These authors contributed equally to this work.

The Pathfinder model was developed to fill a perceived gap within the range of existing simple climate models. Pathfinder is a compilation of existing formulations describing the climate and carbon cycle systems, chosen for their balance between mathematical simplicity and physical accuracy. The resulting model is simple enough to be used with Bayesian inference algorithms for calibration, which enables assimilation of the latest data from complex Earth system models and the IPCC sixth assessment report, as well as a yearly update based on observations of global temperature and atmospheric

Simple climate models (SCMs) typically simulate global mean temperature change caused by either atmospheric concentration changes or anthropogenic emissions of

In our recent research, we have perceived a deficiency within the existing offer of SCMs, in spite of their large and growing number

While these three requirements clearly call for the simplest model possible, as they all need a fast solving model, they also imply a certain degree of complexity. The Bayesian calibration requires an explicit representation of the processes (i.e. the variables) that are used to constrain the model. Coupling with IAMs requires accurately embedding the latest advances of climate sciences to be policy relevant

The Pathfinder model is essentially an integration of existing formulations, adapted to our modelling framework and goals. It is calibrated on Earth system models that contributed to the Coupled Model Intercomparison Project phase 6 (CMIP6), on additional data from the sixth assessment report of the IPCC (AR6), and on observations of global Earth properties up to the year 2021. The calibration philosophy of Pathfinder is to use complex models as prior information and only real-world observations and assessments combining many lines of evidence as constraints.

Compared to other SCMs

Here, we present the first public release of Pathfinder and its source code. We first provide a detailed description of the model's equations. We then describe the Bayesian setup used for calibration, the sources of prior information for it, and the resulting posterior configuration. We end with a validation of the model using standard diagnostic simulations and quantitative metrics for the climate system and carbon cycle.

An overview of Pathfinder is presented in Fig.

Pathfinder in a nutshell. Green blocks represent the carbon cycle, and red blocks represent the climate response. Blue blocks with dotted arrows are impacts that can be derived with the model. Grey blocks are variables that are directly related to anthropogenic activity. Possible inputs of the model are distinguishable through the bold contours of the blocks. In this scheme, arrows correspond to a forward mode where inputs would be

The following presents all equations of the models. Variables are noted using Roman letters and compiled in Tables

The GMST change (

The global ERF is simply the sum of the

The above energy balance model naturally provides the ocean heat content (OHC;

Global SLR has been implemented in Pathfinder as a variable of interest to model climate change impacts. In this version, it is firstly a proof of concept, modelled in a simple yet sensible manner. The total sea level rise (

The thermal expansion contribution scales linearly with the OHC

To model contributions from ice sheets and glaciers, we followed the general approach of

The contribution from glaciers is also a first-order differential equation with an equilibrium inspired by

Following

To calculate the ocean carbon sink, we use the classic mixed-layer impulse response function model from

The ocean sink model in Pathfinder follows the structure of the mixed-layer pulse response function introduced by

The non-linear carbonate chemistry in the mixed layer is emulated in two steps. First, the model's original polynomial function is used to determine the partial pressure of

The flux of carbon between the atmosphere and the ocean (

This flux of carbon entering the ocean is split between the mixed-layer carbon sub-pools, and this added carbon is subsequently transported towards the deep ocean at a rate specific to each sub-pool. This leads to the following differential equations:

While in the real world, ocean acidification is directly related to the carbonate chemistry and the ocean uptake of anthropogenic carbon, we do not have a simple formulation at our disposal that could link it to our ocean carbon cycle module. We therefore use a readily available emulation of the surface ocean acidification (

The land carbon module of Pathfinder is a simplified version of the one in OSCAR

The land sink model in Pathfinder is derived from OSCAR

The vegetation carbon pool (

NPP is expressed as its own preindustrial value multiplied by a function of

Harvesting and mortality fluxes are taken proportional to the carbon pool itself even though in reality the mortality fluxes are climate dependent. For simplicity we assume a constant mortality following the equations in OSCAR

Wildfires emissions are also assumed to be proportional to the vegetation carbon pool, but with an additional linear dependency of the emission rate on

Soil carbon is divided into three pools. The litter carbon pool (

All soil-originating fluxes are taken proportional to their pool of origin and multiplied by a function (

In addition, the function

Finally, the net carbon flux from the atmosphere to the land (

As the land carbon cycle described in the previous section does not account for permafrost carbon, we implemented this feedback using the emulator developed by

The permafrost carbon model in Pathfinder is taken from

The actual thawed fraction (

Thawed carbon is not directly emitted to the atmosphere: it is split into three thawed carbon sub-pools (

The change in atmospheric concentration of

Bayesian inference is a powerful tool for assimilating observational data into reduced-complexity models such as Pathfinder

Such a Bayesian calibration is vulnerable to the possibility that the priors draw on the same information as the constraints. However, given that Pathfinder is a patchwork of emulators whose parameters are obtained independently from one another and following differing experimental setups, we expect that the coherence of information contained within the priors and the constraints is very low. Our choice of using only complex models as prior information and only observations and assessments as constraints also aims at limiting this vulnerability.

Concretely, the posterior probability

The Pathfinder model is a set of differential equations with a number of input parameters, of which

During this Bayesian assimilation, the Pathfinder model is run solely over the historical period (from 1750 to 2021), as the constraints concern only preindustrial or historical years. For the computation, the time-differential system of Pathfinder is solved using an implicit–explicit numerical scheme (also called IMEX), with a time step of a quarter of a year. This solving scheme relies on writing the differential equations of all state variables

The Bayesian procedure itself is implemented using the Python computer language and specifically the PyMC3 package

We use a set of 19 constraints related to all aspects of the model that correspond to the set of observations

Table

Constrained variables in Pathfinder, with values before and after calibration. Variables are noted under their text notation, and Tables

To constrain the temperature response, we use the same five data sets of observed GMST as in Sect.

To further constrain the climate system, we use the mean OHU assessed by the IPCC AR6 over 2006–2018

To better align with the IPCC AR6, we also constrain the ECS of our model (i.e. the

Similarly to what is done with GMST, we constrain the atmospheric

Given its number of parameters and their inconsistent sources, we further constrain the land carbon module by considering present-day (mean over 1998–2002) NPP

To constrain the separate SLR contributions from thermal expansion, GIS, AIS, and glaciers, we use the model-based SLR speed estimates over the recent past (averaged over 2006–2018) reported in the AR6

Contrarily to all other modules, the SLR module is not assumed to start at steady state in 1750, which is represented through the

Out of the model's 77 parameters, 33 are assumed to be fixed (i.e. they are structural parameters), and the remaining

All parameters are summarized in Tables

Parameter distributions before (black lines) and after (blue lines) the Bayesian calibration. Parameters are noted under their text notation, and Tables

All the parameters of the climate module are calibrated. The prior distribution of the radiative parameter

Some parameters from the SLR module are structural: the maximum SLR contribution from glaciers (

The ocean carbon cycle module has a number of structural parameters:

In this version of Pathfinder,

Parameters related to the passive soil carbon pool are taken from

The permafrost module's parameters are recalibrated using the same algorithm as used by

The conversion factor

The structural

The following subsections discuss the adjustments between the prior and posterior parameters that are the results of the Bayesian calibration, as well as the matching of the constraints. These sections constantly refer to Fig.

Distributions of the constrained variables. Dashed lines give the distributions used to constrain. Black lines give the distribution before calibration, while blue lines give the distribution posterior to calibration. Under a variable's name, we give the period over which the constraint is estimated and the data processing method: mean over the period, difference between last and first year, or sum of all the years over the period. (1750 is the preindustrial period). Variables are noted under their text notation, and Tables

Our climate-related constraints lead to adjusting all the parameters of the climate module. As explained in Sect.

The ECS (

Among the dynamic parameters that are adjusted, we note that the deep-ocean heat capacity

Diagnostics of climate and carbon cycle responses in Pathfinder before and after Bayesian calibration. Comparison with AR6

In addition, a number of weak but physically meaningful correlations across the climate module's parameters are found, such as a positive correlation between

Similarly to GMST, the posterior distribution of atmospheric

This is confirmed for the ocean sink, as the posterior central value of

It is also confirmed that the posterior land sink is weaker than the constraint, by 15 % for the central value, which is nevertheless a significant reduction of the prior gap of 34 %. To explain this adjustment, we observe that the

Comparison of SSP scenarios for GMST change projections (w.r.t. 1850–1900) to AR6

Correlation matrix of Pathfinder's parameters after the Bayesian calibration. Parameters are classified according to the equations they are related to, i.e. climate system, sea level, ocean carbon, land carbon, and permafrost carbon. Parameters are noted under their text notation, and Tables

Historical time series of key variables from Pathfinder. Red lines are observations, black lines are the model's outputs before calibration, and blue lines are the same after calibration. Shaded areas and vertical bars correspond to the

Time series of GMST change, integrated land carbon uptake and integrated ocean carbon uptake for idealized experiments (

The land carbon module exhibits significant correlations among posterior parameters. This is likely a consequence of all the constraints combined as they dictate both the preindustrial steady state of the module and its transient response over the historical period. Eliminated configurations are those, for instance, that would show high initial carbon pools that are very sensitive to climate change (as these would lead to a very weak land sink) or that would exhibit a weak

The prior parameters of the SLR module are the least informed of our Bayesian setup. The model initially underestimates the thermal expansion, as well as the GIS and AIS SLR rates. The calibration brings the posterior distributions closer to their respective constraints but it always remains in the lower end of the uncertainty range. The correction is done by adjusting many of the module's parameters (most notably

The historical SLR is markedly corrected by the constraint: from a 19 % gap between the central values of the constraint and the prior estimate, to only 7 % after calibration. Here, we also note that the sum of individual contributions to historical SLR reported in AR6 do not match that total SLR

Because in the Bayesian setup we do not use annual time series of observations as constraints, the posterior distributions given in Fig.

To complete the diagnosis of our model with common metrics used with climate and carbon models, we ran a set of standard idealized experiments, corresponding to the CMIP6

The

Using the

To look more closely at the carbon cycle, we perform two variants of the latter experiment: in

To further validate Pathfinder, we run the five SSP scenarios

Be it on short-, mid- or long-term scales, Pathfinder's projections of GMST are very much in line with the one assessed by the IPCC in the AR6 based on multiple lines of evidence

The ocean carbon storage appears to be overestimated by 5 % to 20 % by Pathfinder across SSP scenarios. This is consistent with the upward adjustment of the ocean carbon sink stemming from our Bayesian calibration. To compare the land carbon storage with CMIP6 models, because our land carbon module does not include land use change processes, we correct the value reported by complex models by the cumulative land use change emissions of each scenario

Our SLR emulator gives estimates (Table

Comparison of SSP scenarios between Pathfinder and AR6 for SLR (with respect to 1995–2014) and SLR speed projections

In this paper, we have presented the Pathfinder model: a simple global carbon–climate model with selected impact variables, carefully designed to balance accuracy of representation and simplicity of formulation and calibrated through Bayesian inference on the latest data from Earth system models and observations. Pathfinder has been shown to perform very well in comparison to complex models, although there remains room for further improvement of the model and its calibration setup. We identify four main avenues to improve the model.

First, some parts of the model may well lean too much on the complexity side of the simplicity–accuracy balance we aimed to strike, owing to the creation process of Pathfinder that mostly compiled existing formulations. Future development should therefore strive to reduce complexity wherever possible. The ocean carbon sub-pools and perhaps the land carbon pools are potential leads in this respect.

Second, the ocean carbon module alone appears to be limited by its structure, which has been inherited from a 25-year-old (yet seminal) article

Third, integration of land use and land cover change in such a model appears warranted, despite the difficulty of doing so in a physically sensible yet simple manner. Given our expertise with the OSCAR model and its bookkeeping module

Fourth, the Bayesian setup can be extended, notably by including more time periods for the existing constraints but also by introducing and constraining entirely new variables such as isotopic ratios

Fifth, although our model is restricted to

In spite of these few shortcomings and potential development leads, Pathfinder v1.0.1 is a powerful tool that fits the niche it has been created for perfectly. We will further demonstrate the strengths and flexibility of Pathfinder in other publications. Meanwhile, we invite the community to seize this open-source model and use it in any study that could benefit from a simple, fast, and accurate carbon–climate model aligned with the latest climate science.

Pathfinder has been developed and run in Python (v3.7.6) (

The calibration procedure takes about 9 h to run on a desktop computer (with a base speed of 3.4

Two relatively benign issues that have been identified during development remain unsolved. First, the model requires a high number of sub-time steps (i.e. high

Brief description of the successive versions of Pathfinder.

Exact same physical equations and numerical values as v1.0. Added best-guess parameters calculated as the average of the posterior distribution, and corresponding historical outputs, for single-configuration runs. Improved README and MANUAL files.

First release. Exact model described in the preprint version of this paper

Calibration to estimate prior

Calibration to estimate prior

Calibration to estimate prior

Calibration to estimate prior

Calibration to estimate prior

Calibration to estimate the prior of GIS SLR module parameters (

Calibration to estimate the prior of glacier SLR module parameters (

Calibration to estimate the prior of AIS SLR module parameters (

Distribution of the logit of the ECS (

Summary of Pathfinder's equation variables in climate, sea level, and ocean carbon modules.

Summary of Pathfinder's equation variables for land carbon, permafrost, and atmospheric modules.

Parameters used in the climate, ocean carbon, and sea level modules.

Parameters used in the permafrost, land carbon modules, and for calibration.

Structural parameter values. Parameters are noted under their code notation, and Tables

Calibrated parameter values before and after Bayesian calibration. Parameters are noted under their code notation, and Tables

The source code of Pathfinder is openly available at

TG developed the model, with contributions from TB regarding sea level rise and PC regarding land carbon cycle. TG coded the model and its calibration. TB ran the diagnostic simulations and made the final figures. TB and TG wrote the manuscript with input from PC.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Andrew H. MacDougall for sharing permafrost simulations made with UVic, and Côme Cheritel for the first version of Figs. 1–4.

Development of Pathfinder was supported by the European Union's Horizon 2020 research and innovation programme under grant agreement no. 820829 (CONSTRAIN project) and by the Austrian Science Fund (FWF) under grant agreement no. P31796-N29 (ERM project).

This paper was edited by Marko Scholze and reviewed by Ian Enting and one anonymous referee.