L. Lindkvist et al. Agricultural and Forest Meteorology 102 2000 51–67 57
along several topographical cross-sections in IDRISI, a GIS software developed by Eastman 1992. The ter-
rain categories that were defined in this way include five principal types: 1 convex terrain: ∩, 2 linear
sloping terrain: ↓, 3 linear flat: ↔, 4 wide concave terrain: ∪ and 5 narrow concave terrain: ∨. In Table
2 information regarding the altitude and type of terrain are included for each sampling location.
Convex terrain peaks and ridges are found mainly at elevations above 800 m. Four areas are identified in
Fig. 2. Two broad peaks above 1100 m Mts. Anå and Lill , a ridge reaching 1000 m north of the peaks and
a broad convex area east of the peaks at 800–950 m. Two types of valleys, narrow and wide concavities,
are present in the same figure. The former varies in width between 1.0 and 2.0 km, whereas the latter is
2.5–4.0 km wide. The narrow type is located between Mt. Anå and Mt. Lill and the wider type follows
the two rivers and surrounds L. Grundsjön and L. Särvsjön.
The areas surrounding L. Grundsjön are considered flat, i.e. less than 3
◦
inclination. A 5–7 km wide level area divides the concavities in the northern part of the
study area. Open level ground is also identified in the southeast corner of Fig. 2. Thus, this terrain type vir-
tually bisects the study area from north to southeast, leaving valleys and hilltops at each side. The remain-
ing terrain type, i.e. slopes more than 3
◦
inclination, connects the convex hilltops and ridges with concav-
ities and flat valley floors. Differences in slope incli- nation are not considered due to the fact that sloping
terrain is known to show only small variations in frost susceptibility compared to the variations that occur
across different terrain types defined in this study.
Table 2 Altitude and dominating terrain form at 38 sampling locations. Each location is assigned one of the five major terrain forms that were
defined according to its curvature; convex, concave, linear sloping and linear flat. The symbols are included in order to simplify the comparison between tables and figures
a
Sampling site 1
2 3
4 5
6 7
8 9
10 11
12 13
14 15
16 17
18 Altitude m a.s.l.
710 700
670 680
870 680
630 600
710 670
680 710
670 720
660 710
830 780
740 Terrain form
↔ ↓
↔ ↓
∩ ↓
↔ ↔
↓ ↓
↓ ↔
∨ ↓
↔ ↓
↔ ∨
∨ Sampling site
19 20
22 23
24 25
26 27
28 29
30 31
32 33
34 35
36 37
38 Altitude m a.s.l.
950 1120
950 1110
680 710
830 995
680 700
790 650
660 700
1080 680
880 780
940 Terrain form
↓ ∩
↓ ∩
↓ ∨
∩ ∩
∪ ↔
∩ ↓
∪ ↓
∩ ↓
∩ ∩
↓
a
↔ : flat area, surface inclined 3
◦
; ↓: slope, surface inclined 3
◦
; ∩: convex area; ∪: wide concave area U-shaped valley; V = narrow concave area V-shaped valley.
4. Results and discussion
4.1. The influence of elevation and local terrain on the establishment of low temperatures
Investigations have shown that local topography may cause large temperature variations to develop
e.g., Marth 1986; Toritani 1990; Bogren and Gus- tavsson 1991. Several studies have been concerned
with investigations of temperature variations at low points in the terrain, e.g. Catchpole 1963; Dight
1967; Doran and Horst 1983; Miller et al. 1983. A division based on weather conditions is necessary
in order to assess the role of different topographical parameters. The following discussion is based on the
variation expected during the two most extreme situ- ations; cloudy, windy nights and clear calm nights. In
order to benefit from frost risk models it is necessary that different weather types are treated accordingly.
During cloudy, windy conditions the counter- radiation and high percentage of diffuse radiation will
lead to a smoothing of local temperature variation. Furthermore, the wind itself will aid in this process
by increasing turbulence. The temperature stratifica- tion can be described as neutral or unstable, i.e. a
temperature decrease with increasing height above the ground. For frost risk studies, this weather type
is of major significance as advective frost can occur during these situations.
The differences in minimum temperature for each of the loggers versus the reference station for specific
situations have been extracted from the recordings. The situations are when a nighttime net radiation
above an average of 20 W m
− 2
and a mean nighttime
58 L. Lindkvist et al. Agricultural and Forest Meteorology 102 2000 51–67
Fig. 3. Relationship between the differences in minimum temperature at different locations and that at the reference site and altitude above sea level of the measurement location during cloudy and windy conditions. y = −0.0062x + 6.74, R
2
= 0.925.
wind speed exceeding 6 m s
− 1
have been attained. The averages of these differences versus altitude have
been plotted for each logger location, Fig. 3. There is a very good correlation R
2
= 0.93 This implies that
the general decrease in temperature during this type of weather controls the differences between the sta-
tions. The lowest nighttime temperatures are found at the highest altitudes and the warmest temperatures are
located in the lowest areas. This relationship accords well with the physical processes described above and
has been shown to be valid in several studies, e.g. Kalma et al. 1986.
During clear, calm nights the variation in mini- mum temperature is to a large extent controlled by
the possibility of cold air accumulation. Cold air accumulates at low points such as valley bottoms
due to the fact that cold air is denser than warm air. Other factors, which have been shown to be of great
importance, are the degree of shelter. In a sheltered location the mixing of surface cold air with warmer
air aloft is reduced. The effect resulting from this has been demonstrated in studies by e.g. Tabony 1985;
Gustavsson 1995; Gustavsson et al. 1998. Shelter can be both in the form of topography, e.g. narrow
valleys, small hills, or in the form of vegetation, i.e. tree stands or lines of trees.
The variation in minimum temperature for clear nights radiation −0 W m
− 2
; wind speed 8 m s
− 1
has been analysed in a similar way as for the cloudy, windy situations described above. In Fig. 4, the av-
erage difference in minimum temperature versus the reference station has been plotted against altitude of
the site. As can been seen in the figure the scatter be- tween the stations is large. A trend can be seen that the
low-lying stations are in general colder than the higher sited ones. However, the temperature values clearly
show that a linear correlation would only describe a small part of the variation, R
2
= 0.40 for the linear fit
with Y = 0.0147X–14.87. In Fig. 4 the local topography for each site has been
included following Table 2. However, the concave ar- eas have been treated as one. It is clear from this figure
that the local topography has a much larger influence on the temperature difference than the height above
sea level.
In several studies, for example Kalma et al. 1986, 1992; Laughlin and Kalma 1987, 1990, it has been
concluded that elevation can be used as a good pre- dictor of the minimum temperature during clear, calm
nights. If this should be the case then the variation between stations sited at the same height should be
small during clear, calm nights. In Fig. 5 the mini- mum air temperature for Loggers 17 and 37 is plotted
for the studied time period. The data shows that Log- ger 17 is much colder during clear nights compared
with Station 37, cf. Table 1. Station 17 is located in a valley and Logger 37 on a hilltop. From this it can
be concluded that the topographical situation around
L. Lindkvist et al. Agricultural and Forest Meteorology 102 2000 51–67 59
Fig. 4. Relationship between the differences in minimum temperature at different locations and that at the reference site and altitude above sea level of the measurement location during clear and calm weather conditions.
the station is of greater importance compared with elevation.
To further study the influence of local topography on the variation in minimum temperature during clear
nights the data from Fig. 4 have been used. The mean variation has been calculated for each type of location
as described above see Table 2. To be able to com- pare the data presented in this study with the figures
Fig. 5. Comparison between two temperature sampling locations at the same height above sea level but in different terrain. Station 17 is sited in a narrow concave valley. Station 37 is located on a ridge see Fig. 2
presented in the studies by Kalma et al. and Laugh- lin and Kalma, the same elevation interval has been
included in the table, i.e. 160 m. From the data presented in Table 2 it is clear that
the local topography has a major influence on the dif- ference in minimum temperature compared with the
reference station. The concave areas are, for example, 8.5
◦
C colder than the convex areas. From this it can be
60 L. Lindkvist et al. Agricultural and Forest Meteorology 102 2000 51–67
argued that a frost assessment model in complex ter- rain should not just be based on altitude for prediction
of minimum temperature during clear nights. In the above-cited studies the residuals were used to describe
the influence from topography, especially drainage of cold air to valleys. However, for the type of topogra-
phy classified in this study, this kind of method ex- plains only a smaller part of the variation. This is due
to the fact that the scatter among the linear fit is too large and that the residuals can amount to nearly 6
◦
C, i.e. more than half of the total variation in nighttime
minimum temperature versus the reference station. The reason why a good correlation was obtained be-
tween minimum temperature and elevation in several previous studies is probably due to the fact that only
one major valley system has been studied. If one plots the temperature variation for a single profile including
the terrain types, convex, slope and concave, a good correlation is achieved. However, if a more complex
system is studied then different types may be found at different altitudinal levels and thereby the correlation
between elevation and temperature variation is nulli- fied. These findings are in agreement with previous
studies that have shown that there is a low correlation between elevation and minimum temperature during
clear, calm nights, e.g. Avissar and Mahrer 1988, Gustavsson 1995.
Another important aspect regarding the modelling of temperature variations is the ability to handle the
Fig. 6. Differences in minimum temperature at different wind speeds calculated for Station 17 and the reference station 1020 m.
large number of weather situations that occur between the two extremes already discussed. Several studies
have focused upon the task of determining the rela- tionships between wind speed and amount of cloud
with net radiation. In studies by Gustavsson 1990 and Bogren and Gustavsson 1991, the importance of
the prevailing wind speed and cloud cover for the de- velopment of large temperature differences has been
discussed. Others e.g., Bootsma 1976 and Laughlin 1982 have related temperature differences in com-
plex terrain to both radiation and wind speed by em- pirical formulae. The influence from cloud ought to be
the same between different areas as it basically con- trols the amount of outgoing long-wave radiation. The
influence from the prevailing wind speed, on the other hand, is more likely to show a large variation from
place to place owing to such factors as topography and local wind shelters.
In Fig. 6 the temperature differences for all 60 nights with varying wind speed have been plotted. A very
sharp difference can be seen for nights with mean nighttime wind speeds exceeding 8 m s
− 1
and nights with those below. Comparison for other stations in the
area gives similar results. The reason for such high limits 8 m s
− 1
and the fact that below that limit the temperature variations are very scattered can proba-
bly be found in the setting of the reference station, which is convex and well exposed. In narrow valleys
on the other hand the topography gives a sheltering
L. Lindkvist et al. Agricultural and Forest Meteorology 102 2000 51–67 61
effect which significantly reduces the turbulent mix- ing between the surface cooled air and warmer aloft.
The data in Fig. 6 clearly demonstrates the importance of wind shelter as a major factor for the development
of large temperature differences.
4.2. Frost under different weather conditions In the previous section it was shown that the local
topography together with altitude acts as a controlling factor for temperature variation. Normally two types
of frost situation can be distinguished, radiation and advective frost. Advection frost occurs during situa-
tions where cold air intrudes into an area. This results in the lowest temperatures at the elevated sites, i.e. a
temperature decrease with height as already discussed. Radiation frost on the other hand, occurs during situa-
tions with in-situ cooling, i.e. clear sky and low wind speed. The temperature, wind speed and net radiation
for the study period is presented in Table 1 along with the number of stations with frost situations at night-
time. All frost situations that occurred during this time period can be classified as radiation frost.
One way of estimating the local frost risk for a spe- cific location is by adding the number of situations
with temperatures below 0
◦
C. Calculations of frost sum and coldness sum day-degrees below a certain
threshold value are commonly used methods for quan- tifying the frost risk at different areas cf. Tuhkanen,
1980. However, such indices are general in their char- acter in that they do not refer to any specific type of
risk.
Lindkvist and Chen 1999 introduced a more comprehensive index. Their summation is based on
occasions with temperatures below 0. Furthermore, each situation was subdivided with relation to local
conditions such as the time period of freezing temper- atures as well as a division into temperature levels.
The temperature level division is used to take into account how much the temperature will fall below 0.
Furthermore, a weight factor is applied to each of the levels according to the increasing risk of plant injury
given that, successively colder situations are more im- portant to consider. The index was found to account
for more than 80 of the variability in mortality, mea- sured 5 years after re-planting in the above mentioned
study. Therefore, the index composes a measure of frost intensity and provides an indication of the risk
of plant re-growth failure in different types of terrain. In order to analyse the relation between the local
topography and local frost risk, the Lindkvist–Chen index was calculated for radiation frost situations and
presented together with topographical information in Table 4. The table combines information concerning
altitude and terrain form at the different sampling lo- cations. The number of frosts and the calculated index
were also included. The table was sorted according to increasing index values. The relationship between
terrain curvature and low summer night temperatures becomes obvious in this arrangement.
4.3. Terrain classification and temperature estimation To be able to estimate the spatial variation in frost
risk of radiation Type A grid was applied to the study area and the pixels were classified according to the ter-
rain curvature. The symbols used for different curva- ture are superimposed on a terrain map and presented
in Fig. 7. The grid pixels cover an area of 2.52.5 km except for those along the borderline of the study area,
which are half in size. Consequently, the choice of terrain curvature that defines a pixel follows the dom-
inating type.
Index values Lindkvist–Chen index were calcu- lated for each one of the 38 measuring stations. It
was then possible to group a specific index value, or a range of values, with a defined terrain form for
pixels with temperature observations. In order to ex- tend this procedure to cover all the pixels in the study
area 121 index values where estimated by applying a geostatistical method to the data-set containing the
calculated index. The method kriging uses a local weighted moving average technique that offers a way
in which to minimise the variance of the estimation error. This technique is thoroughly described in sev-
eral textbooks e.g., Issacs and Srivastava, 1989 and Englund and Sparks, 1988. Furthermore, applications
similar to the one used in the present study are avail- able in Söderström and Magnusson, 1995 and Lind-
kvist and Lindqvist 1997.
Due to the large topographical variability in the study area, ordinary point kriging was used to min-
imise the smoothing of data. This method attempts to estimate the observed values as correctly as possi-
ble by avoiding averaged estimates, i.e. ordinary block kriging. The method involves a risk of receiving higher
62 L. Lindkvist et al. Agricultural and Forest Meteorology 102 2000 51–67
Fig. 7. A classification of terrain elements in the study area. Single arrows indicate the major drainage direction on slopes. 1 ∩ = convex. 2 ↓ = slope. 3 ↔ = flat. 4 ∪ = wide concave U-shaped valley. 5 ∨ = narrow concave terrain V-shaped valley.
kriging standard deviations, therefore it was decided to exclude estimates showing kriging SD higher than
10. To verify the results from the Kriging procedure,
where an average of three to four estimates were pro- duced for each observed value, a cross-validation was
carried out between the 38 calculated index values based on temperature observations and 38 index val-
ues based on estimations at the same locations. Fig. 8 gives the results of the difference between observed
and estimated values. A very high correlation coeffi- cient 0.97 was achieved, which indicates a low level
of estimation error in the interpolation.
Since more extreme observations will show a greater difference to the estimated values in the
interpolation a 1 : 1 line was included in the fig- ure to explain the degree of smoothing. A slight
over-estimation occurs for values in the lower range of the index and under-estimation is present in the
upper range. However, the deviations are generally very small. Since there is an abundance of stations
showing over-estimation, a small increase in the index average from 17 to 19 is noted.
Extreme values are found in two different types of terrain. One consists of hilltops and ridges at elevations
from 800 m and they are almost entirely exempted from radiation frost. The other is narrow concavities.
The change from low towards high index estimates appears to be progressive in character up to values
around 40, thereafter the change takes place in steps. The change is connected to transitions in the terrain
from flat to concave form. The influence of terrain
L. Lindkvist et al. Agricultural and Forest Meteorology 102 2000 51–67 63
Fig. 8. Comparison of estimated against observed frost index values. R
2
= 0.98.
transitions can be seen in Table 3, e.g., the calculated index leaps from 26 to 52 when concave terrain is
encountered. A stepwise increase can be explained by the fact that sampling of intermediate terrain forms is
avoided in order to isolate the main types in this study. A lower slope that is entering gradually a concave
curvature is an example on intermediate terrain form.
It is evident from Table 3 that different terrain types can be connected to specific ranges of frost intensity.
If such a relationship is highly significant it can allow a simplification of frost risk assessment. In order to
investigate the possibility of applying the established relation in the entire study area, the estimated index
was subject to cluster analysis.
Different methods for cluster analysis exist, how- ever they are all designed to minimise the within-group
deviation of the observed variable and to maximise the between-group deviation. Euclidean distance in this
case, a summation of the difference squared between frost index values separated by a known distance was
Table 3 Calculated mean difference in minimum temperature versus the
reference station for four types of topographical locations. Site
Convex Slope
Flat Concave
Altitude interval 160 m −
0.8 −
3.6 −
5.0 −
8.4 Altitude interval 520 m
− 0.1
− 2.7
− 4.9
− 8.5
used to calculate the dissimilarity between the esti- mated index values. The clusters were obtained ac-
cording to Ward’s method, see for example Burrough 1986.
In the analysis each index value was given a terrain ‘signature’ according to the five terrain categories
defined. Three main clusters A–C are prominent Table 5. Also, two sub-groups in each cluster low
and high are apparent. The terrain categories are identified at the sub-group level. By the cluster anal-
ysis it is inferred that different types of flat areas should be treated separately depending on their loca-
tion relative to the surrounding terrain. Furthermore, convex terrain together with exposed and elevated
slopes should obviously form a common group while lower and shaded slopes should compose a separate
entity. Wide and narrow concavities remain as pre- vious in two unique groups. However, as discussed
by previous authors, e.g. DeGaeteno and Schulman 1990, there is no definite method to decide on for
an optimum number of clusters andor sub-clusters. Thus, it is to a certain degree an arbitrary choice.
It was assumed that three to five groups of distinct terrain forms with possible sub-divisions would be an
appropriate number in order to distinguish between areas of different frost risk. As an objective selection
method for the introduction of new clusters, a simple technique of looking at mean values and standard de-
viations SD was applied to find out the final number of clusters to be considered.
Table 4 lists the mean, range and standard deviation of the clusters that were examined. The terrain classes
are included in the table to compare between ranges of index values and type of terrain. Clusters B and
C show the two most evident sub-groups, which are accompanied by a distinct lowering of the SD. For
Cluster A, the SD is already low, however a further reduction motivates a division. Terrain categories and
clusters are matched in the table and three main groups and two sub-levels in each group are formed, each
with a unique frost risk.
4.4. Frost risk evaluation The cluster analysis generated six intervals from
very low to extremely high frost risk. In Fig. 9 a map is manually compiled by assigning to each pixel in the
64 L. Lindkvist et al. Agricultural and Forest Meteorology 102 2000 51–67
Table 4 The relationship between altitude, terrain form, number of frost occasions and frost index values for 38 sampling locations in the study
area. The index values are compiled with the method presented by Lindkvist and Chen 1999 Site
Altitude m a.s.l. No. of frosts
Index value Convex ∩
Slope ↓ Flat ↔
Concave wide ∪ Concave narrow ∨
4 870
∩ 30
790 ∩
36 880
∩ 37
780 ∩
26 830
∩ 27
995 ∩
34 1080
∩ 23
1110 ∩
20 1120
∩ 38
940 ↓
22 950
↓ 19
950 ↓
24 680
2 2
↓ 9
670 2
3 ↓
8 710
2 6
↓ 31
650 2
7 ↓
10 680
3 7
↓ 1
700 3
7 ↓
35 680
3 8
↓ 15
710 4
8 ↓
3 680
4 9
↓ 5
680 4
9 ↓
33 700
4 9
↓ 16
830 6
11 ↔
710 4
12 ↔
13 720
4 13
↓ 6
630 5
14 ↔
2 670
4 14
↔ 11
710 4
16 ↔
7 600
5 17
↔ 14
660 5
20 ↔
29 700
5 22
↔ 28
680 10
37 ∪
18 740
11 39
∨ 32
660 13
45 ∪
25 710
16 50
∨ 12
670 18
58 ∨
17 780
19 62
∨
grid net of terrain categories Fig. 7 a specific cluster of the estimated frost index according to the relation-
ship shown in Table 5. Thereafter, the borderlines of clusters that are members of the same sub-group are
delineated. Thus, the map in Fig. 9 show six different patterns equivalent to six levels of frost risk. Three
unique patterns are used, one for each of the main frost risk levels. By differentiating the intensity of each pat-
tern, it is possible to distinguish between main groups and sub-groups according to the legend in the figure.
Note that areas above 850 m are excluded due to the fact that forestry is not normally practised above this
height in Sweden. The following information can be extracted from
the figure: very low frost risk is found in convex ter- rain and exposed upper slopes while lower slopes es-
pecially those adjacent to the basin areas, show low frost risk. Intermediate risk is connected to flat ground
commonly located in the lower parts of the study area, e.g., in the immediate surrounding of lakes. Whereas
L. Lindkvist et al. Agricultural and Forest Meteorology 102 2000 51–67 65
Fig. 9. Map over the study area showing 6 levels of frost risk. 1 very low, 2 low, 3 intermediate, 4 high, 5 very high and 6 extreme risk.
Table 5 Results of cluster analysis on frost index values and basic statistics for the different clusters. The main cluster levels are marked A–C.
Low and high marks the sub-cluster Main clusters and subtypes
Index mean Index range
SD Terrain class
Frost Risk evaluation A
9 2–12
3.0 ∩
,↓ low
Low 4
2–7 2.0
∩ , ↓
High 10
9–12 1.0
↓ B
24 16–35
6.4 ↔
intermediate Low
19 16–22
2.0 ↔
High 31
28–35 2.7
↔ C
49 39–64
7.2 ∪
,∨ high
Low 43
39–48 3.0
∪ High
56 53–64
3.9 ∨
66 L. Lindkvist et al. Agricultural and Forest Meteorology 102 2000 51–67
the high-risk level are highly related to the surround- ing terrain, i.e. areas located adjacent to cold valleys
that open up into wider terrain and in the transition from wide concavities to flat areas.
The second highest risk level very high is syn- onymous with broad concave areas, which are mainly
a part of large ‘U-shaped’ valley bottoms. Beside in situ production of cold air, accumulation from the sur-
rounding slopes and plateaux account for the high level of frost risk in these areas. Narrow ‘V-shaped’ con-
cavities are found to be extremely frost prone due to the high degree of wind shelter and accumulation of
cold air.
5. Conclusions
