# LCA calculations¶

## Normal static LCA¶

The actual LCA class (bw2calc.LCA) is more of a coordinator then an accountant, as the matrix builder is doing much of the data manipulation. The Static life cycle assessment class only has to do the following:

• Translate the functional unit into a demand array
• Find the right parameter arrays, and ask matrix builder for matrices
• Solve the linear system $$Ax=B$$ using SuperLU or UMFpack.
• Multiply the result by the LCIA CFs, if a LCIA method is present

Note

Due to licensing conflicts, recent versions of SciPy do not include UMFpack. UMFpack is faster than SuperLU, especially for repeated calculations. Python wrappers for UMFpack must be installed separately using scikits.umfpack.

The LCA class also has some convenience functions for redoing some calculations with slight changes, e.g. for uncertainty and sensitivity analysis. See the “redo_*” and “rebuild_*” methods in the LCA class.

## Specifying a functional unit¶

The functional unit for any LCA calculation is a dictionary of keys and amounts:

{
("a database", "pi"): 3.14159265358979
}


However, you can also use a Activity proxy:

In [1]: from brightway2 import *

In [2]: activity = Database("ecoinvent 3.2 cutoff").random()

In [3]: type(activity), activity
Out[3]:
(bw2data.backends.peewee.proxies.Activity,
'quicklime production, milled, packed' (kilogram, CH, None))

In [4]: lca = LCA({activity: 1})

In [5]: lca.demand
Out[5]: {'quicklime production, milled, packed' (kilogram, CH, None): 1}


How does this work? It is quite simple - the Activity proxy knows how to pretend to be a key tuple:

In [7]: activity[0], activity[1]
Out[7]: ('ecoinvent 3.2 cutoff', 'ab2f7a551a06a59de9191065128233e4')

In [8]: activity == ('ecoinvent 3.2 cutoff', 'ab2f7a551a06a59de9191065128233e4')
Out[8]: True


This is an instance of duck typing - if it walks like a duck and quacks like a duck, then we can treat it like a duck.

If you are interested in the details, see how bw2data.proxies.ActivityProxyBase defines __getitem__ and other __ magic methods.

## Turning processed data arrays in matrices¶

A parameter array is a NumPy structured or record array, where each column has a label and data type. Here is an sample of the parameter array for the US LCI:

input output row col type amount
9829 9829 4294967295 4294967295 0 1.0
9708 9708 4294967295 4294967295 0 1.0
9633 9633 4294967295 4294967295 0 1.0
9276 9276 4294967295 4294967295 0 3.0999
8778 8778 4294967295 4294967295 0 1.0
9349 9349 4294967295 4294967295 0 1000.0
5685 9349 4294967295 4294967295 2 14.895
9516 9349 4294967295 4294967295 1 1032.7
9433 9349 4294967295 4294967295 1 4.4287
8838 9349 4294967295 4294967295 1 1.5490

There are also some columns for uncertainty information, but these would only be a distraction for now. The complete spec for the uncertainty fields is given in the stats_arrays documentation.

We notice several things:

• Both the input and output columns have numbers, but we don’t know what they mean yet
• Both the row and col columns are filled with a large number
• The type column has only a few values, but they are also mysterious
• The amount column is the only one that seems reasonable, and gives the values that should be inserted into the matrix

### Input and Output¶

The input and output columns gives values for biosphere flows or transforming activity data sets. The mapping is used to translate keys like ("Douglas Adams", 42) into integer values. So, each mapping number uniquely identifies an activity dataset.

If the input and output values are the same, then this is a production exchange - it describes how much product is produced by the transforming activity dataset.

Warning

Integer mapping ids are not transferable from machine to machine or installation to installation, as the order of insertion (and hence the integer id) is more or less at random. Always .process() datasets on a new machine.

### Rows and columns¶

The row and col columns have the data type unsigned integer, 32 bit, and the maximum value is therefore $$2^{32} - 1$$, i.e. 4294967295. This is just a dummy value telling Brightway2 to insert better data.

The method MatrixBuilder.build_dictionary is used to take input and output values, respectively, and figure out which rows and columns they correspond to. The actual code is succinct - only one line - but what it does is:

1. Get all unique values, as each value will appear multiple times
2. Sort these values
3. Give them integer indices, starting with zero

For our example parameter array, the dictionary from input values to row would be:

{5685: 0,
8778: 1,
8838: 2,
9276: 3,
9349: 4,
9433: 5,
9516: 6,
9633: 7,
9708: 8,
9829: 9}


And the dictionary from output to col would be:

{8778: 0,
9276: 1,
9349: 2,
9633: 3,
9708: 4,
9829: 5}


The method MatrixBuilder.add_matrix_indices would replace the 4294967295 values with dictionary values based on input and output. At this point, we have enough to build a sparse matrix using MatrixBuilder.build_matrix:

row col amount
9 5 1.0
8 4 1.0
7 3 1.0
3 1 3.0999
1 0 1.0
4 2 1000.0
0 2 14.895
6 2 1032.7
5 2 4.4287
2 2 1.5490

Indeed, the coordinate (coo) matrix takes as inputs exactly the row and column indices, and the values to insert.

Of course, there are some details for specific matrices - technosphere matrices need to be square, and should have ones by default on the diagonal, etc. etc., but this is the general idea.

### Types¶

The type column indicates whether a value should be in the technosphere or biosphere matrix: 0 is a transforming activity production amount, 1 is a technosphere exchange, and 2 is a biosphere exchange.

## Stochastic LCA¶

The various stochastic Monte Carlo LCA classes function almost the same as the static LCA, and reuse most of the code. The only change is that instead of building matrices once, random number generators from stats_arrays are instantiated directly from each parameter array. For each Monte Carlo iteration, the amount column is then overwritten with the output from the random number generator, and the system solved as normal. The code to do a new Monte Carlo iteration is quite succinct:

def next(self):
self.rebuild_technosphere_matrix(self.tech_rng.next())
self.rebuild_biosphere_matrix(self.bio_rng.next())
if self.lcia:
self.rebuild_characterization_matrix(self.cf_rng.next())

self.lci_calculation()

if self.lcia:
self.lcia_calculation()
return self.score
else:
return self.supply_array


This design is one of the most elegant parts of Brightway2.

Because there is a common procedure to build static and stochastic matrices, any matrix can easily support uncertainty, e.g. not just LCIA characterization factors, but also weighting, normalization, and anything else you can think of; see Defining a new Matrix - example of Weighting and Normalization matrices.

## Brightway2 LCA Reports¶

Note

The Brightway2 report data format is evolving, and this section should not be understood as definitive.

LCA reports calculated with bw2analyzer.report.SerializedLCAReport are written as a JSON file to disk. It has the following data format:

{
"monte carlo": {
"statistics": {
"interval": [lower, upper values],
"median": median,
"mean": mean
},
"smoothed": [  # This is smoothed values for drawing empirical PDF
[x, y],
],
"histogram": [  # This are point coordinates for each point when drawing histogram bins
[x, y],
]
},
"score": LCA score,
"activity": [
[name, amount, unit],
],
"contribution": {
"hinton": {
"xlabels": [
label,
],
"ylabels": [
label,
],
"total": LCA score,
"results": [
[x index, y index, score], # See hinton JS implementation in bw2ui source code
],
},
"treemap": {
"size:" LCA score,
"name": "LCA result",
"children": [
{
"name": activity name,
"size": activity LCA score
},
]
}
"herfindahl": herfindahl score,
"concentration": concentration score
},
"method": {
"name": method name,
"unit": method unit
},
"version": report data format version number (this is 1),
"type": "Brightway2 serialized LCA report",
"uuid": the UUID of this report,
"online": URL where this report can be accessed. Optional.
}
}


## Graph traversal¶

To generate graphs of impact like supply chain or Sankey diagrams, we need to traverse the graph of the supply chain. The GraphTraversal class does this in a relatively intelligent way, assessing each inventory activity only once regardless of how many times it is used, and prioritizing activities based on their LCA score. It is usually possible to create a reduced graph of the supply chain, with only the most relevant pathways and flows included, in a few seconds.

### Illustration of graph traversal¶

It’s easiest to understand how graph traversal is implemented with a simple example. Take this system:

• This system has four nodes, which are LCI processes, also called transforming activities. Each node has one reference product, and a set of zero or more technosphere inputs. By convention, node A produces one unit of product A.
• This system has four edges which define the inputs of each node. An edge has a start, an end, and an amount.
• We consider solving this system for a functional unit of one unit of A.

As we traverse this supply chain, we will keep different data for the nodes and the edges. For nodes, we are interested in the following:

• amount: The total amount of this node needed to produce the functional unit.
• cum: The cumulative LCA impact score attributable to the needed amount of this node, including its specific supply chain.
• ind: The individual LCA impact score directly attributable to one unit of this node, i.e. the score from the direct emissions and resource consumption of this node.

For edges, we want to know:

• to: The row index of the node consuming the product.
• from: The row index of the node producing the product.
• amount: The total amount of product from needed for the amount of to needed.
• exc_amount: The amount of from needed for one unit of to, i.e. the value given in the technosphere matrix.
• impact: The total LCA impact score embodied in this edge, i.e. the individual score of from times amount.

Our functional unit is one unit of A. Before starting any calculations, we need to set up our data structures. First, we have an empty list of edges. We also have a heap, a list which is automatically sorted, and keeps track of the nodes we need to examine. nodes are identified by their row index in the technosphere matrix. Finally, we have a dictionary of nodes, which looks up nodes by their row indices.

nodes, edges, heap = {}, [], []


We create a special node, the functional unit, and insert it into the nodes dictionary:

nodes[-1] = {
'amount': 1,
'cum': total_lca_score,
'ind': 1e-6 * total_lca_score
}


The cumulative LCA impact score is obviously the total LCA score; we set the individual LCA score to some small but non-zero value so that it isn’t deleted in graph simplification later on.

We next start building our list of edges. We start with all the inputs to the functional unit, which in this case is only one unit of A. Note that the functional unit can have multiple inputs.

for node_id, amount in functional_unit:
edges.append({
"to": -1,  # Special id of functional unit
"from": node_id,
"amount": amount,
"exc_amount": amount,
"impact": LCA(node_id, amount).score,  # Evaluate LCA impact score for node_id/amount
})


Finally, we push each node to the heap:

for node_id, amount in functional_unit:
heappush(heap, (abs(1 / LCA(node_id, amount).score), node_id))


This is not so easy to understand at first glance. What is 1 / LCA(node_id, amount).score? Why the absolute value? What is this heappush thing?

We want one divided by the LCA impact score for node A because our heap is sorted in ascending order, and we want the highest score to be first.

We take the absolute value because we are interested in the magnitude of node scores in deciding which node to process next, not the sign of the score - leaving out the absolute value would put all negative scores at the top of the heap (which is sorted in ascending order).

heappush is just a call to push something on to the heap, which is our automatically sorted list of nodes to examine.

After this first iteration, we have the following nodes and edges in our graph traversal:

nodes = {-1: {'amount': 1, 'cum': some number, 'ind': some small number}}
edges = [{
'to': -1,
'from': 0,  # Assuming A is 0
'amount': 1,
'exc_amount': 1,
'impact': some number
}]
heap = [(some number, 0)]


After this, it is rather simple: pull off the next node from the heap, add it to the list of nodes, construct its edges, and add its inputs to the heap. Iterate until no new nodes are found.

Because the heap is automatically sorted, at each iteration we will take the node with the highest impact that hasn’t yet been assessed.

There are two more things to keep in mind:

• We use a cutoff criteria to stop traversing the supply chain - any node whose cumulative LCA impact score is too small is not added to the heap.
• We only visit each node once. The is functionality in bw2analyzer to “unroll” the supply chain so that afterwards each process can occur more than once.