Aggregation of carbon dioxide sequestration storage assessment units

Abstract

The U.S. Geological Survey is currently conducting a national assessment of carbon dioxide (CO₂) storage resources, mandated by the Energy Independence and Security Act of 2007. Pre-emission capture and storage of CO₂ in subsurface saline formations is one potential method to reduce greenhouse gas emissions and the negative impact of global climate change. Like many large-scale resource assessments, the area under investigation is split into smaller, more manageable storage assessment units (SAUs), which must be aggregated with correctly propagated uncertainty to the basin, regional, and national scales. The aggregation methodology requires two types of data: marginal probability distributions of storage resource for each SAU, and a correlation matrix obtained by expert elicitation describing interdependencies between pairs of SAUs. Dependencies arise because geologic analogs, assessment methods, and assessors often overlap. The correlation matrix is used to induce rank correlation, using a Cholesky decomposition, among the empirical marginal distributions representing individually assessed SAUs. This manuscript presents a probabilistic aggregation method tailored to the correlations and dependencies inherent to a CO₂ storage assessment. Aggregation results must be presented at the basin, regional, and national scales. A single stage approach, in which one large correlation matrix is defined and subsets are used for different scales, is compared to a multiple stage approach, in which new correlation matrices are created to aggregate intermediate results. Although the single-stage approach requires determination of significantly more correlation coefficients, it captures geologic dependencies among similar units in different basins and it is less sensitive to fluctuations in low correlation coefficients than the multiple stage approach. Thus, subsets of one single-stage correlation matrix are used to aggregate to basin, regional, and national scales.

Appendices

Appendix 1: R Code description and example

The aggregation routines are written in R, which is an open source code language. The latest version may be found at www.r-project.org. The input files used in this example are assumed to be in csv format.

1.1 Introduction

There are two input data sets:

• 1. SAUs simulation runs. These are in a t × n matrix of independently generated results, where t is the number of simulations (trials) and n is the number of SAUs to be aggregated.

• 2. User specified pairwise n × n “correlation” matrix

There are three output data sets:

• 1. A t × n matrix of sample numbers needed to induce the user specified correlation. This is an intermediate data set.

• 2. A t × (n + 1) matrix where the first column is the aggregate sum of the n SAU values for the t trials. The remaining n rows are the corresponding sample numbers.

• 3. A vector of summary statistics.

There are four R-programs (functions):

• 1. matrixp.fn checks to see if the user specified pairwise correlation matrix is a symmetric, positive semidefinite and a proper correlation matrix. If it is not a proper correlation matrix, the closest correlation matrix using the Frobenius norm is computed using the method of Higham (2002). The resultant proper correlation matrix is written as a csv file.

• 2. sampnum.fn generates sample numbers using a Cholesky decomposition of the proper correlation matrix to induce the specified correlation among SAUs.

• 3. COagg.fn aggregates the simulation data using the user specified or adjusted correlation matrix.

• 4. COaggsum.fn creates summary statistics from the COagg.fn output.

1.2 Small examples

Two examples are presented. Both use three SAUs, called V1–V3. The first has a correlation matrix Mt1cor.csv, which is positive semidefinite; the second, Mt2cor.csv, is not. Both use the same data set, respectively called Mt1dat.csv and Mt2dat.csv consisting of t = 10 trials.

1.2.1 Example: Mt1

Input data files:

File: Mt1dat.csv and Mt2dat.csv

V1	V2	V3
54	375	43
63	546	49
28	468	61
99	422	55
87	336	22
86	307	33
45	296	29
26	186	41
22	418	29
20	416	51

File: Mt1cor.csv

V1	V2	V3
1	0.9	0.2
0.9	1	0.3
0.2	0.3	1

The functions are executed as follows:

matrixp.fn(Mt1)

[1] “Fnorm = ” “0”     ” Max abs diff = ” “0”

The above result implies that the correlation matrix Mt1cor is positive semidefinite. No adjustment was needed, however, a new matrix Mt1sq.csv, identical to Mt1cor.csv was written to the user’s directory for use by function sampnum.fn.

sampnum.fn(Mt1,23,10)

Note that 23 above is a random number seed and t = 10 is the number of trials; the default number of trials is 10,000. If this program executes correctly there is no output in R, however the following sample number file, Mt1sn.csv is written to the user’s directory.

File: Mt1sn.csv

V1	V2	V3
4	5	6
1	3	2
2	2	1
5	4	8
6	7	4
3	1	5
8	6	7
9	8	10
7	9	3
10	10	9

COagg.fn(Mt1,2)

Note that 2 above implies using user specified correlation. Other options are to specify independence (1) or fractile additivity (3). Fractile additivity assumes a large sample size. If this program executes correctly there is no output in R, however, the following aggregation and sample number file Mt1ResC.csv is written to the user’s directory. Note that C in the file name refers to the user specified correlation matrix.

File: Mt1ResC.csv

Sum	sn1	sn2	sn3
446	4	5	6
356	1	3	2
340	2	2	1
432	5	4	8
505	6	7	4
253	3	1	5
551	8	6	7
570	9	8	10
560	7	9	3
700	10	10	9

COaggsum.fn(Mt1ResC)

If this program executes correctly there is no output in R, however the following summary statistics file Mt1ResCsum.csv is written to the user’s directory and is shown below.

Mt1ResCsum.csv

	Value
min	253
P05	292
P25	375
P50	476
P75	558
P95	642
Max	700
Mean	471
Std dev	132
n	10

1.2.2 Example: Mt2

The major difference between this and the previous (Mt1) example is that Mt2cor is not a positive semidefinite matrix. Thus only this part will be illustrated. The simulation results Mt2dat are the same as Mt1dat. The correlation matrix Mt2cor.csv is:

Mt2cor.csv

V1	V2	V3
1	0.9	−0.3
0.9	1	0.3
−0.3	0.3	1

matrixp.fn(Mt2)

[1] “Fnorm = ” “0.08792” “ Max abs diff = ” “0.04884”

A non-zero Fnorm implies that the input correlation matrix is not positive semidefinite. A new proper correlation matrix Mt2sq.csv is written to the user’s directory for use by function sampnum.fn. It is shown below.

File: Mt2sq.cov

V1	V2	V3
1	0.85	−0.27
0.85	1	0.27
−0.27	0.27	1

The R-code to execute these functions is given in text file: Appendix 2.

Appendix 2: R Code