo-observations up to 2020, 2050, and 2080, reduction of uncertainty is evident: e.g., the end-of-the-century uncertainty range (95% prediction intervals) of global mean temperature anomaly decreases from 1.9 °C (when projected in 2020) to 1.0 °C (when projected in 2050), and further to 0.6 °C (when projected in 2080) on average under SSP2-4.5; under SSP5-8.5, the uncertainty range on average reduces from 2.7 °C in 2020 to 1.2 °C in 2050, and further to 0.7 °C in 2080. Such analyses illustrate how the future reduction of climate change uncertainty can be predicted.

The state space representation can also facilitate additional study objectives with adjustments on its parametric form. One example is to further use the SSM to investigate the posterior distributions of different physical parameters with observational constraints (e.g., as Fig. 7 suggests, using the GCM-informed prior distribution based on GCMs of CMIP6 leads to large ECS uncertainty, greater than the results from directly using the literature prior distribution). Additionally, the method can be extended to the assessment of regional impacts based on the findings from the studies like Arnell et al. (2019) and He et al. (2022) and using additional methods such as pattern scaling (Tebaldi and Arblaster 2014) and modeling of regional climate responses (Beusch et al. 2020).

The SSM method presented in this work can support engineering decision-making for climate change adaptation by projecting temperature and assessing the uncertainty of such projections. The results presented in this work, particularly the uncertainty reduction for projecting global mean temperature anomaly from 2020 to 2050 and from 2050 to 2080, highlight the predicted learning of climate change due to the progressive processing of observations and underscore the potential benefits of flexible adaptation strategies.