1 

### Habitat Equivalency Analysis: An Overview Damage Assessment and Restoration Program National Oceanic and Atmospheric Administration Department of Commerce
March 21, 1995 (Revised October 4, 2000 and May 23, 2006)

1. Introduction 1.1 Goals of the paper Natural resource trustees are authorized to act on behalf of the public to protect the resources of the nation’s environment. Serving as a trustee for coastal and marine resources, NOAA determines the damage claims to be filed against parties responsible for injuries to natural resources resulting from discharges of oil, releases of hazardous substances, or physical injury such as vessel groundings. 1 Habitat equivalency analysis (HEA) is a methodology used to determine compensation for such resource injuries. The principal concept underlying the method is that the public can be compensated for past losses of habitat resources through habitat replacement projects providing additional resources of the same type. Natural resource trustees have employed HEA for groundings, spills and hazardous waste sites. Habitats involved in these analyses include seagrasses, coral reefs, tidal wetlands, salmon streams, and estuarine soft-bottom sediments. The goals of this paper are to present an overview of HEA and illustrate the method with a simple, hypothetical example. In section 1.2 below, we outline briefly the natural resource damage context for HEA applications and the conditions for use of HEA. An example of how HEA is used to estimate the appropriate level of compensation for injuries to natural resources is presented in section 2. Appendices A through C pr esent an algebraic representation of the HEA calculations and provide detailed tables from the example.

1 The Under Secretary for Oceans and Atmosphere (NOAA Administrator) acts on behalf of the Secretary of Commerce as a Federal trustee for natural resources under the Comprehensive Environmental Response, Compensation, and Liability Act (“CERCLA”; 42 U.S.C. § 9601 et seq .), the Clean Water Act (33 U.S.C. § 1251 et seq .) , the National Marine Sanctuaries Act (16 U.S.C. § 1431 et seq .), and the 1990 Oil Pollution Act (“OPA”; 33 U.S.C. § 2701 et seq .).2 1.2 Use of HEA in natural resource damage assessments Natural resource damage claims have three basic components: (1) the cost of restoring 2 the injured resources to baseline, or “primary restoration,” (2) compensation for the interim loss of resources from the time of injury until the resources recover to baseline plus (3) the reasonable costs of performing the damage assessment. 3 Following statutory requirements, all recovered damages are used to restore, replace, rehabilitate or acquire the equivalent of the injured resources (or to cover the costs of assessments). Consequently, recoveries for interim losses are spent on “compensatory restoration” actions providing resources and services equivalent to those lost. To ensure full compensation for interim losses, the trustees determine the scale of the proposed compensatory restoration actions for which the gains provided by the actions equal the losses due to the injury. The damage claim then is the cost of implementing the selected primary and compensatory restoration actions (plus the costs of the assessment) or alternatively, the responsible parties may be allowed to implement the projects themselves, subject to performance criteria established by the trustees. To develop the restoration plan, trustees must determine and quantify injury, develop restoration alternatives that consist of primary and compensatory actions, scale restoration alternatives, and select a preferred restoration alternative. This paper examines a method for scaling restoration alternatives, HEA. 4 For compensatory restoration actions, the scaling question is: what scale of compensatory restoration action will compensate for the interim loss of natural resources and services from the time of the incident until full recovery of the resources? The scale of compensatory restoration actions is conditional upon the choice of primary restoration actions. Consequently, for each

2 Restoration refers to human actions taken after the removal of the cause of injury (e.g., after remediation of a hazardous waste site, removal of the vessel in the event ofa grounding), to return an injured resource to its preinjury conditions. We use the term in its broad sense, to encompass the statutory concepts of “restoration, rehabilitation, replacement, and/ or acquisition of the equivale nt” of the injured resources.

3 At any point in time, baseline refers to the condition of th e natural resources and services that would have existed had the incident not occurred. If the resources are not expected to recover fully, interim losses will be calculated in perpetuity.

4 This description characterizes the process outlined in the natural resource damage assessment (NRDA) regulations implementing OPA (15 CFR Part 990) and in the proposed statutory changes to the CERCLA NRDA provisions (43 CFR Part 11).3 restoration alternative under consideration, the type and scale of the primary restoration actions are
to be identified first. 5 Then the compensatory components of restoration alternatives can be scaled. The process of scaling a project involves adjusting the size of a restoration action to ensure that the present discounted value of project gains equals the present discounted value of interim losses. There are two major scaling approaches: the valuation approach and the simplified service-to-service approach, which applies under certain conditions. HEA is an example of the service-to-service approach to scaling. The implicit assumption of HEA is that the public is willing to accept a one-to-one trade-off between a unit of lost habitat services and a unit of restoration project services (i.e. the public equally values a unit of services at the injury site and the restoration site). 6 HEA does not necessarily assume a oneto-one trade-off in resources, but instead in the services they provide. Consider a marsh as the resource and primary productivity a resource service. Suppose the replacement project provides only 50 percent of the productivity per acre of marsh as the injured site would have provided, but-for the injury. In order to restore the equivalent of lost productivity per year, then, the replacement project requires twice as many acres of marsh. Habitat equivalency analysis is applicable so long as the services provided are comparable. The assumption of comparable services between the lost and restored habitats may be met when, in the judgment of the trustees, the proposed restoration action provides services of the same type and quality, and of comparable value as those lost due to injury. In this context, there is a one-to-one tradeoff between the resource services at the compensatory restoration site and the injury site. Therefore, the scaling analysis simplifies to determining the scale of a restoration action that provides a quantity of discounted replacement services equal to the quantity of discounted services lost due to the injury. In cases where services at the compensatory restoration site are not of the same type and quality or of comparable value to those injured, then the assumption of a one-to-one trade-off between the resources at the injury site and the compensatory restoration site may be

5 This includes identifying the recovery trajectory from primary restoration.

6 The concept of services refers to functions a resource se rves for other resources and for humans. For example, a wetland habitat may provide on-site ecol ogical services such as faunal food a nd shelter, sediment stabilization, nutrient cycling, and primary production. Off-site services may include commercial and/or recreational fishing, bird watching along the flyway, wate r quality improvements due to on-site wate r filtration, and storm protection for on-4 inappropriate. In these cases, NOAA recommends that trustees evaluate whether the conditions for HEA are met and consider using the valuation approach as an alternative to determining the trade-off between injuries and compensatory restoration actions. Necessary conditions for the applicability of HEA include that (1) a common metric (or indicator) can be defined for natural resource services that captures the level of services provided by the habitats and captures any significant differences in the quantities and qualities of services provided by injury and replacement habitats, and (2) the changes in resources and services (due to the injury and the replacement project) are sufficiently small that the value per unit of service is independent of the changes in service levels. 7 When choosing a metric to evaluate the quantity and quality of services provided per unit of habitat, the trustees should examine the capacity , opportunity and the payoff ( i.e. benefits) of the services being provided as well as equity issues involved with the potential compensation projects ( i.e. who loses and who gains as a result of the injury and the potential compensation projects). On-site biophysical characteristics (e.g., soil, vegetative cover, and hydrology) affect the capacity of an ecosystem to provide ecological and human services. Landscape context affects whether the ecosystem will have the opportunity to supply many of the ecological and human services and strongly influences whether humans will value the opportunities for services. 8 Consider, for example, the wetland function of sediment trapping. A wetland’s capacity to provide this function depends on such factors as slope and vegetative cover. The opportunity for the wetland to trap sediments depends on the expected flow of sediments from adjacent land, which will depend upon types of upland land uses (i.e., landscape context). The total value generated from water quality improvements due to sediment trapping will depend upon the uses

shore properties due to the creation of wave breaks. Hu man services include both use and non-use services, so the HEA approach measures and accounts for non-use services in the damage claim. 7 A counterexample shows when this condition is not satis fied. Consider the value of harvesting another salmon when salmon are in abundant supply versus the value of a nother salmon when the harvest has failed in Alaska. The value of providing another pound of salmon may be substantia lly greater when the salmon are in scarce supply, all else equal.

8 For a further discussion of these issues, see, Scaling Compensatory Restoration Actions, Guidance Document for Natural Resource Damage Assessment Under the Oil Pollution Act of 1990, National Oceanic and Atmospheric Administration, Damage Assessment and Restoration Program, 1997 and King, Dennis M., Comparing Ecosystem Services and Values , Report prepared for the National Oceanic and Atmospheric Admi nistration, Damage Assessment and Restoration Program, January 1997.5 of the affected downstream water bodies: the valu e will be greater if there are nearby shellfish beds and finfish spawning areas than if the water flows into a fast-moving river. The choice of a metric to characterize services is essential to determining whether HEA is applicable in a given context. On-site ecological attributes, such as stem density, canopy structure (density multiplied by height), or fish density, are sometimes used as a proxy for services; however, they are primarily indicators of capacity . It is critical to evaluate the role of landscape context to evaluate the opportunity to provide off-site, as well as on-site, ecological
and human services.

2. Habitat Equivalency Analysis: An Example In this section we provide a simplified example to illustrate the method. To complement the example, we provide the algebraic formula for solving an HEA in Appendix A. We construct the following hypothetical scenario. 9 A heavy fuel oil released from a grounded tanker covered 20 acres of marsh composed primarily of smooth cordgrass ( Spartina alterniflora ) in 2000. The oil smothered significant portions of the marsh, penetrating the sediments in many areas and killing much of the biota. This injury impairs the function of the marsh habitat; the marsh provides food and shelter for animals, water quality improvements for downstream resources, shoreline stabilization and other natural resource services. In addition, the loss of marsh affects human services. For example, marsh habitat supports off-site human services through the production of fish that provide recreational and commercial services and through nutrient filtration that provides water quality enhancement. Trustees identified a feasible restoration action for compensation: transplanting Spartina alterniflora at the injury site for primary restoration and transplanting Spartina alterniflora along with some minor regrading at a nearby site. The projects are expected to restore the same type and quality of resources and services. Further, given the similar landscape context of the injury and restoration sites, the trustees judged the projects would restore resources and services of comparable value as those lost.

9 The size and the description of the hypothetical injury ar e not based on actual events and have been chosen simply to demonstrate the HEA calculation.6 3. Under these conditions, HEA applies as a framework for scaling compensatory restoration. The basic steps for implementation include: 1. Document and estimate the duration and extent of injury, from the time of injury until the resource recovers to baseline, or possibly to a maximum level below baseline; 2. Document and estimate the services provided by the compensatory project, over the full life of the habitat; Calculate the size of the replacement project for which the total increase in services provided by the replacement project equals the total interim loss of services due to the injury; and 4. Calculate the costs of the replacement project, or specify the performance standards in cases where the responsible party will be implementing the compensatory habitat project.

In the first two steps, trustees must specify numerical values for ecological parameters for both the injured site and the compensatory project site. For each point in time at both sites, the level of services must be characterized as a percent of the baseline level of services at the injured site. As previously noted, the baseline of services is the level of services that would have been provided at the injured site but for the injury. In our example, we assume that local experts consider grass shrimp ( Palaemonetes pugio ) to be a very important (or key) species in this habitat and they believe that the presence of grass shrimp is highly correlated with many services provided by the marsh. The presence and density of grass shrimp may indicate the general health of the marsh vegetation and the availability of food for higher trophic levels. Therefore, we assume that service levels for the injured site and for the compensatory project site are a function of the baseline mean density of grass shrimp in the marsh. Studies indicate that the spill reduced the mean density of grass shrimp by approximately 50%. Using the mean density of grass shrimp as a metric for marsh services, we assume that the service level of the injured marsh prior to any restoration actions is 50% of its baseline service level. 10

10 Depending on the exact nature and extent of an injury, th e mean density of grass shrimp relative to the baseline density may or may not serve as a good metric for the services provided by the ma rsh. Additional potential indicators of marsh services might include macrofaunal abundance, fish utili zation, vegetative density and percent vegetative cover.7 In step three, we calculate the size of the compensatory project for which the total increase in services provided by the replacement project just equals the total interim loss of services due to the injury. Because losses and gains are occurring in different years, we discount the losses and gains so that units reflect what they are worth in the present year, 2000. This makes units from different time periods comparable. The discount rate incorporates the standard economic assumptions that people place a greater value on having resources available in the present than on having their availability delayed until the future. This process is analogous to financial calculations where, if a dollar is put into the bank today at 3% interest, there will be $1.03 in one year. A person is willing to deposit money in such an interest bearing account only if having $1.03 is (at least) as good as having $1 today. There are a variety of discounting approaches, but mean accounting was applied in the example in this document. Mean accounting involves taking the arithmetic mean of service levels at the beginning and end of each period and crediting that resulting service level as accruing at the midpoint of the period. The annual discount rate used in a HEA calculation represents the public’s preference towards having a restoration project in the present year, rather than waiting until next year. The economics literature supports a discount rate of approximately 3%. 11 We list below the parameters necessary to complete a simple HEA.

Injured Area Parameters:

• Baseline level of services at the injury site;

• Extent and nature of the injury: the spatial extent of injury (in acres for example) and the initial reduction in service level from baseline at the injured site (characterized as a percent of the baseline level of services). These parameters may be combined to measure the “service-acres” of an injury; 12

11 For a further discussion of discounting see: National Oceanic and Atmospheric Ad ministration (1999) Discounting and the Treatment of Uncertainty in Natural Resource Damage Assessment . Damage Assessment and Restoration Program, Damage Assessment Center, Resource Valuation Branch. Technical Paper 99-1. Silver Spring, MD, February.

12 Service-acres may be illustrated with an example. If 30% services remain on an injured 100 acre site, the injury totals 70 service-acres (100 * (1-0.3) = 70). Note that the percent is represented by its decimal equivalent.8 • Injury recovery function (with primary restoration or natural recovery): the rate of (incremental) service recovery and the maximum level of services to be achieved (characterized as a percent of the baseline level of services);

• Recovery period for injured resources: the dates when recovery starts and when
maximum level of services will be achieved.

Replacement Area Parameters:

• Initial level of services at the replacement project site, measured as a percent of baseline services at injury site;

• Replacement project maturity function: the rate of (incremental) service growth and the maximum level of services at the replacement project site (as a percent of the baseline level of services at injury site);

• Maturity period for replacement resources: the dates when services begin to increase and when the maximum level of services will be achieved;

• Replacement/creation project duration: lifetime of increased services.

Discount Rate

• Annual real discount rate

In the following section, we walk through the each of the steps and show how ecological parameters are developed from the injury and how the HEA equation is solved.

Step 1: Quantifying the losses from the injury. For our example, parameter values characterizing the injury are listed in the table below. As shown, we denote the injury to 20 acres of marsh function by specifying that, after injury, 20 acres provide 50% of the services relative to baseline at the time of the injury (2000). The site is projected to maintain a 50% service level until the primary restoration project (transplanting Spartina alterniflora at the9 injury site) is completed in 2001. The injured area is then projected to recover in eight years following a linear growth path to baseline. 13

Table 1: Injury Parameter Values Baseline Information of the Injured Resource: Habitat type injured: Marsh Year of injury 2000 # of injured acres: 20 Level of services in injury year (relative to baseline services): 50%

Recovery of Injured Habitat following Primary Restoration: Year restoration project ends and recovery starts: 2002 Years until full recovery: 8 Services at maximum recovery (relative to baseline): 100% Shape of recovery function: Linear
Discount Rate:
Real annual discount rate 3.0%

The recovery of services provided by the injured habitat is illustrated in Figure 1. On thevertical axis is the level of services provided by the injured resource, measured in “serviceacres”. The service-acres for a given year are calculated as the product of the percent of baseline marsh services provided by an acre of the injured site multiplied by the number of acres injured. 14 When the injury occurs, in year 2000, the number of service-acres drops from 20 to 10, because 50% services remain at the site. Services increase along a linear path beginning in 2002, until full recovery to the baseline at the end of 2009. Interim losses are represented in the diagram by the area labeled “L”.

13 The length and shape of the recovery function are chosen in order to simplify the presentation. An alternative recovery function, such as a constant growth rate or other non-linear growth path, and an alternative length of recovery, could be chosen if app licable to the injured resource.

14 In the multiplication, the percent is represented by the decimal equivalent, so the baseline level of acres is (1.00*20)=20. In 2005, the site is projected to operate at 75% of baseline, so the effective service level isFigure 1:

Time Resource Services

Baseline Services (if injury had not occurred)

Full Recovery Start of Recovery 

## L

2000 Oill Spill 2002 2009

Resource Service Levels at the Injury Site

Recovery Function

To calculate the measure of interim loss in present value terms, we must apply the yearly discount factor to the losses in each year. We calculate an interim loss of 55.106 discounted service-acre-years by summing over all years of the injury. Appendix B presents the specific steps for calculating the discounted interim loss in services.

Step 2: Quantifying the gains from the habitat replacement project. The parameters characterizing the habitat creation project are listed in the table below. Prior to the compensation project, the nearby site offers 25% marsh services relative to the pre-injured marsh site. Service flows from compensation project commence when the project is completed in 2002. We project that marsh services increase during a 10-year growth period along a linear path and reach a maximum service level equal to 100% of the baseline service level of the injured site. We further project that the site will continue to function at the maximum service level in perpetuity.

10 ((1-.75) * 20) = 5.11 Table 2. Replacement Project Parameters Replacement Project Characteristics: Replacement habitat type: Marsh
Initial level of services 25% Year creation/replacement project starts 2001 Year services start increasing 2002 Year in which maximum service level is reached (end of period) 2011
Maximum service level 100% Shape of recovery function Linear Expected length of service increase Infinity Replacement Project Comparison Parameter: Ratio of maximum services per acr e at the compensatory site and the baseline services per acre at injured habitat. 1:1

The increase of services at the habitat creation site is illustrated in Figure 2. The vertical axis measures the services per acre of a replacement project as a percent of the baseline services per acre at the injured site . As shown, services begin at 25% and start increasing in 2002, following a linear path until the services reach full maturity in 2011. The services continue to function at the maximum level in perpetuity. The total increase or gain in services per acre, is shown as area “G”, which is the area between the maturity function and the 25% service level.Figure 2:



# G

Time Resource Service per Acre

Maximum service level of Created Habitat

Grounding Compensatory Restoration Services Mature 2000 2002 2011 Compensatory Restoration Services Begin (relative to the injury site baseline) Resource Service Levels at Replacement Project Site

Maturity Function 100%

25%

To calculate service gains in the present value terms, we must apply the yearly discount factor to the gains in each year and sum over the lifetime of the replacement project. This calculation, presented in more detail in Appendix C, indicates that each acre of replacement project provides 21.015 discounted service-acre-years.

Step 3: Determining the Size of the Replacement Project. To determine the size of the compensatory project needed to compensate for the losses, we divide the total loss in discounted service-acre-years by the gain per acre of replacement and get 2.62 acres, as outlined in Table 3.

1213 Table 3. Determining the Size of a Project to Compensate for Interim Losses

• Injured Area = 20 acres Present discounted interim losses = 55.106 service-acre-years (See Appendix B)

• Present discounted lifetime gains per acre of replacement project = 21.015 service-acre-years per acre (See Appendix C)

• Let R = # replacement habitat acres required for compensation.

• Equating lost services and replacement project gains:
55.106 lost service-acre-years = 21.015 service-acre-years/ acre * R acres

• Solving for the size R of the replacement project yields:
R = 55.106/21.015
= 2.62 acres of replacement habitat

The top graph in Figure 3 illustrates the discounted service losses resulting from the injury and the bottom graph illustrates the discounted service gains resulting from the replacement project. At the time of the incident, 2000, service losses occur and, although recovery doesn’t start until the year restoration is completed in 2002, the value of future losses decreases in the year 2001 because the losses are discounted. The discounted losses reach zero in the year 2009, when the recovery of services at the injured site is complete. The total discounted service losses are equal to area “A” in the top graph. The replacement project begins providing service gains in the year 2002, the year the compensation project is completed. In 2011, the compensation project reaches maturity and continues providing services at the same level in perpetuity. However, the value of these services declines over time, eventually approaching a value very close to zero (the value of the service gains approaches zero asymptotically) because the value of service gains is discounted. The total discounted service gains are equal to area “B” in the bottom graph. A replacement project of 2.62 acres will provide just enough service gains to equal the service losses resulting from the injury. That is, area “B” in the bottom graph of Figure 3 is made equivalent to area “A” in the top graph.Figure 3:

Discounted Service Losses

TimeTotal Discounted AcreYears of Services Lost Due to Injury



### A

2000 Incident 2002 Begin Recovery of Injured and Comp. Site 2009 Recovery of Services at Injured Site 2011 Full Maturity of Services at Compenation Site

Time Total Discounted A cre-Years of Services Provided by Replacement Project



### B

2000 Incident 2002 Begin Recovery of Injured and 2009 Recovery of Services at Comp. Site Injured Site 2011 Full Maturity of Services at Compenation Site Discounted Service Gains

Step 4: Calculating the Cost of the Replacement Project. Step four of HEA, which would be required for any damage assessment and restoration plan regardless of the methodology used in the assessment, occurs after the trustees have calculated the scale of the project. The damages claim is based on the costs of the replacement project. 15 Categories of project costs include the following:

14 15 Again, it should be noted that the responsible parties may perform the replacement project, subject to performance criteria established by the trustees.15 • planning and design

• environmental impact assessment

• permitting

• construction

• monitoring

• mid-course corrections Some of the categories of cost can be characteri zed on a per-acre basis; others impose fixed costs (permitting). We do not calculate project costs in this example.Appendix A: Algebra of HEA

Below, we outline the generic formula employed to calculate the appropriate scale of the compensation project. We first provide the notation for the HEA calculations.

Let t refer to time (in years), where the following events occur in the identified years:
t=0 , the injury occurs
t=C , the base for discounting (when discount factor = 1.0)
t=B , the injured habitat recovers to baseline t=N, the injured resource reaches maximum service provision
t=I , compensatory project begins to provide services
t=M , compensatory project reaches full maturity
t=L , compensatory project stops yielding services

Other variables in the analysis include:

j V , the value per acre-year of the services provided by the injured habitat (without injury)

p 

### V , the value per acre-year of the services provided by the replacement habitat

x t

j , the level of services per acre provided by the injured habitat at the end of year t


### b j , the baseline (without injury) level of services per acre of the injured habitat 16



### x t

p , the level of services per acre provided by the compensatory project at the end of year t


### b p , the initial level of services per acre of the compensatory projects

r, is the discount rate for the time period
J , the number of injured acres
P , the size of the replacement project

We select a metric, x , for capturing overall level of habitat services, or habitat function, which could represent a single service flow from the resource or an index that represents a

16 We simplify the representation of the baseline to be constant through time. Seasonal or inter-annual (or other) forms of variation could be incorporated, by adding time subscripts to the baseline variable b

.

16weighted average of multiple service flows. In the chosen metric, we define: x t j as the level of services per acre provided by the injured habitat at the end of year t, and b j as the baseline level of services of the injured habitat; consequently, ( b j - x t j ) is the extent of injury in year t

. 17 Analogously, we define 

### x t p , as the level of services provided by the replacement habitat at the end of year t, and b p as the initial level of services of the replacement habitat, prior to any enhancement activities; consequently, ( 

### x t p - b p ) represents the increment in resource services provided by the replacement project - which is the relevant measure for our analysis. In our discussion in the text in the body of this paper, however, we referred to habitat services as a percent of the baseline level of services of the injured habitat, b j; in this format, ( b j - x t j )/ b j represents the percent reduction in services per acre at the injured site from the injured site baseline, and ( 

### x t p - b p ) / b j represents the percent increase in services per acre, relative to the injured site baseline, for the replacement site. To translate the quantity in year t

into its appropriate value in the year of the claim, C , we apply the discount factor based upon the annual discount rate, r . Finally, the number of injured
acres is J. The goal of the habitat equivalency analysis is to solve for the size of the replacement project, P. The equation equating the sum of the present discounted value of the services lost at the injured site with the sum of the present disc ounted value of the services provided at the replacement site is: 

## ( ) 

# ( ) ( ) 

## ∑

+

= +− +=

− − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

− + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +− + 1

0 )1( 1 1 1* 1 * 5.0 *1** N

t NC j j Nt j

j j t j t j tC j r r b xb b xxb r VJ



## ( ) 

# ( ) ( ) 

## ∑

+

= +− +=

− − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

− + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −+ + 1 )1( 1 1 1* 1 * 5.0 *1** M

It MC j p p Mt j pp

t p t tC p r r b bx b bxx r VP

Under the assumption that the per unit value of replacement habitat services, , is equal to the
per unit value of injury habitat services, , the calculation to solve for the size of the p V

j V

17 17 More precise estimates of the level of discounted serv ice flows could be obtained by using smaller time periods (e.g. semi-annual or monthly). If smaller time periods are used the discount rate should be adjusted to keep the annual discount rate unchanged.replacement project is simplified because the term 1 =

p j V V . The equation to solve for the amount

of compensatory restoration required is:



## ( ) 

# ( ) ( )
( ) 

# ( ) ( ) 

# ∑ ∑

+

= +− +=

− − +

= +− +=

− −

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

− + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −+ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

− + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +− +

= 1 )1( 1 1 1

0 )1( 1 1

1* 1 * 5.0 *1 1* 1 * 5.0 *1

** M

It MC j p p Mt j pp

t p t tC N

t NC j j Nt j

j j t j t j tC

p j

r r b bx b bxx r r r b xb b xxb r

V V JP

The ratio of

p j V V is greater than one if the per unit value of the injured services is greater than the

per unit value of the replacement services. S ubsequently, more of the replacement project habitat would be needed than if the per unit values were equal. Less of the replacement project habitat would be needed if the per unit value of th e injury habitat is less than the per unit value of the replacement habitat.

1819 Appendix B: Interim Losses from a Marsh Oiling

Table B1 documents the injury and recovery of services on an annual basis and presents
the sum of total discounted service-acre-years lost . There is a unique row for each year between
the time of injury and the time when the resource returns to baseline conditions, and these rows
are designated in Column 1 by year. Column 2 provides some descriptive information on the
status of the resource. Columns 3 and 4 provide information about the percent service loss at the
beginning and end of each period, respectively. Note habitat services grow for eight years
following a linear recovery path, starting in 2002. The beginning of period service loss in each
period is equal to the end of period service loss in the previous period, except for the year of
initial injury. The beginning percent service loss in the first period is equal to the service loss
experienced by the resource immediately following the injury. The end of period service loss,
Column 4, declines as the resource recovers toward baseline conditions. See the algebraic
notations that follow Table 1 for the precise calculation used in Column 4. Column 5 is the
arithmetic mean percent service loss experienced over the period, and is accurate if the overall
recovery function is linear or can be reasonably approximated as linear within each period.
Column 6 is the number of service-acre-years lost in each period, and is the product of the mean
percent service loss and the area of injury. Column 7 is the discount factor, which is multiplied
by the number of service-acre-years lost to yi eld the discounted service-acre-years lost in

Column 8.2012345678PeriodProjectStatus% Service LossService-Acre-Years LostDiscountFactorDiscounted Service-Acre-Years LostBeginningof PeriodEndof PeriodMean200050.00%50.00%50.00%10.0001.00010.0002001PrimaryRestoration50.00%50.00%50.00%10.0000.9719.7092002RecoveryBegins50.00%43.75%46.875%9.3750.9438.837200343.75%37.50%40.625%8.1250.9157.436200437.50%31.25%34.375%6.8750.8886.108200531.25%25.00%28.125%5.6250.8634.852200625.00%18.75%21.875%4.3750.8373.664200718.75%12.50%15.625%3.1250.8132.541200812.50%6.25%9.375%1.8750.7891.4802009RecoveryComplete6.25%0.00%3.125%0.6250.7660.47920100.00%0.00%0.000%0.0000.7440.00020110.00%0.00%0.000%0.0000.7220.000Total Discounted Service-Acre-Years Lost =55.106Table B1. Interim marsh service loss calculations.Algebraic notation for Table B1 calculations:

Column 3: = j j t j b xb 1

− − , except for t=0 which = initial loss after injury

Column 4: = j j t j b xb

−

Column 5: = 

# ( )

j j t j t j b xxb +−

− 1 5.0 OR 0.5 * ( Column 3 + Column 4)

Column 6: = 

# ( )

j j t j t j b xxb J +−

− 1 5.0 * O R J * Column 5

Column 7: = 

## ( ) Ct r − + 1 1

Column 8: = 

# ( ) ( ) Ct j j t j t j

r b xxb J

− − + +− 1 1 * 5.0 * 1 OR Column 6 * Column 7

2122 Appendix C: Service Gains from Compensatory Restoration Project

In Table C1, the increase in services of the compensatory habitat is calculated per acre of replacement project. The first five columns in Table C1 contain information analogous to that in Columns 1 through 5 of Table B1. As the benefits of compensatory restoration are always quantified per unit area (acres in this example), Table C1 does not contain a column similar to the service-acre-years lost. Instead, the annual discounted service-acre-years of gains per acre of compensatory restoration (Column G) are derived by multiplying the mean percent service level (Column E) by the discount factor (Column F). At the bottom of the table, the total discounted service-acre-years per acre are summed.23ABCDEFGPeriodProject Status% Service GainDiscountFactorDiscounted Service-Acre-Years Gained perAcre of RestorationBeginningof PeriodEndof PeriodMean20000.0%0.0%0.00%1.0000.0002001ReplacementProject Begins0.0%0.0%0.00%0.9710.0002002Service IncreaseBegins0.0%7.5%3.75%0.9430.03520037.5%15.0%11.25%0.9150.103200415.0%22.5%18.75%0.8880.167200522.5%30.0%26.25%0.8630.226200630.0%37.5%33.75%0.8370.283200737.5%45.0%41.25%0.8130.335200845.0%52.5%48.75%0.7890.385200952.5%60.0%56.25%0.7660.431201060.0%67.5%63.75%0.7440.4742011Services ReachMaximum67.5%75.0%71.25%0.7220.515Beyond201175.0%75.0%75.00%--18.061Total Discounted Service-Acre-Years Gained per Acre of Restoration =21.015Table C1. Calculation of marsh service provision gains from compensatory restoration.Algebraic notation for Table C1 calculations:

Column C: = j pp t b bx

−

− 1

Column D: = j pp t b bx

−

Column E: = 

# ( )

j pp

t p t b bxx −+

− 1 5.0 OR 0.5 * ( Column 3 + Column 4)

Column F: = 

## ( ) Ct r − + 1 1

Column G: = 

# ( ) ( ) Ct j pp

t p t
r b bxx

−

− + −+ 1 1 * 5.0 1 OR Column 5 * Column 6

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