Aligning sustainability and regional earthquake hazard mitigation planning: integrating greenhouse gas emissions and vertical equity

Concerns about the potential economic consequences of earthquakes have increased in recent years as scientifically based probabilities of future earthquakes in many large urban areas have risen. These hazards disproportionately impact low-income communities as wealth disparities limit their capacity to prepare and recover from potentially disastrous events. In addition to major economic losses, the activities related to building recovery result in significant greenhouse gas emissions contributing to climate change. This article develops a framework that quantifies the complex relationships between pre-earthquake retrofit activities and their economic, environmental and equity implications to promote informed decision-making, using the city of San Francisco, California as a case study. This research consists of two sections. In the first section, a bi-objective optimization model is proposed to identify optimal earthquake risk mitigation policies to minimize total earthquake-related economic and environmental costs, simultaneously. Decisions entail the seismic retrofit, combined seismic and energy retrofit or complete reconstruction of building-type groups. The benefits of increased energy efficiency of the upgraded buildings are incorporated to evaluate decisions from a holistic perspective. In the second section, the model is extended to address the issue of inequitable budget allocation from a public-sector perspective. Vertical equity considerations are incorporated as an optimization constraint to distribute available resources aiming to limit the discrepancy of expected losses as a fraction of income between households across income groups. The tradeoff between equity and economic efficiency is explored. Results show that life-cycle environmental impacts constitute an informative performance metric to regional risk mitigation decision-makers, in addition to the more customarily used monetary losses. Although construction costs primarily dictate optimal decisions from an economic perspective, energy considerations largely impact optimal decisions from an environmental perspective.


Introduction
Earthquakes pose an enormous and growing economic and safety risk to infrastructure systems (Porter 2021). Building-related earthquake losses in the United States are estimated to be $6.1 billion per year, of which over 61% is concentrated in California (FEMA 2017). The seismic risk in the San Francisco Bay Area, near California's San Andreas and Hayward faults, is high, with a 98% probability of a damaging earthquake (of magnitude 6 or more) in the next 30 yr (Field and WGCEP 2015). San Francisco is particularly vulnerable to large earthquake damage concentration due to its high seismic hazard and exposure of its existing building stock. Studies have commonly used San Francisco as a testbed due to its high seismicity for quantifying the seismic risk for building portfolios (Porter et al 2006, Bonstrom and Corotis 2015a, 2015b, 2016, Kotha et al 2018, Markhvida et al 2020. Earthquake impacts can be reduced through pre-earthquake retrofits, which are technical interventions in the structural systems, such as additions of new shear walls and steel bracings or column reinforcements that improve their strength and ductility. The retrofits depend on the building's structural type and vulnerability and come at a high upfront cost. Aiming to tackle those challenges, optimal regional earthquake risk mitigation planning for buildings has received increasing attention in the past decades using linear programming techniques (Shah et al 1992, Dodo et al 2005, Vaziri et al 2010, Motamed et al 2014, Zolfaghari and Peyghaleh 2015, Sadeghi et al 2017, cost-benefit assessments (Smyth et al 2004, Takahashi et al 2004, Porter et al 2006, Liel and Deierlein 2013, Kang et al 2019, Yi et al 2020 and reliability-based methods (Bonstrom and Corotis 2015a, 2015b, 2016.
The scientific literature on the allocation of earthquake risk mitigation resources has focused on minimizing economic losses (Ramirez et al 2012, Polese et al 2015, Bostenaru Dan 2018, casualties (Furukawa et al 2010, So and Spence 2013, Ceferino et al 2018 and downtime (Comerio 2006, Paul et al 2018, Molina Hutt et al 2022. However, in addition to major economic and social losses, earthquakes cause significant greenhouse gas (GHG) emissions contributing to climate change. For example, the 2011 Great East Japan Earthquake damaged 1.12 million buildings and the activities related to building recovery were estimated to sum to $122 billion, which was the equivalent of 2.2% of the gross domestic product of Japan at the time (Wei et al 2016). The same recovery work generated 26.3 million tons of GHG emissions, an amount equal to 2.1% of the total GHG emissions of Japan in 2010 (Wei et al 2016). The environmental impacts of earthquakes have been highlighted by Gonzalez et al (2022) using the 2010/2011 Canterbury Earthquake Sequence as a case study. The purpose of post-earthquake repairs is to restore the seismic performance of damaged buildings to their pre-event level. Earthquake-related GHG emissions are therefore dependent on the materials needed for the repair, or in many cases, replacement of the damaged buildings. Depending on the level of damage, the GHG emissions associated with repairs can be significant, as construction materials emit large amounts of GHG emissions from cradle-to-grave; they are responsible for 10% of the annual global GHG emissions Architecture2030 (2020). In addition to structural rehabilitation, the repair of non-structural components largely contributes to total post-earthquake GHG emissions, often being the dominant factor (Simonen et al 2018, Huang and Simonen 2020, Gonzalez et al 2022. Earthquake-induced GHG emissions can be reduced through structural retrofitting. However, the materials needed for those retrofits result in their own energy demand and embodied GHG emissions. These include GHG emissions that arise from extracting, transporting, manufacturing, and installing building materials on site. Hence, one needs to counterbalance the environmental impacts of pre-earthquake retrofits with the environmental benefits resulting from the enhanced seismic performance to identify optimal retrofit strategies from a life-cycle environmental perspective. A number of recent studies have focused on this topic through case study analyses on a building-by-building basis (Comber et al 2012, Chiu et al 2013, Wei et al 2015, 2016, Sassu et al 2017, Bostenaru Dan 2018, Gkournelos et al 2019, Caruso et al 2020, 2021b, Keskin et al 2021. These are useful to individual homeowners for improving the safety and sustainability of their properties. However, risk mitigation decisions are often made on a city-scale level by local governments aiming to proactively reduce losses in their communities. Assessing their sustainability requires the quantification of the environmental benefit potential of pre-earthquake retrofits on a regional level, involving a variety of building types and sizes. Therefore, more work is needed to holistically evaluate the environmental impacts of mitigation actions by modeling prototype buildings that represent the entire building stock on a city-scale level at which decisions are made.
The improvement in seismic performance of a building retrofit depends on the building's current condition and can be quantified as a percentage reduction in building damages. Given the high economic and environmental costs of pre-earthquake retrofitting, a question that arises is whether complete building demolition and reconstruction might be preferred. A seismic retrofit typically results in larger uncertainties in the building performance relative to a complete reconstruction, especially when evaluated across large building portfolios. Nevertheless, demolishing a building before reaching its target life results in residual economic and environmental value loss. There exists a tradeoff between retrofit and reconstruction, largely dependent on the building age and estimated lifetime. Additionally, a new building will typically be more energy-efficient as a result of energy efficiency regulations and improved design practices, providing additional benefits in terms of energy savings (Levinson 2016). According to Levinson (2016), residential buildings in California constructed after 1990 use 10%-15% less electricity and 25% less natural gas than those built before California's building codes were enacted in 1978. Similar energy-efficiency benefits can be achieved through energy upgrades in existing buildings (Less et al 2021). The reduction in both electricity and natural gas consumption due to building upgrade retrofit or reconstruction leads to considerable GHG emissions savings and is an important factor to consider in regional mitigation infrastructure planning when evaluating environmental sustainability. Building energy consumption is a major contributor to climate change, generating 28% of annual global GHG emissions. Furthermore, the potential energy savings might lead to cost savings for the impacted households. Although the benefits of integrated seismic and energy building retrofit interventions have been reported in the literature (Belleri and Marini 2016, Calvi et al 2016, Sassu et al 2017, Welsh-Huggins and Liel 2017, Lamperti Tornaghi et al 2018, Gkournelos et al 2019, Asadi et al 2020, Caruso et al 2020, 2021a, 2021b, Keskin et al 2021, Menna et al 2021, the energy savings due to energy efficiency improvements have not been considered in previous resource allocation frameworks for regional risk mitigation. Numerous regional risk assessment studies have shown that hazards disproportionately impact low-income communities (Peacock et al 2014, Bolin and Kurtz 2018, Meerow et al 2019, Markhvida et al 2020, Burton 2022. Therefore, addressing social inequity has become a highly relevant concern in applications dealing with resilient infrastructure planning and risk management (Zolfaghari and Peyghaleh 2015, Bibri and Krogstie 2017, Cariolet et al 2019, Marana et al 2019, Ribeiro and Pena Jardim Gonçalves 2019, Karakoc et al 2020, Markhvida et al 2020. The concept of equity can be divided into horizontal and vertical equity (Karakoc et al 2020). Horizontal equity assumes that people with similar needs and abilities should be treated equally (Litman 2002). On the other hand, vertical equity is defined as the unequal, but fair treatment of unequals, and can be expressed as providing each group in society a varying amount of resources that is proportional with the level of their needs and vulnerabilities (Joseph et al 2016, Karakoc et al 2020. In other words, vertical equity assumes that disadvantaged groups should receive a greater share of resources (Litman 2002). Horizontal inequity in regional seismic risk planning has been addressed by Zolfaghari and Peyghaleh (2015). The authors proposed an optimization model for the allocation of regional earthquake risk mitigation funds, incorporating inequity as an optimization constraint to model fair financial resource allocation towards seismic risk management. Their equity model is set horizontally, aiming to allocate the pre-earthquake funds such that the post-earthquake damage losses are distributed equally among different income groups. To our knowledge, vertical inequity has not been addressed in the earthquake risk mitigation literature thus far.
Building retrofitting offers opportunities for reducing earthquake-related costs and environmental impacts but requires large upfront investments in terms of materials and construction activities. This research seeks to identify optimal retrofit policies for the residential building stock at a regional scale leading to a combined reduction of economic and environmental impacts associated with earthquake damages. The option of demolition and complete building reconstruction is also considered as a means of further improving the seismic performance, relative to the retrofit option. The earthquake-induced repair activities are evaluated through seismic loss assessment, in the as-built and post-reconstruction configurations, respectively. The environmental impacts of the pre-earthquake retrofit options and post-earthquake repair activities are assessed by means of a life-cycle assessment (LCA) approach. The benefits of increased energy efficiency of the upgraded buildings are incorporated to evaluate decisions from a holistic perspective. As such, this research explores optimal earthquake retrofit policies for a hazard-prone area, the city of San Francisco, by providing a joint economic and environmental assessment of earthquake mitigation alternatives. Vertical inequity is incorporated into the optimization model to model fair allocation of pre-earthquake mitigation resources, such that expected costs for all income groups are proportional to their income.
Discussions of the environmental impact of earthquakes on buildings have recently received increased attention. Researchers have conducted several LCA studies on the effects of structural hazard vulnerability on buildings' lifetime sustainability, focusing particularly on individual buildings (Comber et al 2012, Comber and Poland 2013, Menna et al 2013, 2021 Wei et al 2015, 2016, Padgett and Li 2016, Sassu et al 2017, Simonen et al 2018, Caruso et al 2020. Nevertheless, this is the first study to identify optimal building retrofit policies from an environmental perspective at a regional scale by modeling prototype buildings that represent a large number of building types and sizes. The authors suggest that life-cycle environmental impacts constitute an informative performance metric to regional risk mitigation decision-makers, in addition to the more customarily used monetary losses. Vertical inequity is, for the first time, integrated into optimal earthquake mitigation planning, recognizing that wealth disparities limit the ability of low-income communities to adapt and recover from disastrous events.

Methods
Earthquake damages can be mitigated through retrofitting or reconstructing the existing building stock. In this research, optimal retrofit strategies are identified using optimization models that aim to minimize monetary and environmental objectives. The proposed models specify the optimal retrofit actions across the building stock by minimizing total pre-earthquake and expected post-earthquake losses over a suite of considered earthquake events. The models are developed from the public-sector perspective with a focus on regional earthquakes and address the issue of inequity in resource allocation.
The integrated economic and environmental cost serves as a holistic metric to identify optimal retrofit strategies. The bi-objective earthquake risk mitigation optimization framework is described in section 2.1. In section 2.2, the optimization model is extended to consider the benefits of increased energy efficiency of the upgraded buildings, achieved through combined seismic and energy retrofit or building reconstruction. Section 2.3 addresses vertical inequity across income groups in regional earthquake mitigation resource allocation.
The residential buildings of the city of San Francisco (138 000 buildings, 47 million sq.m. of floor space) have been selected as a case study. Figures 1(a)-(e) visualize the building type, height, seismic design level, occupancy type and soft-story condition of the buildings in the study area. The seismic design level is dependent on construction age, following the HAZards United States (HAZUS) methodology definitions (FEMA 2021). The planning horizon is 10 yr.
A probabilistic set of 300 earthquake events, in the form of spectral acceleration (SA) maps, are used as seismic inputs (Kavvada et al 2022). A SA map is a vector of SA values for different spectral periods at a set of grid locations within the study area. Each SA map is associated with an occurrence rate such that the compact set of 300 SA maps effectively represents the regional seismic hazard. SA maps are commonly employed in the literature to assess regional infrastructure damages (Jayaram and Baker 2010, Vaziri et al 2012, Christou et al 2018. Efficient representation reduces the number of required SA maps to decrease computational demands without compromising the accuracy of the estimated hazard. The set of SA maps are generated following the approach by Kavvada et al (2022). Conditional to each SA map, building performance is then quantified through the damage level, ranging from no damage to complete damage, using the HAZUS fragility curves (FEMA 2021) for the non-soft-story buildings, and the community action plan for seismic safety (CAPSS) fragility curves (SFDBI 2010) for the soft-story buildings. For details on the building stock and the seismic hazard, please consult supplementary information (SI) appendix I.

Bi-objective optimization model
In this section, a bi-objective optimization model is defined to identify optimal pre-earthquake strategies to minimize total earthquake-related economic and environmental costs, simultaneously. Buildings can either (a) remain at their current design level, (b) can be seismically retrofitted, (c) can be seismically and energy retrofitted, or (d) can be fully reconstructed. In bi-objective optimization, the optimal solution for one objective (e.g. economic) might differ from the optimal solution for the second objective (e.g. environmental). Moreover, a solution that is better with respect to one objective requires a compromise in the other objective (Deb 2011). In such cases, these problems generate a set of tradeoff optimal solutions, known as Pareto-optimal solutions (Deb 2011). A typical way to solve the problem is to use the weighted sum method, where the set of objectives are merged into a single metric by multiplying each of the objectives by a weight, forming a convex combination of objectives (de Weck and Kim 2004). The weights reveal the relative importance of each objective and are parametrically varied to obtain the Pareto front. The two objectives are normalized by the true intervals of their variation over the Pareto optimal set to account for differences in their magnitude. For details, see SI appendix II.
The bi-objective optimization model is defined as follows (de Weck and Kim 2004): where z is the decision vector denoting the building groups to be retrofitted or reconstructed, w ∈ [0, 1] is the weighting factor, and J 1 and J 2 are the two normalized single-objective functions. Each non-normalized objective function J 1 and J 2 is defined as the sum of the pre-earthquake needs and the expected post-earthquake damages over the set of earthquake events in economic and environmental terms, respectively. For details, see SI appendix III.

Construction costs
The unit costs of pre-earthquake seismic retrofits, in $/sq.m., are obtained from Fung et al (2018) through linear regression models as a function of the building characteristics including building age, height, size and structural type. The regression models of this study cannot be applied to the soft-story buildings as they do not account for their vulnerability. For the soft-story building retrofits, costs are estimated based on material quantities depending on the building size from the CAPSS study (SFDBI 2010) (see SI tables S2 and 3). In the case of building reconstruction, costs are set equal to the building residual value. The demolition of a building, before reaching its target lifetime, results in loss of residual value. This loss depends on the building age, remaining lifetime and building replacement cost (figure 2). The remaining building lifetime is estimated using California-specific depreciation curves (California State Board of Equalization 2021). The minimum remaining building lifetime is set equal to the 10 yr planning horizon of the optimization model, assuming that the existing building stock will be in place for that period. The replacement cost is dependent on the building structural type, height and size and is estimated through a prototype building approach. Prototype buildings are defined to represent the entire building stock, one for each building group. Material quantities and costs for the replacement of the prototype buildings are sourced from the RSMeans Square Foot Estimator database (RSMeans 2022). For details, see SI appendix VIII and tables S4-S28. After an earthquake, the building stock is likely to sustain varying levels of damage, calling for repair strategies to restore the seismic performance of damaged buildings to their pre-event level. Damage states can be translated into monetary losses through damage ratios as a function of the building replacement costs (see SI appendix VIII). Earthquakes are assumed to be equally likely to occur within the planning horizon. Future damage costs are discounted accordingly using a 3% rate (United States Department of Energy 2021). To account for variability across space, all costs are adjusted to local conditions using location adjustment factors from the RSMeans database (RSMeans 2022).

Embodied GHG emissions
Now we consider embodied GHG emissions. Seismic vulnerability has been reported in the literature to impact infrastructure sustainability from a life-cycle perspective (Comber et  analysis of the environmental impacts of earthquake building repairs is limited. The sustainability of buildings subject to earthquake risks can be evaluated through LCA frameworks that probabilistically consider the expected damages and the environmental impacts associated with building repairs.
The environmental assessment of building damages consists of accounting for the impacts of repair materials over their full life-cycle from cradle-to-grave, following a process-based LCA approach. With this in mind, researchers have recently conducted process-based LCA studies to assess the GHG emissions related to seismic damages, using a comprehensive inventory of building materials for both structural and non-structural components. These studies focus on case studies of a single building (Menna et  , and not on a regional scale including diverse building types and sizes. The lack of comprehensive inventories on the material quantities associated with repair strategies has led numerous researchers to rely on Economic Input-Output LCA (EIO-LCA) frameworks for the environmental assessment of building damages (Comber et al 2012, Comber and Poland 2013, Simonen et al 2015, 2018, FEMA 2018, Hasik et al 2018, Asadi et al 2019, Huang and Simonen 2020, Caruso et al 2021a, 2021b. EIO-LCA requires product or activity cost information to be used within available tools that translate industry sector-specific costs into the corresponding environmental impacts (Caruso et al 2020). EIO-LCA is a well-documented method, however, its application requires deep insights into how the model and decision-support tool (eiolca.net) could be used correctly and accurately.
EIO-LCA has been used for the assessment of buildings by either estimating the total cost of an entire building applied to a single economic sector best representing the building type (Caruso et al 2021a, 2021b), or by applying individual component costs (e.g. concrete) to more focused manufacturing sectors (e.g. the ready-mixed concrete sector) (Comber et al 2012, Comber and Poland 2013, Simonen et al 2015, 2018, Hasik et al 2018, Huang and Simonen 2020. In both cases, environmental impacts are determined as a function of building costs rather than quantities, resulting in reduced accuracy (Comber et al 2012). Adding to EIO-LCA weaknesses, EIO-LCA data are based on industry averages and are not tailored to specific products which oftentimes cannot be reliably mapped into the predefined economic sectors (FEMA 2018) (e.g. structural steel is very different from an average product from the steel sector). Furthermore, the EIO-LCA tool relies on old data, from 2002, and reflects national averages that are no longer representative (Simonen et al 2015). As a result, the data cannot be customized to reflect specific spatial and temporal settings (Hasik et al 2018). In addition, the EIO-LCA data are cradle-to-gate and thus do not include transportation, on site construction and end-of-life impacts (FEMA 2018). Hence, the accuracy of EIO-based building damage assessments is questionable. All the mentioned limitations point to the inevitable conclusion that EIO-LCA should not be used nowadays for building damage assessments such as those needed for decision-making about optimal earthquake retrofits.
In an effort to address these limitations, an alternative approach can be used for the environmental assessment of building damages. Several researchers have conducted a damage ratio approach, in which they applied economic cost ratios to relate initial environmental impacts to repair-related environmental impacts (Chiu et al 2013, Arroyo et al 2015, Feese et al 2015, Alirezaei et al 2016, Calvi et al 2016, Padgett and Li 2016, Wei et al 2016, Caruso et al 2021a, 2021b, Lanza et al 2022. In other words, the environmental impacts due to repair activities are calculated by multiplying the damage ratio, equal to the ratio of damage costs to replacement costs, with the replacement value in terms of environmental impacts (FEMA 2021, Caruso et al 2021b. A significant assumption in this approach is the hypothesis of a uniform distribution of labor and material costs from initial construction to repair stages (Caruso et al 2021b).
In this research, a hybrid approach is used to estimate the environmental impacts of building repairs, combining the damage ratio approach with process-based LCA methods. The embodied GHG emissions of the building replacement are used as a basis for the damage-related impact assessment, using damage ratios to convert from replacement to repair impacts (see SI appendix VIII). A key element in this approach is therefore the generation of a comprehensive inventory of building materials to best estimate the embodied GHG emissions. The bill of materials data for the prototype buildings are sourced from the RSMeans Square Foot Estimator database (RSMeans 2022), as in the cost estimation methodology (section 2.1.2). Embodied GHG emissions are then obtained by linking material quantities and life-cycle emission factors from environmental product declaration (EPD) reports (see SI appendix VIII and tables S4-S28).
For soft-story buildings, seismic retrofit-related environmental impacts are estimated using a process-based approach given pre-earthquake material quantities suggested by CAPSS (SFDBI 2010) and life-cycle GHG emissions from EPD reports (see SI appendix VIII and tables S2, S3). For the remaining buildings, a cost ratio approach is employed due to lack of systematic data for the material needs to retrofit the variety of building typologies observed at the city-level scale (SI appendix VIII). In the case of building reconstruction, environmental impacts are expressed as the embodied GHG emissions associated with the building residual value (figure 2).

Integrating energy efficiency
In this section, we additionally consider the energy savings as a result of the building upgrades. The improved energy performance of energy upgraded buildings has a positive effect on limiting not only costs but also GHG emissions resulting from energy consumption. These benefits are observed in the case of (a) combined seismic and energy retrofits and (b) building reconstruction. Energy retrofit costs are obtained from Less et al (2021). The single objective functions J i are modified to incorporate the energy savings in economic and environmental terms. Energy savings are estimated as a percent reduction from current building energy consumption due to electricity and natural gas use, the two most common energy sources used in residential buildings (US EIA 2022). Energy upgraded buildings are assumed to use 15% less electricity and 25% less natural gas (Levinson 2016). Due to lack of data on the actual energy consumption of the San Francisco building stock, a chained gradient boosting regression tree machine learning model ( figure 3) is employed to predict current electricity and natural gas consumption using the 2019 California Residential Appliance Saturation Study dataset (California Energy Commission 2019). Instead of training two independent models, one for electricity and one for natural gas, a chained machine learning model is used to exploit the correlation between electricity and natural gas consumption and further enhance its predictive performance. The chained model consists of two models that are trained sequentially. In the first model, the building characteristics are used as features to predict total energy consumption. In the second model, the output of the first model (total energy consumption) as well as the building features are used to predict electricity consumption. Natural gas consumption is then estimated as the difference between total energy and electricity consumption. For details, see SI appendix IV.
A 10 yr planning horizon is used as a basis to quantify the energy savings from building reconstruction as a reasonable estimate given the rapid changes in energy efficiency standards. Energy costs and GHG emissions are location-specific, dependent on the energy provider which for the case of San Francisco is Pacific Gas & Electric (PG&E). Electricity costs are set as 0.3147 $/kWh and natural gas costs as 0.0199 $/MJ (PG&E 2022a). Future costs are discounted using a 3% rate (United States Department of Energy 2021). The electricity-related GHG emissions are estimated to be 0.143 kgCO 2 e/kWh using PG&E's electricity mix (PG&E 2020) and life-cycle electricity generation emission factors (Horvath and Stokes 2011). Expected changes in the electricity mix are considered by adjusting the electricity mix for future years as suggested by (Grubert et al 2020). The natural gas emission factor is set as 0.0578 kgCO 2 e/MJ (PG&E 2022b).

Integrating vertical inequity
The spatial distribution of seismic damages is not uniform. Hence, parts of the population are disproportionately burdened economically after an earthquake. Additionally, disparities in wealth place many low-income communities at greater risk and limits their capacity to prepare and recover from a disastrous event (San Mateo County Office of Sustainability, Planning and Building 2021). Often the needs of the most vulnerable are not prioritized, nor are they included in decision-making processes (Meerow et al 2019). In this section, the regional risk mitigation optimization model described in section 2.2 is expanded to address economic inequities. Specifically, optimal mitigation decisions are based on minimizing total costs while limiting economic disparities between income groups. Available risk mitigation funds are allocated across income groups considering not only their hazard vulnerabilities but also their economic capacity to recover from an earthquake event. Rather than distributing costs equally across income groups (horizontal equity), the model addresses vertical inequity by limiting differences in the cost-to-income ratio, defined as the ratio of annualized household post-retrofit costs to income: cos t − to − income ratio = annualized household post retro f it costs annual household income where: annualized household post retro f it cos ts = annualized household earthquake damage costs

−annual household energy cost savings
Income groups are defined using median household income quartiles (United States Census Bureau 2021). Households living in the same building group are assumed to have the same income, equal to the median census tract income, and suffer the same unit damage costs ($/sq.m.). The Gini coefficient is used as a measure of inequality (Yang et al 2016): where p f and p h are the populations of income groups f and h, respectively, δ f is the cost-to-income ratio of income group f,δ is the average cost-to-income ratio across all income groups and G max represents the inequality tolerance or maximum acceptable Gini coefficient. A Gini coefficient (G) equal to 1 represents complete inequity and equal to 0 represents complete equity across income groups, schematically shown in figure 4. Note that using inequality (3) as an optimization constraint results in a non-linear program. However, the inequality can be reformulated and take the standard form for linear programs which can be solved efficiently. For details, see SI appendix VIIa.

Earthquake risk mitigation optimization framework
The bi-objective optimization model, described in section 2.1, generates a set of tradeoff optimal solutions, or Pareto-optimal solutions, illustrated in figure 5(a). The baseline solution for the case where no retrofit/reconstruction occurs (light blue marker) serves as a benchmark for comparison. The two corner solutions of the pareto-optimal front, shown in purple and orange, correspond to the two single-objective optimization models, focusing on economic and environmental costs, respectively. The resulting monetary costs and GHG emissions of the two corner solutions and their distribution between pre-and post-earthquake phases are illustrated in figures 5(b) and (c), respectively. The aggregated costs of the two corner solutions are shown on the top of figure 5(b), calculated as the sum of the costs arising from the suggested retrofit and reconstruction activities, and the expected post-earthquake damages. Similarly, figure 5(c) focuses on GHG emissions as a result of the two corner solutions. Total pre-earthquake costs are 28% lower for the embodied GHG emissions model relative to the cost model. The GHG emissions model suggests 1.2 times higher retrofit and 2.5 times lower reconstruction costs, leading to a 12% increase in total economic costs and a 6% decrease in total GHG emissions. Hence, oftentimes, from an economic point of view, building demolition and reconstruction is preferable over seismic retrofit, even though from an environmental perspective it would be better to pre-earthquake retrofit. This highlights that building retrofits are relatively costly due to high soft costs associated with non-tangible items such as design, fees, taxes, and insurance. These costs do not result in GHG emissions; therefore, the carbon footprint of building retrofits is relatively low.

Integrating energy efficiency
Next, we examine the impact of considering more energy-efficient operation when a building is energy upgraded or fully reconstructed. The pareto-optimal solutions of the bi-objective optimization model (section 2.1), that considers the energy efficiency benefits, are illustrated in figure 6(a). The resulting monetary costs and GHG emissions of the two corner solutions are shown in figures 6(b) and (c), respectively. The optimal costs model presents a 21.8% reduction in monetary costs and the optimal GHG model a 11% reduction in environmental impacts, relative to the existing building stock performance, used as a baseline. Results are overlaid on the results from section 3.1 that ignored potential energy efficiency savings, shown with gray color. The cost-based corner solution is not significantly affected by the energy cost savings; the low costs of energy lead to rather identical solutions between the two cost-based models, resulting in only a slight reduction in total costs of 1.3%. On the other hand, the energy efficiency component is critical for the GHG-based corner solution and offers an opportunity for high reductions in total GHG emissions. Focusing on the two GHG-based corner solutions, total GHG costs are reduced by 24%. Results show that energy cost savings are a small fraction (1.8%) of the post-earthquake repair costs for the optimal costs model. However, the energy-related GHG emissions savings are significant, 2.9 times as high as the embodied post-earthquake GHG emissions for the optimal GHG solution. It can be concluded that although energy costs are rather inexpensive compared to construction costs, the GHG emissions associated with energy use constitute a large portion of the life-cycle GHG emissions of buildings. The impact of energy efficiency considerations in the optimal solutions is also evident by the differences in the bar lengths between the two models, with and without energy efficiency gains ( figure 6(b)). The cost-based models do not suggest any combined seismic and energy retrofits even when considering potential energy-cost savings. The energy cost savings are not enough to justify the additional costs of the energy retrofit and as a result, the cost-based optimal decisions of the two models are quite similar. However, the integration of energy-related GHG savings emphasizes the benefit of combined seismic and energy-efficiency   retrofits. Considering potential energy savings, the GHG-based model converts all retrofits into combined seismic and energy retrofits and leads to 2.7 and 4.2 times higher retrofit and reconstruction costs, respectively. The differences between the two GHG-based solutions, with the without energy considerations, show that in many cases, the environmental benefits of reduced energy consumption outweigh the higher embodied GHG emissions of the energy retrofit or even reconstruction. Any point on pareto-front shown in figure 6(a) is considered pareto-optimal. By moving along the curve, costs are minimized at the expense of GHG emissions, or GHG emissions are minimized at the expense of costs, but the objectives cannot be improved both at once. Optimal solutions are therefore dependent on the preferences of the decision-makers, expressed as the weights assigned to each of the objectives. Previous research points out that the knee, the point with the greatest marginal utility on the pareto-front, is often attractive to decision-makers . The point refers to the solution for which a small improvement in one objective will lead to serious degradation of the other objective and can be calculated following the approach by Satopaa et al (2011). It corresponds to a middle-ground solution between the two corner solutions as illustrated in figures 6(b) and (c) and results in four-fifths of the costs and two-fourths of the GHG emissions of the baseline no-action solution. The knee point solution considering energy efficiency leads to 7.1% higher monetary costs and 45.3% lower GHG emissions relative to the knee point without energy considerations, highlighting the GHG emissions savings from the operational use phase.
The knee-point optimal solutions, describing the percentage of building group area to be upgraded, are shown in figure 7(a). The upgrade can take the form of seismic retrofit, combined seismic and energy retrofit or complete reconstruction, illustrated in orange, turquoise and purple color, respectively. The majority of upgrades are targeted towards soft-story wood-frame buildings, representing 98% of total upgraded building area. Building age is also a critical factor, with 82.5% and 14% of total upgraded area involving buildings constructed before 1950 and between 1950 and 1970, respectively. The distribution of upgrades across census tracts is illustrated in figure 7(b), where darker colors represent the census tracts that require high upfront investments. Investments are largely correlated with the concentration of soft-story wood-frame buildings constructed before 1970, as opposed to total building area density (figure S2).
Note that in this analysis future damages costs are discounted using the average real return rate, equal to 3%, consistent with other studies in the literature (Paté-Cornell 1984, Cowing et al 2004, Liel and Deierlein 2013, Valcárcel et al 2013, Khalilian et al 2021, United States Department of Energy 2021. The choice of the discount rate is a policy decision, subject to ongoing debate (Bayer 2003, Cowing et al 2004, Rackwitz et al 2005, Jawad and Ozbay 2006, Sanchez-Silva et al 2012. The sensitivity of optimal solutions to the discount rate is explored in SI appendix V using the commonly debated 2.2% and 7% discount rate values, proposed by Multi-Hazard Mitigation Council (2019) and United States Office of Management and Budget (2022).
Recognizing that policies involving building demolition and reconstruction might not unenforceable due to political challenges and other constraints, the analysis was repeated excluding building reconstruction as an option, and only considering seismic retrofits and integrated seismic and energy retrofits as alternative strategies. In that case, the integration of energy-efficiency savings has no impact on the cost-based model. The cost-based model does not suggest any combined seismic and energy retrofits as the increase in upfront retrofit costs is higher than the expected energy-cost savings. However, the GHG-based model is largely affected by the energy-efficiency considerations resulting in 5.5 times higher upfront retrofit costs, invested solely in combined seismic and energy retrofits. For details, see SI appendix VI.

Integrating vertical inequity
This section explores the equity implications of mitigation strategies by extending the model to consider the socio-economic and physical earthquake vulnerability of the population when allocating pre-earthquake mitigation resources. Analysis was performed to investigate whether low-income communities are disproportionately impacted by earthquakes. The population of the city of San Francisco was disaggregated into four income groups, defined based on median census tract household income quartiles ( figure 8(a)). The distribution of annualized household earthquake damage costs under current conditions across income groups is shown in figure 8(b). Costs are not equally distributed among income groups, heavily impacting the lower-middle and high-income groups. This metric solely focuses on the magnitude of the impact and fails to incorporate the ability of the affected households to recover after an earthquake. Annualized costs can be expressed as a fraction of the household income through the cost-to-income ratio. Figure 8(c) shows the cost-to-income ratio distribution across income groups, revealing a clear trend. Low and lower-middle income groups will be disproportionately burdened after an earthquake as expected annualized earthquake costs represent a high percentage of their income. The mean cost-to-income ratio of low-income households is significantly higher than high-income ones, about 3.4 times as high, motivating the use of vertical equity to drive budget allocation decisions for earthquake risk mitigation.
Assuming that governmental resources are available to support seismic and energy retrofits, we next seek to determine the optimal distribution of resources across households. Vertical equity considerations are incorporated in the optimization model as a constraint to limit disparities in the cost-to-income ratio between income groups. The level of inequity is defined by the maximum acceptable Gini coefficient (G max ), generating a set of pareto-optimal solutions (figure 9(a)). The red dashed line corresponds to the lowest G max for which the vertical inequity constraint is inactive. Therefore, for G max ⩾ 0.3, including the inequity constraint has no impact on the optimization solution and both the objective function cost and the pre-quake budget curves are flat. The yellow dashed line, at G max = 0.15, corresponds to the knee point solution. These two solutions, yellow and red, are selected to showcase the impact of the equity constraint on the analysis, denoting the with and without equity model, respectively. As expected, integrating inequity comes at a cost; the equity model results in a 6% increase in the objective value, representing city-wide total costs, and creates the need for a 23% higher upfront budget ( figure 9(a)). Nevertheless, the cost of equity largely depends on the desired inequity threshold, set by the decision-maker; satisfying a higher degree of equity is increasingly expensive. Decision-makers can evaluate the equity-economic efficiency tradeoff to identify optimal decisions based on their budget constraints. The model with equity considerations identifies the optimal budget allocation to constraint differences in the cost-to-income ratio across income groups, redistributing available resources from the upper-middle and high income groups to the low-income groups ( figure 9(b)). Based on the equity-constrained model, 60% and 28% of the budget is distributed to low-income and lower-middle households, respectively. In other words, the equity model allocates available resources to primarily support low-income groups. The impact of the equity constraint on the distribution of damage costs and cost-to-income ratio across income groups is shown in figures 9(c) and (d), respectively. The equity model (yellow boxplots) largely reduces post-earthquake costs for the low-income group and allows higher costs for the higher income groups, aiming to reduce differences between the cost-to-income ratio for all income groups. Note the median household cost-income ratios (orange lines). The equity constraint notably reduces the median household cost-to-income ratio for the low-income group, at the sacrifice of the upper half of the upper-middle and high-income groups. The allocation redistribution leads to a decrease of 33% in the mean cost-to-income ratio for low-income households and an increase of 18% and 25% for the upper-middle and high-income groups, respectively. Additional information on the allocation of available resources across building types and census tracts, based on the equity model, is presented in SI appendix VIIb. Note that the equity-constrained and unconstrained models serve different purposes. The equity-constrained model intends to distribute available funding across households through an equity lens. Nevertheless, households that are expected to experience high damage costs but do not receive funding can be subsequently identified through the unconstrained model.

Conclusions
In this study, the environmental impact of the building life-cycle is shown to be as a meaningful performance metric for earthquake mitigation planning. The integrated regional optimization framework expands on the traditionally used monetary objectives to achieve a combined reduction of economic and environmental impacts. It is implemented for the first time at a regional scale using prototype buildings to represent the large number of building types and sizes in the entire building stock. Results confirm the findings of previous studies (Comber et al 2012, Chiu et al 2013, Wei et al 2015, 2016, Sassu et al 2017, Bostenaru Dan 2018, Gkournelos et al 2019, Caruso et al 2020, 2021b, Keskin et al 2021 that negative environmental impacts mitigated by retrofitting can outweigh the expenditures for the retrofit itself. Hence, the significance of pre-event hazard mitigation is highlighted in both economic and environmental terms. Pre-earthquake action plans are assessed holistically considering the energy benefits as a result of the building upgrades. Even though the contribution of energy cost savings to total costs is negligible, energy considerations largely impact optimal pre-earthquake decisions from an environmental perspective. This underlines the contribution of energy use to life-cycle GHG emissions of buildings. Vertical inequity is, for the first time, incorporated into the earthquake risk mitigation optimization model to identify how to allocate limited resources considering the socio-economic and physical earthquake vulnerability. The approach expands on previous research by Zolfaghari and Peyghaleh (2015) that targeted horizontal inequity, thereby requiring post-earthquake damage costs to be equally distributed among income groups. A horizontal inequity constraint fails to incorporate the ability of the affected households to recover from disastrous events and solely focuses on the magnitude of the impact. The vertical equity considerations lead to an increase in both the objective function value, representing city-wide costs, and the required pre-earthquake budget. Decision-makers are therefore called to balance the economic-environmental and equity-efficiency tradeoffs of their policies and make optimal decisions based on their priorities and capacities, not just economic efficiency. Costs are inarguably one of the most critical factors when it comes to risk mitigation planning, however, the GHG emissions and inequity implications are important to consider in the context of broader efforts to make cities more sustainable.
The lack of material inventory data for building reconstruction and retrofit poses a major challenge in integrating the environmental impacts pre-and post-earthquake actions in regional mitigation planning. The majority of studies on the impacts of natural hazards on building sustainability are focused on single building case studies. The case study-focused work does not offer the potential to generalize to a larger scale, which is a major shortcoming identified in the current literature. Recognizing this gap, researchers have employed EIO-based LCA models for the environmental assessment of the building damages. The impacts are determined as a function of building costs rather than quantities and are therefore of insufficient quality to be used for regional mitigation planning. In this study, environmental impacts are estimated using material inventories for building reconstruction and retrofit from the RSMeans and CAPSS databases (SFDBI 2010, RSMeans 2022) and life-cycle emission factors from EPDs. The energy savings as a result of building reconstruction are assessed using a machine learning model trained on the RASS database (California Energy Commission 2019). Income information is collected on a census tract level, assuming that all households within a census tract have the same income, equal to the median census tract income. The authors recognize the accuracy limitations of the modeling approach but were limited by data availability. Nonetheless, this study aims to develop a generalizable modeling framework applicable to any location and scale, given that appropriate building and demographic data are available. Future work could extend our analysis to include additional retrofit alternatives, e.g. options for retrofit materials and retrofit levels.
The integration of economic and environmental impacts in the bi-objective optimization framework gives rise to a set of tradeoff optimal solutions; a set of nondominated solutions where neither objective can be improved without sacrificing the other. The knee point, where a small improvement in one objective leads to large decrease in the other, is selected as the optimal solution for detailed analysis. An alternative popular approach for integrating economic and environmental objectives is through the social cost of carbon which is an estimate of the economic costs of emitting one additional metric ton of carbon dioxide into the atmosphere. The current estimate of the social cost of carbon is about $60/mt CO 2 e (US EPA 2017). However, many experts agree that the estimate does not include all of the widely recognized and accepted scientific and economic impacts of climate change, and thus it is far lower than the true costs of carbon pollution (EDF 2022). Even greater uncertainties exist in the social cost of carbon (SCC) estimation of future GHG emissions, as in the case of seismic damages. Additionally, construction costs in San Francisco are rather high, about 1.4 times higher than average US prices (RSMeans 2022). These reasons led to the social cost of carbon associated with construction in San Francisco to be about 1% of the actual construction cost estimates. Hence, under current estimates, the impact of the SCC on the results of the San Francisco economic costs model is not significant. However, the large uncertainties in SCC estimation, in addition to uncertainties in the carbon footprint of repair activities, may result in a drastically different economic evaluation of the environmental damages. On the other hand, the quality of the economic data used in this study is relatively high, generating a smaller uncertainty range when it comes to building repair costs. This means that the contribution of the SCC to total economic costs can vary widely and thus, the SCC was not considered an appropriate performance metric for this study.
This work focused on the sustainability of building infrastructure systems impacted by earthquakes, addressing its three pillars: economic, environmental and social sustainability. It is widely recognized that achieving a completely seismically resistant community is cost-prohibitive (National Research Council 2011). Yet many communities seem to have an earthquake investment gap, paying billions more annually on average to recover from earthquakes than they invest to prevent losses beforehand (Porter 2021). Decision-makers have the responsibility to develop risk mitigation plans and enhance community sustainability through mitigation and pre-disaster preparation. Although, local governments have successfully used Community Development Block Grant funds to retrofit residential buildings (Association of Bay Area Governments 2016), they typically have limited funds to invest in risk mitigation at a regional scale. Nevertheless, there exist federal programs, such as FEMA's Hazard Mitigation Grant Program (FEMA 2022), that provide financial support for communities to implement mitigation activities. These programs require local governments to submit hazard mitigation plans in their application process. The suggested framework can be a valuable tool in the development of risk mitigation plans and deliver insights on how to allocate risk mitigation funds through an equity lens. Alternatively, in the absence of available funding resources, the framework can be used to identify vulnerable building types and effective risk mitigation strategies to help inform policy decisions. These decisions can take the form of mandated retrofit policies (e.g. 2013 San Francisco Mandatory Soft Story Retrofit Program and 1992 San Francisco Unreinforced Masonry Building Program) or voluntary programs providing financial incentives, in the form of subsidies or tax credits, to eligible homeowners to perform building retrofits (e.g. California Earthquake Brace and Bolt Program). A case in point for such financial incentives is the Inflation Reduction Act (2022), one goal of which is to increase energy efficiency upgrades for residential buildings (The White House 2022). Additionally, once vulnerable building types and households have been identified, model results can inform policy decisions to (a) increase property insurance penetration, which has been demonstrated to be an effective risk-reduction strategy (Markhvida et al 2020), and (b) provide insurance premium discounts to vulnerable households through an equity perspective (e.g. California Homeowners Earthquake Insurance Policies, California Earthquake Authority 2022).

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.