208 G. Hudson, R.V. Birnie Agricultural and Forest Meteorology 101 2000 203–216
2.3. Risk assessment In this paper, the interpretation of risk follows that
outlined by Warner 1993, who defines risk as the probability that a particular adverse event occurs dur-
ing a stated period of time. An adverse event or haz- ard is defined as an occurrence that produces harm,
i.e. loss consequent on damage, where damage is the loss of some inherent quality suffered by a physical
or biological entity.
Following Warner’s terminology, risk assessment is a general term used to describe the study of decisions
subject to uncertain consequences and comprises three stages — risk estimation, risk evaluation and risk man-
agement. In this study only risk estimation is covered. To do this, we identified hazard with the Non-Prime
land category and estimated the probabilities of oc- currence of the categories Prime and Non-Prime over
time. There are a number of ways that such a time series of annual switches between climatic Prime and
Non-Prime land can be summarised. For example, the proportion of time spent, or the mean number of con-
secutive years in the climatic Prime category could be quoted. In this study, probabilities of occurrence were
estimated from the transitions between climatic Prime and Non-Prime land in consecutive years. This is a
simple form of Markov-chain analysis and has, for ex- ample, been applied to daily rainfall Gabriel and Neu-
mann, 1962. The transitions between climatic Prime and Non-Prime land are calculated using a sequence
of years rather than days as used for rainfall, but the technique is applied in identical fashion. The principal
result is a matrix of transition probabilities. From this matrix, assuming stationarity of the series no trend,
we can estimate two forms of risk: the probability of continuous sequences of years that land is in the cli-
matic Prime category and the mean return period in years to climatic Prime land from Non-Prime land.
3. Analyses
Four separate analyses were carried out by allocat- ing 1961–1980 climate data to climatic LCA classes
and PrimeNon-Prime categories. First, we compared the climatic LCA for the two periods 1958–1978 and
1961–1980 to test the robustness of the classification. Then we quantified the inter-annual variability from
1961–1980 in climatic LCA class at the MORECS stations. Thirdly, the transition probabilities between
climatic Prime and Non-Prime categories were esti- mated. Finally, the geographic variation in mean return
times to the climatic Prime category was mapped.
3.1. Effect of different climate periods Comparing the climatic LCA classes for the periods
1961–1980 and 1958–1978, we tested the null hypoth- esis that climatic LCA class is not sensitive to the cli-
mate period used. Comparable data were available for 12 stations. The climatic LCA classes from 1958–1978
and those derived from 1961–1980 are summarised in Table 3. The climatic LCA class is lower in 1961–1980
than in 1958–1978 for seven of the stations shown in bold type. Additionally, in Fig. 1, the change in po-
sition of each station is shown with a line connecting its position in 1958–1978 to that in 1961–1980. This
shows that some stations move more than others do and that there is a general shift to cooler and wetter in
most stations. If the second period of data were used to construct the class partitions their positions would
be shifted on the graph.
To test the null hypothesis that there was no differ- ence in the LCA class for the two periods, the classes
were compared using a randomisation test. For each station, the pair of climatic LCA values 1958–1978
Table 3 Long-term climatic Land Capability for Agriculture for 1958–1978
and 1961–1980
a
Climate station Elevation m
Climatic LCA 1958–1978
b
1961–1980 Haddington
49 1
1 Leuchars
10 2
2 Auchincruive
48 2
2
Dumfries 49
2 3
1
Faskally 94
2 3
1
Wick 36
3
1
3
2
Fort Augustus 21
3
1
4
1
Kirkwall 26
4
2
4
2
Braemar 339
4
1
4
2
Stornoway 15
3
2
5
Prabost 67
4
2
5 Eskdalemuir
242 5
5
a
Stations which do not change LCA class are highlighted in bold.
b
Bibby et al., 1982.
G. Hudson, R.V. Birnie Agricultural and Forest Meteorology 101 2000 203–216 209
and 1961–1980 was randomly allocated to the two periods. This gave a randomised set of 12 pairs of
climatic LCA classes. The mean difference in cli- matic LCA classes between the two periods was then
calculated for this randomised set of 12 pairs. This procedure was repeated a 1000 times to generate a
distribution of mean differences which was then com- pared with the observed mean difference. The results
showed that there is a 99.5 chance of the stations having different long-term climatic LCA classes.
The result demonstrates that the climatic LCA class is sensitive to the period from which climatic data are
selected and the null hypothesis is therefore rejected. The implications for the construction of the climatic
LCA class partitions will be discussed later. The inac- curacy arising from using the empirical classification
derived with 1958–1978 climate data to allocate LCA classes with the 1961–1980 climate data has minimal
impact on the following results.
3.2. Inter-annual variability in climatic LCA While the land capability system is a long-term
classification, it is sensitive to weather variability. We have chosen to quantify that inter-annual variability
using climatic LCA. Each year’s MaxPSMD and AT0 were used to allocate the station to a notional an-
nual climatic LCA class. For each station, time series of the annual climatic LCA classes and the 20-year
trajectory of each station’s AT0 versus MaxPSMD were plotted onto the 1958–1978 climatic LCA class
partition of the AT0 and MaxPSMD feature space. Examples are given in Fig. 3a and b for Auchincruive
and Braemar. The graphs show that the annual values follow a complex trajectory through the climatic LCA
class partitions, such that the long-term climatic LCA class rarely occurs in any individual year, except for
stations near each end of the LCA range in Classes 1 and 5. In addition, plotting the trajectories of several
stations on a single graph not shown here revealed that the relative positions of stations change through
time, indicating that the spatial correlation structures are not constant.
Similar trajectories were plotted for all 23 stations for each year from 1961–1980 and the annual climatic
LCA classes read off. These results are presented in Table 4 which also summarises the number of times
the stations are above +, below − or at = their long-term climatic LCA class.
Table 4 can be used to summarise the LCA class and Prime land categories in several ways. For exam-
ple, the average time that each climatic LCA class is in the Non-Prime category Fig. 4 increases for each
successive climatic LCA Class from 1 to 5 for exam- ple, climatic LCA Class 3
1
is in the Prime category for 64 of the 20 years. The results in Table 4 highlight
the variability in weather that the climatic LCA class is based on, both between stations and between years.
However, they fail to answer the conditional question: ‘After a station leaves a class or category what is the
time taken to return to it?’ The answer will establish risk for each station. Markov-chain analysis is used to
address this question in the next section, by estimat- ing the mean return time back to a class or category,
after leaving it.
3.3. Transition probabilities between land categories The results mentioned earlier highlight the annual
variations in weather that cause the fluctuations in cli- matic LCA class for individual climate stations. Given
the number of possible transitions between the nine climatic LCA classes and subclasses, we decided to re-
duce the number of classes to two states for estimating transition probabilities. The nine climatic LCA classes
were simplified into two broad categories, which have relevance to the study area:
1. climatic Prime category 2. climatic Non-Prime category
Table 4 provides information on those years when the stations were classified in the climatic Prime land
category LCA classes shown in bold and tabulates the number of times the stations were observed in the
climatic PrimeNon-Prime categories.
The Prime and Non-Prime categories for individ- ual years were used to calculate single-step transition
probabilities between the categories for each climate station. Transition probabilities are computed from
the 1961–1980 data presented in Table 3 using Markov-chain analysis. Mean return times to the
Prime or the Non-Prime category can be estimated from these transition matrices and provide an assess-
ment of risk. The categories of climatic Prime and Non-Prime equate to the Markov states P and NP.
210 G. Hudson, R.V. Birnie Agricultural and Forest Meteorology 101 2000 203–216
Fig. 3. The trajectories in the land capability for agriculture feature space followed by Auchincruive and Braemar over the 20 years 1961–1980.
G. Hudson, R.V. Birnie Agricultural and Forest Meteorology 101 2000 203–216 211
Table 4 Climatic land capability for agriculture LCA for 23 Scottish climate stations computed on an annual basis from 1961–1980
a
Station LTLCA 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 Up Down Unchanged P
NP Haddington
1 1
1 2
1 1
1 1
3
1
1 1
1 1
1 1
1 1
1 1
1 1
2 18
20 Turnhouse
1 1
1 3
1
1 3
2
3
1
1 2
2 1
1 1
1 1
1 1
1 1
1 1
5 15
19 1
Kinloss 1
1 1
2 1
2 3
1
1 3
2
1 1
1 1
1 1
1 1
1 1
1 1
4 16
19 1
Leuchars 2
1 1
3
2
1 3
1
3
2
1 3
2
2 2
1 1
1 1
1 1
1 3
1
3
1
2 11
6 3
17 3
Stirling 2
2 1
5 3
1
5 4
1
2 2
4
1
2 1
2 2
1 1
1 1
1 2
1 8
5 7
16 4
Paisley 2
2 1
4
1
2 5
3
2
3
1
1 3
1
1 1
2 4
2
1 1
1 1
1 4
1
1 10
7 3
15 5
Auchincruive 2
2 1
5 2
5 4
2
4
2
3
2
2 2
1 3
1
1 1
1 1
1 1
3
2
1 9
7 4
14 6
Mylnefield 2
1 1
3
2
1 4
2
4
2
1 3
2
3
2
2 3
1
1 1
1 1
1 1
3
1
4
1
2 10
8 2
14 6
Dumfries 3
1
3
1
3
1
5 4
2
6 4
2
5 4
2
3
1
2 2
3
1
2 1
1 1
1 1
4
1
2 9
7 4
13 7
Faskally 3
1
2 2
5 1
3
2
4
1
2 3
1
3
2
2 3
1
2 2
1 2
1 2
2 4
2
2 13
5 2
15 5
Dyce 3
1
2 2
5 1
3
2
5 1
5 3
2
2 1
1 1
2 2
1 3
1
4
1
5 3
1
11 7
2 13
7 Fortrose
3
1
1 3
1
3
1
1 4
1
5 1
4
1
2 3
2
4
2
1 2
2 3
1
1 2
3
2
5 3
2
9 8
3 12
8 Bush House
3
1
2 2
5 2
3
2
5 3
1
4
2
3
2
3
2
3
1
2 3
2
1 1
1 3
1
3
2
4
1
2 8
9 3
11 9
Wick 3
2
2 4
1
4
1
2 5
5 2
5 3
2
3
1
3
1
2 4
1
3
1
2 1
4
1
3
1
6 2
11 8
1 11
9 Fort Augustus
4
1
3
2
4
2
5 3
1
5 5
4
1
3
2
4
1
5 3
2
3
1
6 2
2 1
2 2
5 3
1
11 7
2 8
12 Blyth Bridge
4
1
2 3
2
5 3
2
6 5
4
2
4
2
5 4
1
3
2
3
1
4
2
3
1
2 2
3
2
4
2
5 3
2
10 9
1 5
15 Dundeugh
4
2
3
2
5 5
5 6
5 5
4
2
6 3
1
4
2
5 4
2
2 3
1
1 2
3
1
5 3
2
8 9
3 6
14 Penwhirn
4
2
5 4
2
6 6
6 6
5 4
2
6 3
2
5 5
4
1
4
1
2 2
3
1
2 5
2 8
10 2
5 15
Kirkwall 4
2
4
2
5 5
5 4
2
5 4
1
5 5
3
2
2 3
2
4
1
3
1
3
1
1 4
2
3
2
6 2
10 7
3 5
15 Braemar
4
2
4
1
4
2
6 3
2
5 5
3
2
5 5
6 4
1
4
1
3
2
2 3
2
2 4
2
4
2
6 4
1
10 7
3 2
18 Stornoway
5 4
1
5 5
5 5
5 5
2 5
4
2
4
2
5 5
3
1
4
1
4
1
3
1
3
1
6 3
1
10 1
9 5
15 Prabost
5 5
5 4
1
5 5
5 5
4
2
4
2
4
1
5 6
6 3
2
5 4
2
3
1
4
1
6 2
9 3
8 2
18 Eskdalemuir
5 4
2
6 6
5 6
5 5
4
2
6 3
2
5 5
5 5
3
2
4
1
4
2
3
2
6 4
1
8 5
7 20
Up –
13 9 1
10 0 8
5 2
11 12 11 10 19 18 20 15 13 0 16 193 –
– –
– Down
– 1
5 20 5
19 19 6 14 13 4
3 4
6 1
6 19 1
– 146
– –
– Unchanged
– 9
9 2
8 4
4 9
4 8
8 8
8 7
4 5
3 7
4 4
6 –
– 121
– –
Prime P 13
15 13 4 14 3
3 12 6
8 13 14 16 11 20 19 20 18 15 5
18 – –
– 247 –
Non-PrimeNP 10 8
10 19 9 20 20 11 17 15 10 9
7 12 3
4 3
5 8
18 5 –
– –
– 213
a
The long-term LCA LTLCA is provided for reference and annual variability is summarised in two ways: i instances where annual LCA is updownunchanged relative to the long-term LCA and ii the number of instances of Prime P vs Non-Prime NP. Prime land,
LCA Classes 3
1
–1, is shown in bold. The summaries reveal variability within stations and between years rows and between stations and within years columns.
From state P two transitions are possible in a pair of consecutive years; the state can remain the same,
i.e. P ⇒ P or the state can change to NP, i.e. P ⇒ NP. There are four one-step transitions possible between
the two states in two consecutive years, shown in matrix form as
P ⇒ P P ⇒ NP
NP ⇒ P NP ⇒ NP
1 The number of times each of the four transitions oc-
cur, is counted at each station and to estimate transi- tion probabilities, proportions are calculated for each
starting state. For example, the matrices of transition counts and probabilities in brackets at Auchincruive
and Braemar are
Auchincruive
100.769 30.231 30.500
30.500
Braemar
00.000 21.000 20.118
150.882 2
Assuming that the sequence of states is stationary does not display a trend and manipulating the tran-
sition probabilities according to Markov-chain theory Gabriel and Neumann, 1962, mean return times
between states can be estimated.
The mean return time mrt is the mean time taken, after leaving a state, to return to it from any other
state. When a site never leaves one state the mrt from the other state is null. The matrix of estimated transi-
tion probabilities and the mrts for the 23 climate sta- tions are shown in Table 5. The estimated mrt to the
Prime state ranges from 1 year at Haddington i.e.,
212 G. Hudson, R.V. Birnie Agricultural and Forest Meteorology 101 2000 203–216
Table 5 Estimated transition probabilities and mean return times to Prime and Non-Prime land for each climate station in Scotland
Climate station 1961–1980 climatic Land Capability
Estimated transition probabilities Mean return time in years to
For Agriculture Prime
Non-Prime Haddington
1
1.000 0.000 0.000
0.000 1.00
Null Turnhouse
1
0.944 0.056 1.000
0.000 1.06
18.86 Kinloss
1
0.944 0.056 1.000
0.000 1.06
18.86 Leuchars
2
0.813 0.188 1.000
0.000 1.19
6.32 Stirling
2
0.800 0.200 0.750
0.250 1.27
4.75 Paisley
2
0.714 0.286 0.800
0.200 1.36
3.80 Auchincruive
2
0.769 0.231 0.500
0.500 1.46
3.16 Mylnefield
2
0.692 0.308 0.667
0.333 1.46
3.17 Dumfries
3
1
0.833 0.167 0.286
0.714 1.58
2.71 Faskally
3
1
0.714 0.286 0.800
0.200 1.36
3.80 Dyce
3
1
0.667 0.333 0.571
0.429 1.58
2.71 Fortrose
3
1
0.667 0.333 0.429
0.571 1.78
2.29 Bush House
3
1
0.500 0.500 0.556
0.444 1.90
2.11 Wick
3
2
0.400 0.600 0.667
0.333 1.90
2.11 Fort Augustus
4
1
0.571 0.429 0.333
0.667 2.29
1.78 Blyth Bridge
4
1
0.400 0.600 0.143
0.857 5.20
1.24 Dundeugh
4
2
0.667 0.333 0.154
0.846 3.16
1.46 Penwhirn
4
2
0.750 0.250 0.133
0.867 2.88
1.53 Kirkwall
4
2
0.500 0.500 0.200
0.800 3.50
1.40 Braemar
4
2
0.000 1.000 0.118
0.882 9.47
1.12 Stornoway
5 0.250 0.750
0.267 0.733
3.81 1.36
Prabost 5
0.000 1.000 0.111
0.889 10.01
1.11 Eskdalemuir
5 0.000 0.000
0.000 1.000
Null 1.00
G. Hudson, R.V. Birnie Agricultural and Forest Meteorology 101 2000 203–216 213
Fig. 4. The number of years out of 20 that land at stations is in the Non-Prime category, averaged by land capability for agriculture LCA class.
never leaves the Prime state to null at Eskdalemuir i.e., never reaches the Prime state. All the stations in
climatic LCA Classes 1, 2 and 3 have estimated mrt to Prime land of less than 2 years. Climatic LCA Classes
4 and 5 have mrts to the Prime state of more than 2 years.
As would be expected, the stations classified as be- ing Non-Prime generally have longer mrts. However,
the return times do not equate directly to the long-term climatic LCA class i.e., there is no systematic trend in
the mrt from higher to lower climatic LCA class. For example, Stornoway with a long-term climatic LCA
Class 5 has a shorter mrt to Prime land than Braemar or Blyth Bridge with climatic LCA Classes 4
2
and 4
1
, re- spectively. These stations are in contrasting geograph-
ical locations and it is likely that there is an over-riding spatial structure influencing the climatic variability.
This indicates that whilst it is possible to calculate transition probabilities on the basis of climatic LCA
class the estimated mrts are a property of the geo- graphic location of the station and not a property of
the climatic LCA class. This is a very significant find- ing since it means that the process of interpolating
these mrt estimates must include explicit account of the spatial and temporal structure of the weather data
used to derive the climatic LCA classification. 3.4. Geography of mean return times to Prime land
The preceding section has demonstrated a method for computing mrts to the Prime land category for indi-
vidual climate stations. To map the geography of mrts, land evaluation units could be used, as in the approach
described by van Lanen et al. 1992. An alternative method, which we have used, is to compute mrts from
interpolated weather data. Here, we use Scotland as a case study to illustrate and assess the use of interpo-
lated weather data to map mrts over wide geographic areas. The weather data are interpolated to a coarse
grid, but the same principle would apply to an inter- polated data set with finer resolution.
The interpolated monthly MORECS square data from 74 squares in Scotland were used to derive a geo-
graphic assessment of mrt to the climatic Prime cate- gory. Since this data comprises monthly temperature
means and end-of-month soil moisture deficits, meth- ods to derive predicted AT0 and MaxPSMD for each
square had to be developed. Allocating the monthly mean temperature to each day of the month was used
for the AT0 calculation. For MaxPSMD, the single station MORECS data for 460 stations and year com-
binations was used to construct a third-order poly- nomial regression to predict daily MaxPSMD from
214 G. Hudson, R.V. Birnie Agricultural and Forest Meteorology 101 2000 203–216
Fig. 5. The mean return times to climatic Prime land in 40 km × 40 km Meteorological Office Rainfall and Evaporation Calculation System MORECS squares grouped into four risk classes. The Prime land area in Scotland is shown for comparison.
end-of-month MaxPSMD R
2
= 0.92. The predicted
AT0 and MaxPSMD values for each combination of MORECS square and year were then used to allocate
the squares to the climatic Prime or Non-Prime cat- egory for each year. Then the mrt to climatic Prime
land was estimated using transition probabilities esti- mated for each square. The results are shown in Fig.
5, together with the distribution of Prime land in Scot- land, based on all LCA limitations. Whilst there are
clearly issues concerning the spatial resolution of the
G. Hudson, R.V. Birnie Agricultural and Forest Meteorology 101 2000 203–216 215
data and the comparability of climatic Prime land to Prime land, Fig. 5 reveals the broad pattern of varia-
tion in mrt or zones of risk, compared to the distribu- tion of Prime land.
This analysis demonstrates that, providing there is an adequate network of meteorological stations and a
sufficiently long run of climatic data, it is possible to develop spatial estimates of climatic variability that
can be expressed in terms of zones of risk. The climatic LCA methodology can be adapted to incorporate this.
So, for example, the mrt to Prime land classes shown in Fig. 5 is re-expressed as low 0–2 years, moderate
2–4 years, high 4–8 years and extreme 8 years risks.
4. Concluding remarks
